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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, 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, C2×D4, C2×D4, C2×D4, C2×Q8, C20, C20, C2×C10, C2×C10, C8⋊C4, C4.4D4, C41D4, C2×D8, C2×SD16, C40, C2×C20, C2×C20, C2×C20, C5×D4, C5×Q8, C22×C10, C83D4, C4×C20, C5×C22⋊C4, C2×C40, C5×D8, C5×SD16, D4×C10, D4×C10, D4×C10, Q8×C10, C5×C8⋊C4, C5×C4.4D4, C5×C41D4, C10×D8, C10×SD16, C5×C83D4
Quotients: C1, C2, C22, C5, D4, C23, C10, C2×D4, C2×C10, C41D4, C8⋊C22, C5×D4, C22×C10, C83D4, D4×C10, C5×C41D4, C5×C8⋊C22, C5×C83D4

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