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G = C4×C8⋊D5order 320 = 26·5

Direct product of C4 and C8⋊D5

Series: Derived Chief Lower central Upper central

 Derived series C1 — C10 — C4×C8⋊D5
 Chief series C1 — C5 — C10 — C20 — C2×C20 — C2×C4×D5 — D5×C42 — C4×C8⋊D5
 Lower central C5 — C10 — C4×C8⋊D5
 Upper central C1 — C42 — C4×C8

Generators and relations for C4×C8⋊D5
G = < a,b,c,d | a4=b8=c5=d2=1, ab=ba, ac=ca, ad=da, bc=cb, dbd=b5, dcd=c-1 >

Subgroups: 398 in 142 conjugacy classes, 83 normal (19 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C5, C8, C8, C2×C4, C2×C4, C2×C4, C23, D5, C10, C10, C42, C42, C2×C8, C2×C8, M4(2), C22×C4, Dic5, Dic5, C20, D10, D10, C2×C10, C4×C8, C4×C8, C8⋊C4, C2×C42, C2×M4(2), C52C8, C40, C4×D5, C4×D5, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C22×D5, C4×M4(2), C8⋊D5, C2×C52C8, C4×Dic5, C4×Dic5, C4×C20, C2×C40, C2×C4×D5, C2×C4×D5, C4×C52C8, C408C4, C4×C40, D5×C42, C2×C8⋊D5, C4×C8⋊D5
Quotients: C1, C2, C4, C22, C2×C4, C23, D5, C42, M4(2), C22×C4, D10, C2×C42, C2×M4(2), C4×D5, C22×D5, C4×M4(2), C8⋊D5, C2×C4×D5, D5×C42, C2×C8⋊D5, C4×C8⋊D5

Smallest permutation representation of C4×C8⋊D5
On 160 points
Generators in S160
(1 46 121 94)(2 47 122 95)(3 48 123 96)(4 41 124 89)(5 42 125 90)(6 43 126 91)(7 44 127 92)(8 45 128 93)(9 63 86 142)(10 64 87 143)(11 57 88 144)(12 58 81 137)(13 59 82 138)(14 60 83 139)(15 61 84 140)(16 62 85 141)(17 148 116 36)(18 149 117 37)(19 150 118 38)(20 151 119 39)(21 152 120 40)(22 145 113 33)(23 146 114 34)(24 147 115 35)(25 111 132 52)(26 112 133 53)(27 105 134 54)(28 106 135 55)(29 107 136 56)(30 108 129 49)(31 109 130 50)(32 110 131 51)(65 99 153 73)(66 100 154 74)(67 101 155 75)(68 102 156 76)(69 103 157 77)(70 104 158 78)(71 97 159 79)(72 98 160 80)
(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 35 81 107 73)(2 36 82 108 74)(3 37 83 109 75)(4 38 84 110 76)(5 39 85 111 77)(6 40 86 112 78)(7 33 87 105 79)(8 34 88 106 80)(9 53 104 126 152)(10 54 97 127 145)(11 55 98 128 146)(12 56 99 121 147)(13 49 100 122 148)(14 50 101 123 149)(15 51 102 124 150)(16 52 103 125 151)(17 138 129 66 47)(18 139 130 67 48)(19 140 131 68 41)(20 141 132 69 42)(21 142 133 70 43)(22 143 134 71 44)(23 144 135 72 45)(24 137 136 65 46)(25 157 90 119 62)(26 158 91 120 63)(27 159 92 113 64)(28 160 93 114 57)(29 153 94 115 58)(30 154 95 116 59)(31 155 96 117 60)(32 156 89 118 61)
(1 99)(2 104)(3 101)(4 98)(5 103)(6 100)(7 97)(8 102)(9 82)(10 87)(11 84)(12 81)(13 86)(14 83)(15 88)(16 85)(17 26)(18 31)(19 28)(20 25)(21 30)(22 27)(23 32)(24 29)(33 54)(34 51)(35 56)(36 53)(37 50)(38 55)(39 52)(40 49)(41 160)(42 157)(43 154)(44 159)(45 156)(46 153)(47 158)(48 155)(57 140)(58 137)(59 142)(60 139)(61 144)(62 141)(63 138)(64 143)(65 94)(66 91)(67 96)(68 93)(69 90)(70 95)(71 92)(72 89)(73 121)(74 126)(75 123)(76 128)(77 125)(78 122)(79 127)(80 124)(105 145)(106 150)(107 147)(108 152)(109 149)(110 146)(111 151)(112 148)(113 134)(114 131)(115 136)(116 133)(117 130)(118 135)(119 132)(120 129)

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

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

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

104 conjugacy classes

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

104 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 type + + + + + + + + + image C1 C2 C2 C2 C2 C2 C4 C4 C4 D5 M4(2) D10 D10 C4×D5 C4×D5 C8⋊D5 kernel C4×C8⋊D5 C4×C5⋊2C8 C40⋊8C4 C4×C40 D5×C42 C2×C8⋊D5 C8⋊D5 C4×Dic5 C2×C4×D5 C4×C8 C20 C42 C2×C8 C8 C2×C4 C4 # reps 1 1 2 1 1 2 16 4 4 2 8 2 4 16 8 32

Matrix representation of C4×C8⋊D5 in GL4(𝔽41) generated by

 9 0 0 0 0 9 0 0 0 0 9 0 0 0 0 9
,
 1 0 0 0 0 1 0 0 0 0 10 18 0 0 20 31
,
 6 40 0 0 1 0 0 0 0 0 7 40 0 0 8 40
,
 35 1 0 0 6 6 0 0 0 0 0 35 0 0 34 0
G:=sub<GL(4,GF(41))| [9,0,0,0,0,9,0,0,0,0,9,0,0,0,0,9],[1,0,0,0,0,1,0,0,0,0,10,20,0,0,18,31],[6,1,0,0,40,0,0,0,0,0,7,8,0,0,40,40],[35,6,0,0,1,6,0,0,0,0,0,34,0,0,35,0] >;

C4×C8⋊D5 in GAP, Magma, Sage, TeX

C_4\times C_8\rtimes D_5
% in TeX

G:=Group("C4xC8:D5");
// GroupNames label

G:=SmallGroup(320,314);
// by ID

G=gap.SmallGroup(320,314);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,120,758,58,136,12550]);
// Polycyclic

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

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