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## G = C5×C8.12D4order 320 = 26·5

### Direct product of C5 and C8.12D4

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C4 — C5×C8.12D4
 Chief series C1 — C2 — C22 — C2×C4 — C2×C20 — D4×C10 — C10×SD16 — C5×C8.12D4
 Lower central C1 — C2 — C2×C4 — C5×C8.12D4
 Upper central C1 — C2×C10 — C4×C20 — C5×C8.12D4

Generators and relations for C5×C8.12D4
G = < a,b,c,d | a5=b8=c4=1, d2=b4, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b3, dcd-1=b4c-1 >

Subgroups: 258 in 130 conjugacy classes, 58 normal (26 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C10, C10, C10, C42, C22⋊C4, C2×C8, D8, SD16, Q16, C2×D4, C2×Q8, C20, C20, C2×C10, C2×C10, C4×C8, C4.4D4, C2×D8, C2×SD16, C2×Q16, C40, C2×C20, C2×C20, C2×C20, C5×D4, C5×Q8, C22×C10, C8.12D4, C4×C20, C5×C22⋊C4, C2×C40, C5×D8, C5×SD16, C5×Q16, D4×C10, Q8×C10, C4×C40, C5×C4.4D4, C10×D8, C10×SD16, C10×Q16, C5×C8.12D4
Quotients: C1, C2, C22, C5, D4, C23, C10, C2×D4, C2×C10, C41D4, C4○D8, C5×D4, C22×C10, C8.12D4, D4×C10, C5×C41D4, C5×C4○D8, C5×C8.12D4

Smallest permutation representation of C5×C8.12D4
On 160 points
Generators in S160
(1 143 61 135 53)(2 144 62 136 54)(3 137 63 129 55)(4 138 64 130 56)(5 139 57 131 49)(6 140 58 132 50)(7 141 59 133 51)(8 142 60 134 52)(9 37 20 94 103)(10 38 21 95 104)(11 39 22 96 97)(12 40 23 89 98)(13 33 24 90 99)(14 34 17 91 100)(15 35 18 92 101)(16 36 19 93 102)(25 41 115 124 107)(26 42 116 125 108)(27 43 117 126 109)(28 44 118 127 110)(29 45 119 128 111)(30 46 120 121 112)(31 47 113 122 105)(32 48 114 123 106)(65 83 155 73 147)(66 84 156 74 148)(67 85 157 75 149)(68 86 158 76 150)(69 87 159 77 151)(70 88 160 78 152)(71 81 153 79 145)(72 82 154 80 146)
(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 32 13 69)(2 25 14 70)(3 26 15 71)(4 27 16 72)(5 28 9 65)(6 29 10 66)(7 30 11 67)(8 31 12 68)(17 160 62 115)(18 153 63 116)(19 154 64 117)(20 155 57 118)(21 156 58 119)(22 157 59 120)(23 158 60 113)(24 159 61 114)(33 87 143 48)(34 88 144 41)(35 81 137 42)(36 82 138 43)(37 83 139 44)(38 84 140 45)(39 85 141 46)(40 86 142 47)(49 110 103 147)(50 111 104 148)(51 112 97 149)(52 105 98 150)(53 106 99 151)(54 107 100 152)(55 108 101 145)(56 109 102 146)(73 131 127 94)(74 132 128 95)(75 133 121 96)(76 134 122 89)(77 135 123 90)(78 136 124 91)(79 129 125 92)(80 130 126 93)
(1 8 5 4)(2 3 6 7)(9 16 13 12)(10 11 14 15)(17 18 21 22)(19 24 23 20)(25 67 29 71)(26 70 30 66)(27 65 31 69)(28 68 32 72)(33 40 37 36)(34 35 38 39)(41 85 45 81)(42 88 46 84)(43 83 47 87)(44 86 48 82)(49 56 53 52)(50 51 54 55)(57 64 61 60)(58 59 62 63)(73 122 77 126)(74 125 78 121)(75 128 79 124)(76 123 80 127)(89 94 93 90)(91 92 95 96)(97 100 101 104)(98 103 102 99)(105 151 109 147)(106 146 110 150)(107 149 111 145)(108 152 112 148)(113 159 117 155)(114 154 118 158)(115 157 119 153)(116 160 120 156)(129 132 133 136)(130 135 134 131)(137 140 141 144)(138 143 142 139)

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

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

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

110 conjugacy classes

 class 1 2A 2B 2C 2D 2E 4A ··· 4F 4G 4H 5A 5B 5C 5D 8A ··· 8H 10A ··· 10L 10M ··· 10T 20A ··· 20X 20Y ··· 20AF 40A ··· 40AF order 1 2 2 2 2 2 4 ··· 4 4 4 5 5 5 5 8 ··· 8 10 ··· 10 10 ··· 10 20 ··· 20 20 ··· 20 40 ··· 40 size 1 1 1 1 8 8 2 ··· 2 8 8 1 1 1 1 2 ··· 2 1 ··· 1 8 ··· 8 2 ··· 2 8 ··· 8 2 ··· 2

110 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 type + + + + + + + + image C1 C2 C2 C2 C2 C2 C5 C10 C10 C10 C10 C10 D4 D4 C4○D8 C5×D4 C5×D4 C5×C4○D8 kernel C5×C8.12D4 C4×C40 C5×C4.4D4 C10×D8 C10×SD16 C10×Q16 C8.12D4 C4×C8 C4.4D4 C2×D8 C2×SD16 C2×Q16 C40 C2×C20 C10 C8 C2×C4 C2 # reps 1 1 2 1 2 1 4 4 8 4 8 4 4 2 8 16 8 32

Matrix representation of C5×C8.12D4 in GL4(𝔽41) generated by

 18 0 0 0 0 18 0 0 0 0 37 0 0 0 0 37
,
 26 15 0 0 26 26 0 0 0 0 1 0 0 0 0 1
,
 9 0 0 0 0 9 0 0 0 0 32 2 0 0 0 9
,
 15 26 0 0 26 26 0 0 0 0 1 0 0 0 9 40
G:=sub<GL(4,GF(41))| [18,0,0,0,0,18,0,0,0,0,37,0,0,0,0,37],[26,26,0,0,15,26,0,0,0,0,1,0,0,0,0,1],[9,0,0,0,0,9,0,0,0,0,32,0,0,0,2,9],[15,26,0,0,26,26,0,0,0,0,1,9,0,0,0,40] >;

C5×C8.12D4 in GAP, Magma, Sage, TeX

C_5\times C_8._{12}D_4
% in TeX

G:=Group("C5xC8.12D4");
// GroupNames label

G:=SmallGroup(320,996);
// by ID

G=gap.SmallGroup(320,996);
# by ID

G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-2,589,288,1766,1276,7004,172]);
// Polycyclic

G:=Group<a,b,c,d|a^5=b^8=c^4=1,d^2=b^4,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b^3,d*c*d^-1=b^4*c^-1>;
// generators/relations

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