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G = C5×C22.35C24order 320 = 26·5

Direct product of C5 and C22.35C24

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

Aliases: C5×C22.35C24, C10.1142- 1+4, C4⋊Q810C10, (C4×Q8)⋊9C10, (Q8×C20)⋊29C2, C422C2.C10, C42.C26C10, C22⋊Q8.8C10, C42.39(C2×C10), C20.277(C4○D4), (C2×C10).361C24, (C2×C20).670C23, (C4×C20).280C22, C42⋊C2.12C10, C2.6(C5×2- 1+4), (C22×C10).96C23, C23.13(C22×C10), C22.35(C23×C10), (Q8×C10).272C22, (C22×C20).449C22, (C5×C4⋊Q8)⋊31C2, C4.21(C5×C4○D4), C4⋊C4.69(C2×C10), C2.18(C10×C4○D4), C10.237(C2×C4○D4), C22⋊C4.3(C2×C10), (C2×Q8).59(C2×C10), (C5×C42.C2)⋊23C2, (C5×C22⋊Q8).18C2, (C5×C4⋊C4).248C22, (C22×C4).61(C2×C10), (C2×C4).28(C22×C10), (C5×C422C2).2C2, (C5×C42⋊C2).26C2, (C5×C22⋊C4).149C22, SmallGroup(320,1543)

Series: Derived Chief Lower central Upper central

C1C22 — C5×C22.35C24
C1C2C22C2×C10C2×C20C5×C4⋊C4C5×C422C2 — C5×C22.35C24
C1C22 — C5×C22.35C24
C1C2×C10 — C5×C22.35C24

Generators and relations for C5×C22.35C24
 G = < a,b,c,d,e,f,g | a5=b2=c2=f2=1, d2=g2=b, e2=c, ab=ba, ac=ca, ad=da, ae=ea, af=fa, ag=ga, bc=cb, ede-1=gdg-1=bd=db, fef=be=eb, bf=fb, bg=gb, fdf=cd=dc, ce=ec, cf=fc, cg=gc, eg=ge, fg=gf >

Subgroups: 242 in 192 conjugacy classes, 146 normal (26 characteristic)
C1, C2, C2 [×2], C2, C4 [×2], C4 [×13], C22, C22 [×3], C5, C2×C4 [×2], C2×C4 [×12], C2×C4 [×2], Q8 [×4], C23, C10, C10 [×2], C10, C42 [×2], C42 [×4], C22⋊C4 [×6], C4⋊C4 [×20], C22×C4, C2×Q8 [×2], C20 [×2], C20 [×13], C2×C10, C2×C10 [×3], C42⋊C2, C4×Q8 [×2], C22⋊Q8 [×2], C42.C2, C42.C2 [×4], C422C2 [×4], C4⋊Q8, C2×C20 [×2], C2×C20 [×12], C2×C20 [×2], C5×Q8 [×4], C22×C10, C22.35C24, C4×C20 [×2], C4×C20 [×4], C5×C22⋊C4 [×6], C5×C4⋊C4 [×20], C22×C20, Q8×C10 [×2], C5×C42⋊C2, Q8×C20 [×2], C5×C22⋊Q8 [×2], C5×C42.C2, C5×C42.C2 [×4], C5×C422C2 [×4], C5×C4⋊Q8, C5×C22.35C24
Quotients: C1, C2 [×15], C22 [×35], C5, C23 [×15], C10 [×15], C4○D4 [×2], C24, C2×C10 [×35], C2×C4○D4, 2- 1+4 [×2], C22×C10 [×15], C22.35C24, C5×C4○D4 [×2], C23×C10, C10×C4○D4, C5×2- 1+4 [×2], C5×C22.35C24

Smallest permutation representation of C5×C22.35C24
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 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)
(1 106 26 96)(2 107 27 97)(3 108 28 98)(4 109 29 99)(5 110 30 100)(6 81 16 91)(7 82 17 92)(8 83 18 93)(9 84 19 94)(10 85 20 95)(11 86 156 76)(12 87 157 77)(13 88 158 78)(14 89 159 79)(15 90 160 80)(21 111 31 101)(22 112 32 102)(23 113 33 103)(24 114 34 104)(25 115 35 105)(36 126 46 116)(37 127 47 117)(38 128 48 118)(39 129 49 119)(40 130 50 120)(41 131 51 121)(42 132 52 122)(43 133 53 123)(44 134 54 124)(45 135 55 125)(56 146 66 136)(57 147 67 137)(58 148 68 138)(59 149 69 139)(60 150 70 140)(61 151 71 141)(62 152 72 142)(63 153 73 143)(64 154 74 144)(65 155 75 145)
(1 61 21 56)(2 62 22 57)(3 63 23 58)(4 64 24 59)(5 65 25 60)(6 131 156 126)(7 132 157 127)(8 133 158 128)(9 134 159 129)(10 135 160 130)(11 116 16 121)(12 117 17 122)(13 118 18 123)(14 119 19 124)(15 120 20 125)(26 71 31 66)(27 72 32 67)(28 73 33 68)(29 74 34 69)(30 75 35 70)(36 81 41 76)(37 82 42 77)(38 83 43 78)(39 84 44 79)(40 85 45 80)(46 91 51 86)(47 92 52 87)(48 93 53 88)(49 94 54 89)(50 95 55 90)(96 151 101 146)(97 152 102 147)(98 153 103 148)(99 154 104 149)(100 155 105 150)(106 141 111 136)(107 142 112 137)(108 143 113 138)(109 144 114 139)(110 145 115 140)
(6 11)(7 12)(8 13)(9 14)(10 15)(16 156)(17 157)(18 158)(19 159)(20 160)(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)(96 101)(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)(136 151)(137 152)(138 153)(139 154)(140 155)(141 146)(142 147)(143 148)(144 149)(145 150)
(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 86 66 76)(57 87 67 77)(58 88 68 78)(59 89 69 79)(60 90 70 80)(61 91 71 81)(62 92 72 82)(63 93 73 83)(64 94 74 84)(65 95 75 85)(96 116 106 126)(97 117 107 127)(98 118 108 128)(99 119 109 129)(100 120 110 130)(101 121 111 131)(102 122 112 132)(103 123 113 133)(104 124 114 134)(105 125 115 135)

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

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

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

110 conjugacy classes

class 1 2A2B2C2D4A···4F4G···4Q5A5B5C5D10A···10L10M10N10O10P20A···20X20Y···20BP
order122224···44···4555510···101010101020···2020···20
size111142···24···411111···144442···24···4

110 irreducible representations

dim111111111111112244
type+++++++-
imageC1C2C2C2C2C2C2C5C10C10C10C10C10C10C4○D4C5×C4○D42- 1+4C5×2- 1+4
kernelC5×C22.35C24C5×C42⋊C2Q8×C20C5×C22⋊Q8C5×C42.C2C5×C422C2C5×C4⋊Q8C22.35C24C42⋊C2C4×Q8C22⋊Q8C42.C2C422C2C4⋊Q8C20C4C10C2
# reps112254144882016441628

Matrix representation of C5×C22.35C24 in GL6(𝔽41)

100000
010000
0010000
0001000
0000100
0000010
,
100000
010000
0040000
0004000
0000400
0000040
,
4000000
0400000
0040000
0004000
0000400
0000040
,
28390000
2130000
003102937
001303733
003322823
0039352823
,
3200000
0320000
0090390
0000401
0000320
00040320
,
100000
28400000
001000
000100
0090400
0090040
,
100000
010000
0013900
0014000
0003201
00932400

G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,10,0,0,0,0,0,0,10,0,0,0,0,0,0,10,0,0,0,0,0,0,10],[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],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[28,2,0,0,0,0,39,13,0,0,0,0,0,0,31,13,33,39,0,0,0,0,2,35,0,0,29,37,28,28,0,0,37,33,23,23],[32,0,0,0,0,0,0,32,0,0,0,0,0,0,9,0,0,0,0,0,0,0,0,40,0,0,39,40,32,32,0,0,0,1,0,0],[1,28,0,0,0,0,0,40,0,0,0,0,0,0,1,0,9,9,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,1,0,9,0,0,39,40,32,32,0,0,0,0,0,40,0,0,0,0,1,0] >;

C5×C22.35C24 in GAP, Magma, Sage, TeX

C_5\times C_2^2._{35}C_2^4
% in TeX

G:=Group("C5xC2^2.35C2^4");
// GroupNames label

G:=SmallGroup(320,1543);
// by ID

G=gap.SmallGroup(320,1543);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-5,-2,-2,560,1149,1128,3446,891,2467,304]);
// Polycyclic

G:=Group<a,b,c,d,e,f,g|a^5=b^2=c^2=f^2=1,d^2=g^2=b,e^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,b*c=c*b,e*d*e^-1=g*d*g^-1=b*d=d*b,f*e*f=b*e=e*b,b*f=f*b,b*g=g*b,f*d*f=c*d=d*c,c*e=e*c,c*f=f*c,c*g=g*c,e*g=g*e,f*g=g*f>;
// generators/relations

׿
×
𝔽