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## G = C22×C5⋊C8order 160 = 25·5

### Direct product of C22 and C5⋊C8

Series: Derived Chief Lower central Upper central

 Derived series C1 — C5 — C22×C5⋊C8
 Chief series C1 — C5 — C10 — Dic5 — C5⋊C8 — C2×C5⋊C8 — C22×C5⋊C8
 Lower central C5 — C22×C5⋊C8
 Upper central C1 — C23

Generators and relations for C22×C5⋊C8
G = < a,b,c,d | a2=b2=c5=d8=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c3 >

Subgroups: 164 in 76 conjugacy classes, 54 normal (9 characteristic)
C1, C2, C2 [×6], C4 [×4], C22 [×7], C5, C8 [×4], C2×C4 [×6], C23, C10, C10 [×6], C2×C8 [×6], C22×C4, Dic5, Dic5 [×3], C2×C10 [×7], C22×C8, C5⋊C8 [×4], C2×Dic5 [×6], C22×C10, C2×C5⋊C8 [×6], C22×Dic5, C22×C5⋊C8
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C8 [×4], C2×C4 [×6], C23, C2×C8 [×6], C22×C4, F5, C22×C8, C5⋊C8 [×4], C2×F5 [×3], C2×C5⋊C8 [×6], C22×F5, C22×C5⋊C8

Smallest permutation representation of C22×C5⋊C8
Regular action on 160 points
Generators in S160
(1 104)(2 97)(3 98)(4 99)(5 100)(6 101)(7 102)(8 103)(9 27)(10 28)(11 29)(12 30)(13 31)(14 32)(15 25)(16 26)(17 144)(18 137)(19 138)(20 139)(21 140)(22 141)(23 142)(24 143)(33 105)(34 106)(35 107)(36 108)(37 109)(38 110)(39 111)(40 112)(41 125)(42 126)(43 127)(44 128)(45 121)(46 122)(47 123)(48 124)(49 73)(50 74)(51 75)(52 76)(53 77)(54 78)(55 79)(56 80)(57 136)(58 129)(59 130)(60 131)(61 132)(62 133)(63 134)(64 135)(65 149)(66 150)(67 151)(68 152)(69 145)(70 146)(71 147)(72 148)(81 153)(82 154)(83 155)(84 156)(85 157)(86 158)(87 159)(88 160)(89 114)(90 115)(91 116)(92 117)(93 118)(94 119)(95 120)(96 113)
(1 48)(2 41)(3 42)(4 43)(5 44)(6 45)(7 46)(8 47)(9 67)(10 68)(11 69)(12 70)(13 71)(14 72)(15 65)(16 66)(17 154)(18 155)(19 156)(20 157)(21 158)(22 159)(23 160)(24 153)(25 149)(26 150)(27 151)(28 152)(29 145)(30 146)(31 147)(32 148)(33 49)(34 50)(35 51)(36 52)(37 53)(38 54)(39 55)(40 56)(57 96)(58 89)(59 90)(60 91)(61 92)(62 93)(63 94)(64 95)(73 105)(74 106)(75 107)(76 108)(77 109)(78 110)(79 111)(80 112)(81 143)(82 144)(83 137)(84 138)(85 139)(86 140)(87 141)(88 142)(97 125)(98 126)(99 127)(100 128)(101 121)(102 122)(103 123)(104 124)(113 136)(114 129)(115 130)(116 131)(117 132)(118 133)(119 134)(120 135)
(1 50 81 151 59)(2 152 51 60 82)(3 61 145 83 52)(4 84 62 53 146)(5 54 85 147 63)(6 148 55 64 86)(7 57 149 87 56)(8 88 58 49 150)(9 115 124 106 24)(10 107 116 17 125)(11 18 108 126 117)(12 127 19 118 109)(13 119 128 110 20)(14 111 120 21 121)(15 22 112 122 113)(16 123 23 114 105)(25 141 40 46 96)(26 47 142 89 33)(27 90 48 34 143)(28 35 91 144 41)(29 137 36 42 92)(30 43 138 93 37)(31 94 44 38 139)(32 39 95 140 45)(65 159 80 102 136)(66 103 160 129 73)(67 130 104 74 153)(68 75 131 154 97)(69 155 76 98 132)(70 99 156 133 77)(71 134 100 78 157)(72 79 135 158 101)
(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)

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

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

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

C22×C5⋊C8 is a maximal subgroup of   C10.(C4⋊C8)  Dic5.C42  C5⋊C88D4  C5⋊C8⋊D4  C20⋊C8⋊C2  C5⋊C87D4  (C2×D4).7F5
C22×C5⋊C8 is a maximal quotient of   Dic5.12M4(2)  C5⋊C16.C22

40 conjugacy classes

 class 1 2A ··· 2G 4A ··· 4H 5 8A ··· 8P 10A ··· 10G order 1 2 ··· 2 4 ··· 4 5 8 ··· 8 10 ··· 10 size 1 1 ··· 1 5 ··· 5 4 5 ··· 5 4 ··· 4

40 irreducible representations

 dim 1 1 1 1 1 1 4 4 4 type + + + + - + image C1 C2 C2 C4 C4 C8 F5 C5⋊C8 C2×F5 kernel C22×C5⋊C8 C2×C5⋊C8 C22×Dic5 C2×Dic5 C22×C10 C2×C10 C23 C22 C22 # reps 1 6 1 6 2 16 1 4 3

Matrix representation of C22×C5⋊C8 in GL6(𝔽41)

 40 0 0 0 0 0 0 40 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 40 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
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 40 0 0 1 0 0 40 0 0 0 1 0 40 0 0 0 0 1 40
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 30 24 6 39 0 0 36 22 30 28 0 0 19 11 13 34 0 0 2 17 11 17

G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[40,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,40,40,40,40],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,30,36,19,2,0,0,24,22,11,17,0,0,6,30,13,11,0,0,39,28,34,17] >;

C22×C5⋊C8 in GAP, Magma, Sage, TeX

C_2^2\times C_5\rtimes C_8
% in TeX

G:=Group("C2^2xC5:C8");
// GroupNames label

G:=SmallGroup(160,210);
// by ID

G=gap.SmallGroup(160,210);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-5,48,69,2309,599]);
// Polycyclic

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

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