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G = C5×C23.32C23  order 320 = 26·5

Direct product of C5 and C23.32C23

direct product, metabelian, nilpotent (class 2), monomial, 2-elementary

Aliases: C5×C23.32C23, C10.1092- 1+4, (C2×Q8)⋊9C20, (C4×Q8)⋊5C10, (Q8×C10)⋊29C4, (Q8×C20)⋊25C2, Q8.8(C2×C20), C2.8(C23×C20), C10.81(C23×C4), C4.20(C22×C20), C42.32(C2×C10), (C22×Q8).7C10, C20.224(C22×C4), (C2×C20).710C23, (C4×C20).275C22, (C2×C10).339C24, C42⋊C2.10C10, C2.1(C5×2- 1+4), C22.11(C22×C20), C22.12(C23×C10), C23.31(C22×C10), (Q8×C10).284C22, (C22×C20).442C22, (C22×C10).255C23, (Q8×C2×C10).17C2, C4⋊C4.82(C2×C10), (C2×C4).31(C2×C20), (C5×Q8).47(C2×C4), (C2×C20).376(C2×C4), (C2×Q8).72(C2×C10), C22⋊C4.29(C2×C10), (C5×C4⋊C4).407C22, (C22×C4).52(C2×C10), (C2×C10).266(C22×C4), (C2×C4).135(C22×C10), (C5×C42⋊C2).24C2, (C5×C22⋊C4).160C22, SmallGroup(320,1521)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C2 — C5×C23.32C23
Chief series
C1 — C2 — C22 — C2×C10 — C2×C20 — C5×C4⋊C4 — Q8×C20 — C5×C23.32C23
Lower central
C1 — C2 — C5×C23.32C23
Upper central
C1 — C2×C10 — C5×C23.32C23

Generators and relations for C5×C23.32C23
 G = < a,b,c,d,e,f,g | a5=b2=c2=d2=1, e2=d, f2=g2=c, ab=ba, ac=ca, ad=da, ae=ea, af=fa, ag=ga, ebe-1=bc=cb, bd=db, bf=fb, bg=gb, cd=dc, fef-1=ce=ec, gfg-1=cf=fc, cg=gc, de=ed, df=fd, dg=gd, eg=ge >

Subgroups: 290 in 266 conjugacy classes, 242 normal (12 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C5, C2×C4, Q8, C23, C10, C10, C10, C42, C22⋊C4, C4⋊C4, C22×C4, C2×Q8, C20, C20, C2×C10, C2×C10, C2×C10, C42⋊C2, C4×Q8, C22×Q8, C2×C20, C5×Q8, C22×C10, C23.32C23, C4×C20, C5×C22⋊C4, C5×C4⋊C4, C22×C20, Q8×C10, C5×C42⋊C2, Q8×C20, Q8×C2×C10, C5×C23.32C23
Quotients: C1, C2, C4, C22, C5, C2×C4, C23, C10, C22×C4, C24, C20, C2×C10, C23×C4, 2- 1+4, C2×C20, C22×C10, C23.32C23, C22×C20, C23×C10, C23×C20, C5×2- 1+4, C5×C23.32C23

Smallest permutation representation of C5×C23.32C23
►On 160 points
Generators in S160
(1 2 3 4 5)(6 7 8 9 10)(11 12 13 14 15)(16 17 18 19 20)(21 22 23 24 25)(26 27 28 29 30)(31 32 33 34 35)(36 37 38 39 40)(41 42 43 44 45)(46 47 48 49 50)(51 52 53 54 55)(56 57 58 59 60)(61 62 63 64 65)(66 67 68 69 70)(71 72 73 74 75)(76 77 78 79 80)(81 82 83 84 85)(86 87 88 89 90)(91 92 93 94 95)(96 97 98 99 100)(101 102 103 104 105)(106 107 108 109 110)(111 112 113 114 115)(116 117 118 119 120)(121 122 123 124 125)(126 127 128 129 130)(131 132 133 134 135)(136 137 138 139 140)(141 142 143 144 145)(146 147 148 149 150)(151 152 153 154 155)(156 157 158 159 160)

(1 21)(2 22)(3 23)(4 24)(5 25)(6 11)(7 12)(8 13)(9 14)(10 15)(16 156)(17 157)(18 158)(19 159)(20 160)(26 31)(27 32)(28 33)(29 34)(30 35)(36 41)(37 42)(38 43)(39 44)(40 45)(46 51)(47 52)(48 53)(49 54)(50 55)(56 61)(57 62)(58 63)(59 64)(60 65)(66 71)(67 72)(68 73)(69 74)(70 75)(76 81)(77 82)(78 83)(79 84)(80 85)(86 91)(87 92)(88 93)(89 94)(90 95)(96 111)(97 112)(98 113)(99 114)(100 115)(101 106)(102 107)(103 108)(104 109)(105 110)(116 131)(117 132)(118 133)(119 134)(120 135)(121 126)(122 127)(123 128)(124 129)(125 130)(136 151)(137 152)(138 153)(139 154)(140 155)(141 146)(142 147)(143 148)(144 149)(145 150)

(1 26)(2 27)(3 28)(4 29)(5 30)(6 16)(7 17)(8 18)(9 19)(10 20)(11 156)(12 157)(13 158)(14 159)(15 160)(21 31)(22 32)(23 33)(24 34)(25 35)(36 46)(37 47)(38 48)(39 49)(40 50)(41 51)(42 52)(43 53)(44 54)(45 55)(56 66)(57 67)(58 68)(59 69)(60 70)(61 71)(62 72)(63 73)(64 74)(65 75)(76 86)(77 87)(78 88)(79 89)(80 90)(81 91)(82 92)(83 93)(84 94)(85 95)(96 106)(97 107)(98 108)(99 109)(100 110)(101 111)(102 112)(103 113)(104 114)(105 115)(116 126)(117 127)(118 128)(119 129)(120 130)(121 131)(122 132)(123 133)(124 134)(125 135)(136 146)(137 147)(138 148)(139 149)(140 150)(141 151)(142 152)(143 153)(144 154)(145 155)

(1 21)(2 22)(3 23)(4 24)(5 25)(6 156)(7 157)(8 158)(9 159)(10 160)(11 16)(12 17)(13 18)(14 19)(15 20)(26 31)(27 32)(28 33)(29 34)(30 35)(36 41)(37 42)(38 43)(39 44)(40 45)(46 51)(47 52)(48 53)(49 54)(50 55)(56 61)(57 62)(58 63)(59 64)(60 65)(66 71)(67 72)(68 73)(69 74)(70 75)(76 81)(77 82)(78 83)(79 84)(80 85)(86 91)(87 92)(88 93)(89 94)(90 95)(96 101)(97 102)(98 103)(99 104)(100 105)(106 111)(107 112)(108 113)(109 114)(110 115)(116 121)(117 122)(118 123)(119 124)(120 125)(126 131)(127 132)(128 133)(129 134)(130 135)(136 141)(137 142)(138 143)(139 144)(140 145)(146 151)(147 152)(148 153)(149 154)(150 155)

(1 101 21 96)(2 102 22 97)(3 103 23 98)(4 104 24 99)(5 105 25 100)(6 81 156 76)(7 82 157 77)(8 83 158 78)(9 84 159 79)(10 85 160 80)(11 86 16 91)(12 87 17 92)(13 88 18 93)(14 89 19 94)(15 90 20 95)(26 111 31 106)(27 112 32 107)(28 113 33 108)(29 114 34 109)(30 115 35 110)(36 121 41 116)(37 122 42 117)(38 123 43 118)(39 124 44 119)(40 125 45 120)(46 131 51 126)(47 132 52 127)(48 133 53 128)(49 134 54 129)(50 135 55 130)(56 141 61 136)(57 142 62 137)(58 143 63 138)(59 144 64 139)(60 145 65 140)(66 151 71 146)(67 152 72 147)(68 153 73 148)(69 154 74 149)(70 155 75 150)

(1 66 26 56)(2 67 27 57)(3 68 28 58)(4 69 29 59)(5 70 30 60)(6 131 16 121)(7 132 17 122)(8 133 18 123)(9 134 19 124)(10 135 20 125)(11 116 156 126)(12 117 157 127)(13 118 158 128)(14 119 159 129)(15 120 160 130)(21 71 31 61)(22 72 32 62)(23 73 33 63)(24 74 34 64)(25 75 35 65)(36 86 46 76)(37 87 47 77)(38 88 48 78)(39 89 49 79)(40 90 50 80)(41 91 51 81)(42 92 52 82)(43 93 53 83)(44 94 54 84)(45 95 55 85)(96 136 106 146)(97 137 107 147)(98 138 108 148)(99 139 109 149)(100 140 110 150)(101 141 111 151)(102 142 112 152)(103 143 113 153)(104 144 114 154)(105 145 115 155)

(1 46 26 36)(2 47 27 37)(3 48 28 38)(4 49 29 39)(5 50 30 40)(6 151 16 141)(7 152 17 142)(8 153 18 143)(9 154 19 144)(10 155 20 145)(11 136 156 146)(12 137 157 147)(13 138 158 148)(14 139 159 149)(15 140 160 150)(21 51 31 41)(22 52 32 42)(23 53 33 43)(24 54 34 44)(25 55 35 45)(56 76 66 86)(57 77 67 87)(58 78 68 88)(59 79 69 89)(60 80 70 90)(61 81 71 91)(62 82 72 92)(63 83 73 93)(64 84 74 94)(65 85 75 95)(96 126 106 116)(97 127 107 117)(98 128 108 118)(99 129 109 119)(100 130 110 120)(101 131 111 121)(102 132 112 122)(103 133 113 123)(104 134 114 124)(105 135 115 125)
 
G:=sub<Sym(160)| (1,2,3,4,5)(6,7,8,9,10)(11,12,13,14,15)(16,17,18,19,20)(21,22,23,24,25)(26,27,28,29,30)(31,32,33,34,35)(36,37,38,39,40)(41,42,43,44,45)(46,47,48,49,50)(51,52,53,54,55)(56,57,58,59,60)(61,62,63,64,65)(66,67,68,69,70)(71,72,73,74,75)(76,77,78,79,80)(81,82,83,84,85)(86,87,88,89,90)(91,92,93,94,95)(96,97,98,99,100)(101,102,103,104,105)(106,107,108,109,110)(111,112,113,114,115)(116,117,118,119,120)(121,122,123,124,125)(126,127,128,129,130)(131,132,133,134,135)(136,137,138,139,140)(141,142,143,144,145)(146,147,148,149,150)(151,152,153,154,155)(156,157,158,159,160), (1,21)(2,22)(3,23)(4,24)(5,25)(6,11)(7,12)(8,13)(9,14)(10,15)(16,156)(17,157)(18,158)(19,159)(20,160)(26,31)(27,32)(28,33)(29,34)(30,35)(36,41)(37,42)(38,43)(39,44)(40,45)(46,51)(47,52)(48,53)(49,54)(50,55)(56,61)(57,62)(58,63)(59,64)(60,65)(66,71)(67,72)(68,73)(69,74)(70,75)(76,81)(77,82)(78,83)(79,84)(80,85)(86,91)(87,92)(88,93)(89,94)(90,95)(96,111)(97,112)(98,113)(99,114)(100,115)(101,106)(102,107)(103,108)(104,109)(105,110)(116,131)(117,132)(118,133)(119,134)(120,135)(121,126)(122,127)(123,128)(124,129)(125,130)(136,151)(137,152)(138,153)(139,154)(140,155)(141,146)(142,147)(143,148)(144,149)(145,150), (1,26)(2,27)(3,28)(4,29)(5,30)(6,16)(7,17)(8,18)(9,19)(10,20)(11,156)(12,157)(13,158)(14,159)(15,160)(21,31)(22,32)(23,33)(24,34)(25,35)(36,46)(37,47)(38,48)(39,49)(40,50)(41,51)(42,52)(43,53)(44,54)(45,55)(56,66)(57,67)(58,68)(59,69)(60,70)(61,71)(62,72)(63,73)(64,74)(65,75)(76,86)(77,87)(78,88)(79,89)(80,90)(81,91)(82,92)(83,93)(84,94)(85,95)(96,106)(97,107)(98,108)(99,109)(100,110)(101,111)(102,112)(103,113)(104,114)(105,115)(116,126)(117,127)(118,128)(119,129)(120,130)(121,131)(122,132)(123,133)(124,134)(125,135)(136,146)(137,147)(138,148)(139,149)(140,150)(141,151)(142,152)(143,153)(144,154)(145,155), (1,21)(2,22)(3,23)(4,24)(5,25)(6,156)(7,157)(8,158)(9,159)(10,160)(11,16)(12,17)(13,18)(14,19)(15,20)(26,31)(27,32)(28,33)(29,34)(30,35)(36,41)(37,42)(38,43)(39,44)(40,45)(46,51)(47,52)(48,53)(49,54)(50,55)(56,61)(57,62)(58,63)(59,64)(60,65)(66,71)(67,72)(68,73)(69,74)(70,75)(76,81)(77,82)(78,83)(79,84)(80,85)(86,91)(87,92)(88,93)(89,94)(90,95)(96,101)(97,102)(98,103)(99,104)(100,105)(106,111)(107,112)(108,113)(109,114)(110,115)(116,121)(117,122)(118,123)(119,124)(120,125)(126,131)(127,132)(128,133)(129,134)(130,135)(136,141)(137,142)(138,143)(139,144)(140,145)(146,151)(147,152)(148,153)(149,154)(150,155), (1,101,21,96)(2,102,22,97)(3,103,23,98)(4,104,24,99)(5,105,25,100)(6,81,156,76)(7,82,157,77)(8,83,158,78)(9,84,159,79)(10,85,160,80)(11,86,16,91)(12,87,17,92)(13,88,18,93)(14,89,19,94)(15,90,20,95)(26,111,31,106)(27,112,32,107)(28,113,33,108)(29,114,34,109)(30,115,35,110)(36,121,41,116)(37,122,42,117)(38,123,43,118)(39,124,44,119)(40,125,45,120)(46,131,51,126)(47,132,52,127)(48,133,53,128)(49,134,54,129)(50,135,55,130)(56,141,61,136)(57,142,62,137)(58,143,63,138)(59,144,64,139)(60,145,65,140)(66,151,71,146)(67,152,72,147)(68,153,73,148)(69,154,74,149)(70,155,75,150), (1,66,26,56)(2,67,27,57)(3,68,28,58)(4,69,29,59)(5,70,30,60)(6,131,16,121)(7,132,17,122)(8,133,18,123)(9,134,19,124)(10,135,20,125)(11,116,156,126)(12,117,157,127)(13,118,158,128)(14,119,159,129)(15,120,160,130)(21,71,31,61)(22,72,32,62)(23,73,33,63)(24,74,34,64)(25,75,35,65)(36,86,46,76)(37,87,47,77)(38,88,48,78)(39,89,49,79)(40,90,50,80)(41,91,51,81)(42,92,52,82)(43,93,53,83)(44,94,54,84)(45,95,55,85)(96,136,106,146)(97,137,107,147)(98,138,108,148)(99,139,109,149)(100,140,110,150)(101,141,111,151)(102,142,112,152)(103,143,113,153)(104,144,114,154)(105,145,115,155), (1,46,26,36)(2,47,27,37)(3,48,28,38)(4,49,29,39)(5,50,30,40)(6,151,16,141)(7,152,17,142)(8,153,18,143)(9,154,19,144)(10,155,20,145)(11,136,156,146)(12,137,157,147)(13,138,158,148)(14,139,159,149)(15,140,160,150)(21,51,31,41)(22,52,32,42)(23,53,33,43)(24,54,34,44)(25,55,35,45)(56,76,66,86)(57,77,67,87)(58,78,68,88)(59,79,69,89)(60,80,70,90)(61,81,71,91)(62,82,72,92)(63,83,73,93)(64,84,74,94)(65,85,75,95)(96,126,106,116)(97,127,107,117)(98,128,108,118)(99,129,109,119)(100,130,110,120)(101,131,111,121)(102,132,112,122)(103,133,113,123)(104,134,114,124)(105,135,115,125)>;
 
G:=Group( (1,2,3,4,5)(6,7,8,9,10)(11,12,13,14,15)(16,17,18,19,20)(21,22,23,24,25)(26,27,28,29,30)(31,32,33,34,35)(36,37,38,39,40)(41,42,43,44,45)(46,47,48,49,50)(51,52,53,54,55)(56,57,58,59,60)(61,62,63,64,65)(66,67,68,69,70)(71,72,73,74,75)(76,77,78,79,80)(81,82,83,84,85)(86,87,88,89,90)(91,92,93,94,95)(96,97,98,99,100)(101,102,103,104,105)(106,107,108,109,110)(111,112,113,114,115)(116,117,118,119,120)(121,122,123,124,125)(126,127,128,129,130)(131,132,133,134,135)(136,137,138,139,140)(141,142,143,144,145)(146,147,148,149,150)(151,152,153,154,155)(156,157,158,159,160), (1,21)(2,22)(3,23)(4,24)(5,25)(6,11)(7,12)(8,13)(9,14)(10,15)(16,156)(17,157)(18,158)(19,159)(20,160)(26,31)(27,32)(28,33)(29,34)(30,35)(36,41)(37,42)(38,43)(39,44)(40,45)(46,51)(47,52)(48,53)(49,54)(50,55)(56,61)(57,62)(58,63)(59,64)(60,65)(66,71)(67,72)(68,73)(69,74)(70,75)(76,81)(77,82)(78,83)(79,84)(80,85)(86,91)(87,92)(88,93)(89,94)(90,95)(96,111)(97,112)(98,113)(99,114)(100,115)(101,106)(102,107)(103,108)(104,109)(105,110)(116,131)(117,132)(118,133)(119,134)(120,135)(121,126)(122,127)(123,128)(124,129)(125,130)(136,151)(137,152)(138,153)(139,154)(140,155)(141,146)(142,147)(143,148)(144,149)(145,150), (1,26)(2,27)(3,28)(4,29)(5,30)(6,16)(7,17)(8,18)(9,19)(10,20)(11,156)(12,157)(13,158)(14,159)(15,160)(21,31)(22,32)(23,33)(24,34)(25,35)(36,46)(37,47)(38,48)(39,49)(40,50)(41,51)(42,52)(43,53)(44,54)(45,55)(56,66)(57,67)(58,68)(59,69)(60,70)(61,71)(62,72)(63,73)(64,74)(65,75)(76,86)(77,87)(78,88)(79,89)(80,90)(81,91)(82,92)(83,93)(84,94)(85,95)(96,106)(97,107)(98,108)(99,109)(100,110)(101,111)(102,112)(103,113)(104,114)(105,115)(116,126)(117,127)(118,128)(119,129)(120,130)(121,131)(122,132)(123,133)(124,134)(125,135)(136,146)(137,147)(138,148)(139,149)(140,150)(141,151)(142,152)(143,153)(144,154)(145,155), (1,21)(2,22)(3,23)(4,24)(5,25)(6,156)(7,157)(8,158)(9,159)(10,160)(11,16)(12,17)(13,18)(14,19)(15,20)(26,31)(27,32)(28,33)(29,34)(30,35)(36,41)(37,42)(38,43)(39,44)(40,45)(46,51)(47,52)(48,53)(49,54)(50,55)(56,61)(57,62)(58,63)(59,64)(60,65)(66,71)(67,72)(68,73)(69,74)(70,75)(76,81)(77,82)(78,83)(79,84)(80,85)(86,91)(87,92)(88,93)(89,94)(90,95)(96,101)(97,102)(98,103)(99,104)(100,105)(106,111)(107,112)(108,113)(109,114)(110,115)(116,121)(117,122)(118,123)(119,124)(120,125)(126,131)(127,132)(128,133)(129,134)(130,135)(136,141)(137,142)(138,143)(139,144)(140,145)(146,151)(147,152)(148,153)(149,154)(150,155), (1,101,21,96)(2,102,22,97)(3,103,23,98)(4,104,24,99)(5,105,25,100)(6,81,156,76)(7,82,157,77)(8,83,158,78)(9,84,159,79)(10,85,160,80)(11,86,16,91)(12,87,17,92)(13,88,18,93)(14,89,19,94)(15,90,20,95)(26,111,31,106)(27,112,32,107)(28,113,33,108)(29,114,34,109)(30,115,35,110)(36,121,41,116)(37,122,42,117)(38,123,43,118)(39,124,44,119)(40,125,45,120)(46,131,51,126)(47,132,52,127)(48,133,53,128)(49,134,54,129)(50,135,55,130)(56,141,61,136)(57,142,62,137)(58,143,63,138)(59,144,64,139)(60,145,65,140)(66,151,71,146)(67,152,72,147)(68,153,73,148)(69,154,74,149)(70,155,75,150), (1,66,26,56)(2,67,27,57)(3,68,28,58)(4,69,29,59)(5,70,30,60)(6,131,16,121)(7,132,17,122)(8,133,18,123)(9,134,19,124)(10,135,20,125)(11,116,156,126)(12,117,157,127)(13,118,158,128)(14,119,159,129)(15,120,160,130)(21,71,31,61)(22,72,32,62)(23,73,33,63)(24,74,34,64)(25,75,35,65)(36,86,46,76)(37,87,47,77)(38,88,48,78)(39,89,49,79)(40,90,50,80)(41,91,51,81)(42,92,52,82)(43,93,53,83)(44,94,54,84)(45,95,55,85)(96,136,106,146)(97,137,107,147)(98,138,108,148)(99,139,109,149)(100,140,110,150)(101,141,111,151)(102,142,112,152)(103,143,113,153)(104,144,114,154)(105,145,115,155), (1,46,26,36)(2,47,27,37)(3,48,28,38)(4,49,29,39)(5,50,30,40)(6,151,16,141)(7,152,17,142)(8,153,18,143)(9,154,19,144)(10,155,20,145)(11,136,156,146)(12,137,157,147)(13,138,158,148)(14,139,159,149)(15,140,160,150)(21,51,31,41)(22,52,32,42)(23,53,33,43)(24,54,34,44)(25,55,35,45)(56,76,66,86)(57,77,67,87)(58,78,68,88)(59,79,69,89)(60,80,70,90)(61,81,71,91)(62,82,72,92)(63,83,73,93)(64,84,74,94)(65,85,75,95)(96,126,106,116)(97,127,107,117)(98,128,108,118)(99,129,109,119)(100,130,110,120)(101,131,111,121)(102,132,112,122)(103,133,113,123)(104,134,114,124)(105,135,115,125) );
 
G=PermutationGroup([[(1,2,3,4,5),(6,7,8,9,10),(11,12,13,14,15),(16,17,18,19,20),(21,22,23,24,25),(26,27,28,29,30),(31,32,33,34,35),(36,37,38,39,40),(41,42,43,44,45),(46,47,48,49,50),(51,52,53,54,55),(56,57,58,59,60),(61,62,63,64,65),(66,67,68,69,70),(71,72,73,74,75),(76,77,78,79,80),(81,82,83,84,85),(86,87,88,89,90),(91,92,93,94,95),(96,97,98,99,100),(101,102,103,104,105),(106,107,108,109,110),(111,112,113,114,115),(116,117,118,119,120),(121,122,123,124,125),(126,127,128,129,130),(131,132,133,134,135),(136,137,138,139,140),(141,142,143,144,145),(146,147,148,149,150),(151,152,153,154,155),(156,157,158,159,160)], [(1,21),(2,22),(3,23),(4,24),(5,25),(6,11),(7,12),(8,13),(9,14),(10,15),(16,156),(17,157),(18,158),(19,159),(20,160),(26,31),(27,32),(28,33),(29,34),(30,35),(36,41),(37,42),(38,43),(39,44),(40,45),(46,51),(47,52),(48,53),(49,54),(50,55),(56,61),(57,62),(58,63),(59,64),(60,65),(66,71),(67,72),(68,73),(69,74),(70,75),(76,81),(77,82),(78,83),(79,84),(80,85),(86,91),(87,92),(88,93),(89,94),(90,95),(96,111),(97,112),(98,113),(99,114),(100,115),(101,106),(102,107),(103,108),(104,109),(105,110),(116,131),(117,132),(118,133),(119,134),(120,135),(121,126),(122,127),(123,128),(124,129),(125,130),(136,151),(137,152),(138,153),(139,154),(140,155),(141,146),(142,147),(143,148),(144,149),(145,150)], [(1,26),(2,27),(3,28),(4,29),(5,30),(6,16),(7,17),(8,18),(9,19),(10,20),(11,156),(12,157),(13,158),(14,159),(15,160),(21,31),(22,32),(23,33),(24,34),(25,35),(36,46),(37,47),(38,48),(39,49),(40,50),(41,51),(42,52),(43,53),(44,54),(45,55),(56,66),(57,67),(58,68),(59,69),(60,70),(61,71),(62,72),(63,73),(64,74),(65,75),(76,86),(77,87),(78,88),(79,89),(80,90),(81,91),(82,92),(83,93),(84,94),(85,95),(96,106),(97,107),(98,108),(99,109),(100,110),(101,111),(102,112),(103,113),(104,114),(105,115),(116,126),(117,127),(118,128),(119,129),(120,130),(121,131),(122,132),(123,133),(124,134),(125,135),(136,146),(137,147),(138,148),(139,149),(140,150),(141,151),(142,152),(143,153),(144,154),(145,155)], [(1,21),(2,22),(3,23),(4,24),(5,25),(6,156),(7,157),(8,158),(9,159),(10,160),(11,16),(12,17),(13,18),(14,19),(15,20),(26,31),(27,32),(28,33),(29,34),(30,35),(36,41),(37,42),(38,43),(39,44),(40,45),(46,51),(47,52),(48,53),(49,54),(50,55),(56,61),(57,62),(58,63),(59,64),(60,65),(66,71),(67,72),(68,73),(69,74),(70,75),(76,81),(77,82),(78,83),(79,84),(80,85),(86,91),(87,92),(88,93),(89,94),(90,95),(96,101),(97,102),(98,103),(99,104),(100,105),(106,111),(107,112),(108,113),(109,114),(110,115),(116,121),(117,122),(118,123),(119,124),(120,125),(126,131),(127,132),(128,133),(129,134),(130,135),(136,141),(137,142),(138,143),(139,144),(140,145),(146,151),(147,152),(148,153),(149,154),(150,155)], [(1,101,21,96),(2,102,22,97),(3,103,23,98),(4,104,24,99),(5,105,25,100),(6,81,156,76),(7,82,157,77),(8,83,158,78),(9,84,159,79),(10,85,160,80),(11,86,16,91),(12,87,17,92),(13,88,18,93),(14,89,19,94),(15,90,20,95),(26,111,31,106),(27,112,32,107),(28,113,33,108),(29,114,34,109),(30,115,35,110),(36,121,41,116),(37,122,42,117),(38,123,43,118),(39,124,44,119),(40,125,45,120),(46,131,51,126),(47,132,52,127),(48,133,53,128),(49,134,54,129),(50,135,55,130),(56,141,61,136),(57,142,62,137),(58,143,63,138),(59,144,64,139),(60,145,65,140),(66,151,71,146),(67,152,72,147),(68,153,73,148),(69,154,74,149),(70,155,75,150)], [(1,66,26,56),(2,67,27,57),(3,68,28,58),(4,69,29,59),(5,70,30,60),(6,131,16,121),(7,132,17,122),(8,133,18,123),(9,134,19,124),(10,135,20,125),(11,116,156,126),(12,117,157,127),(13,118,158,128),(14,119,159,129),(15,120,160,130),(21,71,31,61),(22,72,32,62),(23,73,33,63),(24,74,34,64),(25,75,35,65),(36,86,46,76),(37,87,47,77),(38,88,48,78),(39,89,49,79),(40,90,50,80),(41,91,51,81),(42,92,52,82),(43,93,53,83),(44,94,54,84),(45,95,55,85),(96,136,106,146),(97,137,107,147),(98,138,108,148),(99,139,109,149),(100,140,110,150),(101,141,111,151),(102,142,112,152),(103,143,113,153),(104,144,114,154),(105,145,115,155)], [(1,46,26,36),(2,47,27,37),(3,48,28,38),(4,49,29,39),(5,50,30,40),(6,151,16,141),(7,152,17,142),(8,153,18,143),(9,154,19,144),(10,155,20,145),(11,136,156,146),(12,137,157,147),(13,138,158,148),(14,139,159,149),(15,140,160,150),(21,51,31,41),(22,52,32,42),(23,53,33,43),(24,54,34,44),(25,55,35,45),(56,76,66,86),(57,77,67,87),(58,78,68,88),(59,79,69,89),(60,80,70,90),(61,81,71,91),(62,82,72,92),(63,83,73,93),(64,84,74,94),(65,85,75,95),(96,126,106,116),(97,127,107,117),(98,128,108,118),(99,129,109,119),(100,130,110,120),(101,131,111,121),(102,132,112,122),(103,133,113,123),(104,134,114,124),(105,135,115,125)]])
 

170 conjugacy classes

class 1 2A2B2C2D2E4A···4AB5A5B5C5D10A···10L10M···10T20A···20DH
order1222224···4555510···1010···1020···20
size1111222···211111···12···22···2

170 irreducible representations

dim111111111144
type++++-
imageC1C2C2C2C4C5C10C10C10C202- 1+4C5×2- 1+4
kernelC5×C23.32C23C5×C42⋊C2Q8×C20Q8×C2×C10Q8×C10C23.32C23C42⋊C2C4×Q8C22×Q8C2×Q8C10C2
# reps1681164243246428

Matrix representation of C5×C23.32C23 ►in GL6(𝔽41)

100000
0370000
001000
000100
000010
000001
,
100000
0400000
0040000
0004000
000010
0036501
,
100000
010000
0040000
0004000
0000400
0000040
,
4000000
010000
0040000
0004000
0000400
0000040
,
900000
010000
000010
00536139
0040000
00281305
,
100000
0400000
00261500
00151500
00734030
0070150
,
4000000
0400000
000100
0040000
00536139
0050140

G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,37,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,36,0,0,0,40,0,5,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[40,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[9,0,0,0,0,0,0,1,0,0,0,0,0,0,0,5,40,28,0,0,0,36,0,13,0,0,1,1,0,0,0,0,0,39,0,5],[1,0,0,0,0,0,0,40,0,0,0,0,0,0,26,15,7,7,0,0,15,15,34,0,0,0,0,0,0,15,0,0,0,0,30,0],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,0,40,5,5,0,0,1,0,36,0,0,0,0,0,1,1,0,0,0,0,39,40] >;
 

C5×C23.32C23 in GAP, Magma, Sage, TeX 

C_5\times C_2^3._{32}C_2^3
 
% in TeX
 
G:=Group("C5xC2^3.32C2^3");
 
// GroupNames label
 
G:=SmallGroup(320,1521);
 
// by ID
 
G=gap.SmallGroup(320,1521);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-5,-2,-2,1120,1149,568,891,436,2467]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^5=b^2=c^2=d^2=1,e^2=d,f^2=g^2=c,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,a*g=g*a,e*b*e^-1=b*c=c*b,b*d=d*b,b*f=f*b,b*g=g*b,c*d=d*c,f*e*f^-1=c*e=e*c,g*f*g^-1=c*f=f*c,c*g=g*c,d*e=e*d,d*f=f*d,d*g=g*d,e*g=g*e>;
 
// generators/relations
 

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×
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