direct product, metabelian, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: C22×C5⋊C8, C23.4F5, Dic5.14C23, (C2×C10)⋊2C8, C10⋊2(C2×C8), C5⋊2(C22×C8), C2.3(C22×F5), (C22×C10).5C4, C22.19(C2×F5), C10.11(C22×C4), (C2×Dic5).13C4, Dic5.18(C2×C4), (C22×Dic5).8C2, (C2×Dic5).57C22, (C2×C10).19(C2×C4), SmallGroup(160,210)
Series: Derived ►Chief ►Lower central ►Upper central
| C5 — C22×C5⋊C8 |
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
►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⋊C8⋊8D4 C5⋊C8⋊D4 C20⋊C8⋊C2 C5⋊C8⋊7D4 (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