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G = C5×C22.53C24order 320 = 26·5

Direct product of C5 and C22.53C24

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

Aliases: C5×C22.53C24, C10.1682+ 1+4, (D4×C20)⋊51C2, (C4×D4)⋊22C10, (Q8×C20)⋊37C2, (C4×Q8)⋊17C10, C41D4.7C10, C4.4D414C10, C42.53(C2×C10), C20.349(C4○D4), (C2×C10).379C24, (C4×C20).294C22, (C2×C20).967C23, (D4×C10).326C22, C22.D412C10, C23.22(C22×C10), C22.53(C23×C10), (C22×C20).39C22, (Q8×C10).279C22, C2.20(C5×2+ 1+4), (C22×C10).105C23, C4.47(C5×C4○D4), C4⋊C4.79(C2×C10), C2.31(C10×C4○D4), (C2×D4).72(C2×C10), C10.250(C2×C4○D4), (C5×C41D4).14C2, (C5×C4.4D4)⋊34C2, (C2×Q8).67(C2×C10), C22⋊C4.26(C2×C10), (C5×C4⋊C4).404C22, (C2×C4).64(C22×C10), (C22×C4).13(C2×C10), (C5×C22.D4)⋊31C2, (C5×C22⋊C4).91C22, SmallGroup(320,1561)

Series: Derived Chief Lower central Upper central

C1C22 — C5×C22.53C24
C1C2C22C2×C10C2×C20C5×C22⋊C4C5×C4.4D4 — C5×C22.53C24
C1C22 — C5×C22.53C24
C1C2×C10 — C5×C22.53C24

Generators and relations for C5×C22.53C24
 G = < a,b,c,d,e,f,g | a5=b2=c2=1, d2=f2=c, e2=cb=bc, g2=b, ab=ba, ac=ca, ad=da, ae=ea, af=fa, ag=ga, ede-1=bd=db, geg-1=be=eb, bf=fb, bg=gb, fdf-1=cd=dc, ce=ec, cf=fc, cg=gc, dg=gd, ef=fe, fg=gf >

Subgroups: 362 in 236 conjugacy classes, 150 normal (14 characteristic)
C1, C2, C2 [×2], C2 [×4], C4 [×4], C4 [×9], C22, C22 [×12], C5, C2×C4, C2×C4 [×10], C2×C4 [×4], D4 [×10], Q8 [×4], C23 [×4], C10, C10 [×2], C10 [×4], C42, C42 [×4], C22⋊C4 [×12], C4⋊C4 [×6], C22×C4 [×4], C2×D4 [×6], C2×Q8 [×2], C20 [×4], C20 [×9], C2×C10, C2×C10 [×12], C4×D4 [×4], C4×Q8 [×2], C22.D4 [×4], C4.4D4 [×4], C41D4, C2×C20, C2×C20 [×10], C2×C20 [×4], C5×D4 [×10], C5×Q8 [×4], C22×C10 [×4], C22.53C24, C4×C20, C4×C20 [×4], C5×C22⋊C4 [×12], C5×C4⋊C4 [×6], C22×C20 [×4], D4×C10 [×6], Q8×C10 [×2], D4×C20 [×4], Q8×C20 [×2], C5×C22.D4 [×4], C5×C4.4D4 [×4], C5×C41D4, C5×C22.53C24
Quotients: C1, C2 [×15], C22 [×35], C5, C23 [×15], C10 [×15], C4○D4 [×4], C24, C2×C10 [×35], C2×C4○D4 [×2], 2+ 1+4, C22×C10 [×15], C22.53C24, C5×C4○D4 [×4], C23×C10, C10×C4○D4 [×2], C5×2+ 1+4, C5×C22.53C24

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

125 conjugacy classes

class 1 2A2B2C2D2E2F2G4A···4L4M···4Q5A5B5C5D10A···10L10M···10AB20A···20AV20AW···20BP
order122222224···44···4555510···1010···1020···2020···20
size111144442···24···411111···14···42···24···4

125 irreducible representations

dim1111111111112244
type+++++++
imageC1C2C2C2C2C2C5C10C10C10C10C10C4○D4C5×C4○D42+ 1+4C5×2+ 1+4
kernelC5×C22.53C24D4×C20Q8×C20C5×C22.D4C5×C4.4D4C5×C41D4C22.53C24C4×D4C4×Q8C22.D4C4.4D4C41D4C20C4C10C2
# reps14244141681616483214

Matrix representation of C5×C22.53C24 in GL4(𝔽41) generated by

37000
03700
00100
00010
,
40000
04000
0010
0001
,
1000
0100
00400
00040
,
03200
9000
00032
00320
,
32000
0900
00320
00032
,
40000
04000
0001
00400
,
0100
40000
00400
00040
G:=sub<GL(4,GF(41))| [37,0,0,0,0,37,0,0,0,0,10,0,0,0,0,10],[40,0,0,0,0,40,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,40,0,0,0,0,40],[0,9,0,0,32,0,0,0,0,0,0,32,0,0,32,0],[32,0,0,0,0,9,0,0,0,0,32,0,0,0,0,32],[40,0,0,0,0,40,0,0,0,0,0,40,0,0,1,0],[0,40,0,0,1,0,0,0,0,0,40,0,0,0,0,40] >;

C5×C22.53C24 in GAP, Magma, Sage, TeX

C_5\times C_2^2._{53}C_2^4
% in TeX

G:=Group("C5xC2^2.53C2^4");
// GroupNames label

G:=SmallGroup(320,1561);
// by ID

G=gap.SmallGroup(320,1561);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-5,-2,-2,1120,1149,1688,3446,856,1242,304]);
// Polycyclic

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

׿
×
𝔽