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## G = C5×C8⋊3D4order 320 = 26·5

### Direct product of C5 and C8⋊3D4

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C4 — C5×C8⋊3D4
 Chief series C1 — C2 — C22 — C2×C4 — C2×C20 — D4×C10 — C10×SD16 — C5×C8⋊3D4
 Lower central C1 — C2 — C2×C4 — C5×C8⋊3D4
 Upper central C1 — C2×C10 — C4×C20 — C5×C8⋊3D4

Generators and relations for C5×C83D4
G = < a,b,c,d | a5=b8=c4=d2=1, ab=ba, ac=ca, ad=da, cbc-1=b5, dbd=b-1, dcd=c-1 >

Subgroups: 322 in 144 conjugacy classes, 58 normal (22 characteristic)
C1, C2, C2 [×2], C2 [×3], C4 [×2], C4 [×3], C22, C22 [×9], C5, C8 [×4], C2×C4, C2×C4 [×2], C2×C4, D4 [×10], Q8 [×2], C23 [×3], C10, C10 [×2], C10 [×3], C42, C22⋊C4 [×2], C2×C8 [×2], D8 [×4], SD16 [×4], C2×D4, C2×D4 [×2], C2×D4 [×2], C2×Q8, C20 [×2], C20 [×3], C2×C10, C2×C10 [×9], C8⋊C4, C4.4D4, C41D4, C2×D8 [×2], C2×SD16 [×2], C40 [×4], C2×C20, C2×C20 [×2], C2×C20, C5×D4 [×10], C5×Q8 [×2], C22×C10 [×3], C83D4, C4×C20, C5×C22⋊C4 [×2], C2×C40 [×2], C5×D8 [×4], C5×SD16 [×4], D4×C10, D4×C10 [×2], D4×C10 [×2], Q8×C10, C5×C8⋊C4, C5×C4.4D4, C5×C41D4, C10×D8 [×2], C10×SD16 [×2], C5×C83D4
Quotients: C1, C2 [×7], C22 [×7], C5, D4 [×6], C23, C10 [×7], C2×D4 [×3], C2×C10 [×7], C41D4, C8⋊C22 [×2], C5×D4 [×6], C22×C10, C83D4, D4×C10 [×3], C5×C41D4, C5×C8⋊C22 [×2], C5×C83D4

Smallest permutation representation of C5×C83D4
On 160 points
Generators in S160
(1 39 111 31 103)(2 40 112 32 104)(3 33 105 25 97)(4 34 106 26 98)(5 35 107 27 99)(6 36 108 28 100)(7 37 109 29 101)(8 38 110 30 102)(9 22 46 118 93)(10 23 47 119 94)(11 24 48 120 95)(12 17 41 113 96)(13 18 42 114 89)(14 19 43 115 90)(15 20 44 116 91)(16 21 45 117 92)(49 121 137 57 129)(50 122 138 58 130)(51 123 139 59 131)(52 124 140 60 132)(53 125 141 61 133)(54 126 142 62 134)(55 127 143 63 135)(56 128 144 64 136)(65 87 153 73 145)(66 88 154 74 146)(67 81 155 75 147)(68 82 156 76 148)(69 83 157 77 149)(70 84 158 78 150)(71 85 159 79 151)(72 86 160 80 152)
(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 71 127 119)(2 68 128 116)(3 65 121 113)(4 70 122 118)(5 67 123 115)(6 72 124 120)(7 69 125 117)(8 66 126 114)(9 106 158 58)(10 111 159 63)(11 108 160 60)(12 105 153 57)(13 110 154 62)(14 107 155 59)(15 112 156 64)(16 109 157 61)(17 25 73 129)(18 30 74 134)(19 27 75 131)(20 32 76 136)(21 29 77 133)(22 26 78 130)(23 31 79 135)(24 28 80 132)(33 87 137 96)(34 84 138 93)(35 81 139 90)(36 86 140 95)(37 83 141 92)(38 88 142 89)(39 85 143 94)(40 82 144 91)(41 97 145 49)(42 102 146 54)(43 99 147 51)(44 104 148 56)(45 101 149 53)(46 98 150 50)(47 103 151 55)(48 100 152 52)
(1 122)(2 121)(3 128)(4 127)(5 126)(6 125)(7 124)(8 123)(9 10)(11 16)(12 15)(13 14)(17 20)(18 19)(21 24)(22 23)(25 136)(26 135)(27 134)(28 133)(29 132)(30 131)(31 130)(32 129)(33 144)(34 143)(35 142)(36 141)(37 140)(38 139)(39 138)(40 137)(41 44)(42 43)(45 48)(46 47)(49 104)(50 103)(51 102)(52 101)(53 100)(54 99)(55 98)(56 97)(57 112)(58 111)(59 110)(60 109)(61 108)(62 107)(63 106)(64 105)(65 68)(66 67)(69 72)(70 71)(73 76)(74 75)(77 80)(78 79)(81 88)(82 87)(83 86)(84 85)(89 90)(91 96)(92 95)(93 94)(113 116)(114 115)(117 120)(118 119)(145 148)(146 147)(149 152)(150 151)(153 156)(154 155)(157 160)(158 159)

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

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

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

80 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 4A 4B 4C 4D 4E 5A 5B 5C 5D 8A 8B 8C 8D 10A ··· 10L 10M ··· 10X 20A ··· 20H 20I ··· 20P 20Q 20R 20S 20T 40A ··· 40P order 1 2 2 2 2 2 2 4 4 4 4 4 5 5 5 5 8 8 8 8 10 ··· 10 10 ··· 10 20 ··· 20 20 ··· 20 20 20 20 20 40 ··· 40 size 1 1 1 1 8 8 8 2 2 4 4 8 1 1 1 1 4 4 4 4 1 ··· 1 8 ··· 8 2 ··· 2 4 ··· 4 8 8 8 8 4 ··· 4

80 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 4 4 type + + + + + + + + + image C1 C2 C2 C2 C2 C2 C5 C10 C10 C10 C10 C10 D4 D4 C5×D4 C5×D4 C8⋊C22 C5×C8⋊C22 kernel C5×C8⋊3D4 C5×C8⋊C4 C5×C4.4D4 C5×C4⋊1D4 C10×D8 C10×SD16 C8⋊3D4 C8⋊C4 C4.4D4 C4⋊1D4 C2×D8 C2×SD16 C40 C2×C20 C8 C2×C4 C10 C2 # reps 1 1 1 1 2 2 4 4 4 4 8 8 4 2 16 8 2 8

Matrix representation of C5×C83D4 in GL6(𝔽41)

 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
,
 40 0 0 0 0 0 0 40 0 0 0 0 0 0 19 38 0 19 0 0 22 19 19 22 0 0 3 38 25 16 0 0 19 19 3 19
,
 22 40 0 0 0 0 34 19 0 0 0 0 0 0 0 0 1 0 0 0 1 40 1 39 0 0 1 0 0 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 3 40 0 0 0 0 0 0 22 3 0 22 0 0 22 19 19 22 0 0 3 38 25 16 0 0 3 0 3 16

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],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,19,22,3,19,0,0,38,19,38,19,0,0,0,19,25,3,0,0,19,22,16,19],[22,34,0,0,0,0,40,19,0,0,0,0,0,0,0,1,1,0,0,0,0,40,0,0,0,0,1,1,0,0,0,0,0,39,0,1],[1,3,0,0,0,0,0,40,0,0,0,0,0,0,22,22,3,3,0,0,3,19,38,0,0,0,0,19,25,3,0,0,22,22,16,16] >;

C5×C83D4 in GAP, Magma, Sage, TeX

C_5\times C_8\rtimes_3D_4
% in TeX

G:=Group("C5xC8:3D4");
// GroupNames label

G:=SmallGroup(320,997);
// by ID

G=gap.SmallGroup(320,997);
# by ID

G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-2,589,288,1766,1731,436,7004,172]);
// Polycyclic

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

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