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G = C5×C23.36D4order 320 = 26·5

Direct product of C5 and C23.36D4

direct product, metabelian, nilpotent (class 3), monomial, 2-elementary

Aliases: C5×C23.36D4, C4○D42C20, D44(C2×C20), Q84(C2×C20), C4.54(D4×C10), (C2×C20).314D4, C20.461(C2×D4), C4.4(C22×C20), D4⋊C414C10, C23.36(C5×D4), Q8⋊C414C10, C22.44(D4×C10), (C10×M4(2))⋊28C2, (C2×M4(2))⋊10C10, (C2×C40).322C22, (C2×C20).893C23, C20.208(C22×C4), (C22×C10).158D4, C20.155(C22⋊C4), C10.126(C8⋊C22), (D4×C10).288C22, (Q8×C10).252C22, C10.126(C8.C22), (C22×C20).410C22, (C2×C4⋊C4)⋊10C10, (C10×C4⋊C4)⋊37C2, (C5×C4○D4)⋊14C4, (C5×D4)⋊34(C2×C4), (C5×Q8)⋊31(C2×C4), C2.1(C5×C8⋊C22), C4⋊C4.39(C2×C10), (C2×C4).20(C2×C20), (C2×C8).47(C2×C10), (C2×C4○D4).5C10, (C2×C4).123(C5×D4), C4.23(C5×C22⋊C4), (C5×D4⋊C4)⋊37C2, C2.1(C5×C8.C22), (C2×C20).366(C2×C4), (C5×Q8⋊C4)⋊37C2, (C10×C4○D4).19C2, (C2×D4).46(C2×C10), (C2×C10).620(C2×D4), C2.20(C10×C22⋊C4), (C2×Q8).37(C2×C10), C22.5(C5×C22⋊C4), (C5×C4⋊C4).360C22, C10.149(C2×C22⋊C4), (C22×C4).29(C2×C10), (C2×C4).68(C22×C10), (C2×C10).96(C22⋊C4), SmallGroup(320,918)

Series: Derived Chief Lower central Upper central

C1C4 — C5×C23.36D4
C1C2C22C2×C4C2×C20C5×C4⋊C4C5×D4⋊C4 — C5×C23.36D4
C1C2C4 — C5×C23.36D4
C1C2×C10C22×C20 — C5×C23.36D4

Generators and relations for C5×C23.36D4
 G = < a,b,c,d,e,f | a5=b2=c2=d2=1, e4=d, f2=c, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, ebe-1=bd=db, bf=fb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=cde3 >

Subgroups: 274 in 162 conjugacy classes, 82 normal (34 characteristic)
C1, C2 [×3], C2 [×4], C4 [×2], C4 [×2], C4 [×4], C22, C22 [×2], C22 [×6], C5, C8 [×2], C2×C4 [×2], C2×C4 [×4], C2×C4 [×9], D4 [×2], D4 [×5], Q8 [×2], Q8, C23, C23, C10 [×3], C10 [×4], C4⋊C4 [×2], C4⋊C4, C2×C8 [×2], M4(2) [×2], C22×C4, C22×C4 [×2], C2×D4, C2×D4, C2×Q8, C4○D4 [×4], C4○D4 [×2], C20 [×2], C20 [×2], C20 [×4], C2×C10, C2×C10 [×2], C2×C10 [×6], D4⋊C4 [×2], Q8⋊C4 [×2], C2×C4⋊C4, C2×M4(2), C2×C4○D4, C40 [×2], C2×C20 [×2], C2×C20 [×4], C2×C20 [×9], C5×D4 [×2], C5×D4 [×5], C5×Q8 [×2], C5×Q8, C22×C10, C22×C10, C23.36D4, C5×C4⋊C4 [×2], C5×C4⋊C4, C2×C40 [×2], C5×M4(2) [×2], C22×C20, C22×C20 [×2], D4×C10, D4×C10, Q8×C10, C5×C4○D4 [×4], C5×C4○D4 [×2], C5×D4⋊C4 [×2], C5×Q8⋊C4 [×2], C10×C4⋊C4, C10×M4(2), C10×C4○D4, C5×C23.36D4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C5, C2×C4 [×6], D4 [×4], C23, C10 [×7], C22⋊C4 [×4], C22×C4, C2×D4 [×2], C20 [×4], C2×C10 [×7], C2×C22⋊C4, C8⋊C22, C8.C22, C2×C20 [×6], C5×D4 [×4], C22×C10, C23.36D4, C5×C22⋊C4 [×4], C22×C20, D4×C10 [×2], C10×C22⋊C4, C5×C8⋊C22, C5×C8.C22, C5×C23.36D4

Smallest permutation representation of C5×C23.36D4
On 160 points
Generators in S160
(1 98 23 90 15)(2 99 24 91 16)(3 100 17 92 9)(4 101 18 93 10)(5 102 19 94 11)(6 103 20 95 12)(7 104 21 96 13)(8 97 22 89 14)(25 42 116 33 108)(26 43 117 34 109)(27 44 118 35 110)(28 45 119 36 111)(29 46 120 37 112)(30 47 113 38 105)(31 48 114 39 106)(32 41 115 40 107)(49 123 139 57 131)(50 124 140 58 132)(51 125 141 59 133)(52 126 142 60 134)(53 127 143 61 135)(54 128 144 62 136)(55 121 137 63 129)(56 122 138 64 130)(65 81 155 73 147)(66 82 156 74 148)(67 83 157 75 149)(68 84 158 76 150)(69 85 159 77 151)(70 86 160 78 152)(71 87 153 79 145)(72 88 154 80 146)
(1 31)(2 28)(3 25)(4 30)(5 27)(6 32)(7 29)(8 26)(9 108)(10 105)(11 110)(12 107)(13 112)(14 109)(15 106)(16 111)(17 116)(18 113)(19 118)(20 115)(21 120)(22 117)(23 114)(24 119)(33 92)(34 89)(35 94)(36 91)(37 96)(38 93)(39 90)(40 95)(41 103)(42 100)(43 97)(44 102)(45 99)(46 104)(47 101)(48 98)(49 151)(50 148)(51 145)(52 150)(53 147)(54 152)(55 149)(56 146)(57 159)(58 156)(59 153)(60 158)(61 155)(62 160)(63 157)(64 154)(65 127)(66 124)(67 121)(68 126)(69 123)(70 128)(71 125)(72 122)(73 135)(74 132)(75 129)(76 134)(77 131)(78 136)(79 133)(80 130)(81 143)(82 140)(83 137)(84 142)(85 139)(86 144)(87 141)(88 138)
(1 127)(2 128)(3 121)(4 122)(5 123)(6 124)(7 125)(8 126)(9 55)(10 56)(11 49)(12 50)(13 51)(14 52)(15 53)(16 54)(17 63)(18 64)(19 57)(20 58)(21 59)(22 60)(23 61)(24 62)(25 67)(26 68)(27 69)(28 70)(29 71)(30 72)(31 65)(32 66)(33 75)(34 76)(35 77)(36 78)(37 79)(38 80)(39 73)(40 74)(41 82)(42 83)(43 84)(44 85)(45 86)(46 87)(47 88)(48 81)(89 134)(90 135)(91 136)(92 129)(93 130)(94 131)(95 132)(96 133)(97 142)(98 143)(99 144)(100 137)(101 138)(102 139)(103 140)(104 141)(105 146)(106 147)(107 148)(108 149)(109 150)(110 151)(111 152)(112 145)(113 154)(114 155)(115 156)(116 157)(117 158)(118 159)(119 160)(120 153)
(1 5)(2 6)(3 7)(4 8)(9 13)(10 14)(11 15)(12 16)(17 21)(18 22)(19 23)(20 24)(25 29)(26 30)(27 31)(28 32)(33 37)(34 38)(35 39)(36 40)(41 45)(42 46)(43 47)(44 48)(49 53)(50 54)(51 55)(52 56)(57 61)(58 62)(59 63)(60 64)(65 69)(66 70)(67 71)(68 72)(73 77)(74 78)(75 79)(76 80)(81 85)(82 86)(83 87)(84 88)(89 93)(90 94)(91 95)(92 96)(97 101)(98 102)(99 103)(100 104)(105 109)(106 110)(107 111)(108 112)(113 117)(114 118)(115 119)(116 120)(121 125)(122 126)(123 127)(124 128)(129 133)(130 134)(131 135)(132 136)(137 141)(138 142)(139 143)(140 144)(145 149)(146 150)(147 151)(148 152)(153 157)(154 158)(155 159)(156 160)
(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 8 127 126)(2 125 128 7)(3 6 121 124)(4 123 122 5)(9 12 55 50)(10 49 56 11)(13 16 51 54)(14 53 52 15)(17 20 63 58)(18 57 64 19)(21 24 59 62)(22 61 60 23)(25 32 67 66)(26 65 68 31)(27 30 69 72)(28 71 70 29)(33 40 75 74)(34 73 76 39)(35 38 77 80)(36 79 78 37)(41 83 82 42)(43 81 84 48)(44 47 85 88)(45 87 86 46)(89 135 134 90)(91 133 136 96)(92 95 129 132)(93 131 130 94)(97 143 142 98)(99 141 144 104)(100 103 137 140)(101 139 138 102)(105 151 146 110)(106 109 147 150)(107 149 148 108)(111 145 152 112)(113 159 154 118)(114 117 155 158)(115 157 156 116)(119 153 160 120)

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

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

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

110 conjugacy classes

class 1 2A2B2C2D2E2F2G4A4B4C4D4E···4J5A5B5C5D8A8B8C8D10A···10L10M···10T10U···10AB20A···20P20Q···20AN40A···40P
order1222222244444···45555888810···1010···1010···1020···2020···2040···40
size1111224422224···4111144441···12···24···42···24···44···4

110 irreducible representations

dim1111111111111122224444
type+++++++++-
imageC1C2C2C2C2C2C4C5C10C10C10C10C10C20D4D4C5×D4C5×D4C8⋊C22C8.C22C5×C8⋊C22C5×C8.C22
kernelC5×C23.36D4C5×D4⋊C4C5×Q8⋊C4C10×C4⋊C4C10×M4(2)C10×C4○D4C5×C4○D4C23.36D4D4⋊C4Q8⋊C4C2×C4⋊C4C2×M4(2)C2×C4○D4C4○D4C2×C20C22×C10C2×C4C23C10C10C2C2
# reps122111848844432311241144

Matrix representation of C5×C23.36D4 in GL6(𝔽41)

100000
010000
0016000
0001600
0000160
0000016
,
4000000
0400000
000010
000001
001000
000100
,
4000000
0400000
0040000
0004000
0000400
0000040
,
100000
010000
0040000
0004000
0000400
0000040
,
24390000
21170000
0023944
0022374
003737392
004373939
,
1720000
19240000
003923737
0022374
003737392
0037422

G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,1,0,0],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[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],[24,21,0,0,0,0,39,17,0,0,0,0,0,0,2,2,37,4,0,0,39,2,37,37,0,0,4,37,39,39,0,0,4,4,2,39],[17,19,0,0,0,0,2,24,0,0,0,0,0,0,39,2,37,37,0,0,2,2,37,4,0,0,37,37,39,2,0,0,37,4,2,2] >;

C5×C23.36D4 in GAP, Magma, Sage, TeX

C_5\times C_2^3._{36}D_4
% in TeX

G:=Group("C5xC2^3.36D4");
// GroupNames label

G:=SmallGroup(320,918);
// by ID

G=gap.SmallGroup(320,918);
# by ID

G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-2,560,589,1731,7004,3511,172]);
// Polycyclic

G:=Group<a,b,c,d,e,f|a^5=b^2=c^2=d^2=1,e^4=d,f^2=c,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,b*c=c*b,e*b*e^-1=b*d=d*b,b*f=f*b,c*d=d*c,c*e=e*c,c*f=f*c,d*e=e*d,d*f=f*d,f*e*f^-1=c*d*e^3>;
// generators/relations

׿
×
𝔽