direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C10×C22⋊C8, C23⋊2C40, C24.3C20, (C22×C8)⋊1C10, (C22×C10)⋊6C8, C22⋊2(C2×C40), (C22×C40)⋊5C2, C4.69(D4×C10), (C2×C40)⋊42C22, (C2×C20).535D4, C20.474(C2×D4), C2.1(C22×C40), (C23×C20).8C2, (C23×C4).5C10, (C23×C10).11C4, (C22×C20).47C4, C10.54(C22×C8), (C22×C4).11C20, C23.28(C2×C20), C2.3(C10×M4(2)), (C2×C20).981C23, C10.81(C2×M4(2)), (C2×C10).48M4(2), C20.161(C22⋊C4), C22.9(C5×M4(2)), C22.19(C22×C20), (C22×C20).495C22, (C2×C8)⋊10(C2×C10), (C2×C10)⋊13(C2×C8), (C2×C4).59(C2×C20), (C2×C4).145(C5×D4), C2.3(C10×C22⋊C4), C4.31(C5×C22⋊C4), (C2×C20).461(C2×C4), C10.138(C2×C22⋊C4), C22.32(C5×C22⋊C4), (C2×C4).149(C22×C10), (C2×C10).332(C22×C4), (C22×C10).182(C2×C4), (C22×C4).135(C2×C10), (C2×C10).198(C22⋊C4), SmallGroup(320,907)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C10×C22⋊C8
G = < a,b,c,d | a10=b2=c2=d8=1, ab=ba, ac=ca, ad=da, dbd-1=bc=cb, cd=dc >
Subgroups: 290 in 202 conjugacy classes, 114 normal (26 characteristic)
C1, C2 [×3], C2 [×4], C2 [×4], C4 [×4], C4 [×2], C22, C22 [×10], C22 [×12], C5, C8 [×4], C2×C4 [×2], C2×C4 [×6], C2×C4 [×10], C23, C23 [×6], C23 [×4], C10 [×3], C10 [×4], C10 [×4], C2×C8 [×4], C2×C8 [×4], C22×C4 [×2], C22×C4 [×4], C22×C4 [×4], C24, C20 [×4], C20 [×2], C2×C10, C2×C10 [×10], C2×C10 [×12], C22⋊C8 [×4], C22×C8 [×2], C23×C4, C40 [×4], C2×C20 [×2], C2×C20 [×6], C2×C20 [×10], C22×C10, C22×C10 [×6], C22×C10 [×4], C2×C22⋊C8, C2×C40 [×4], C2×C40 [×4], C22×C20 [×2], C22×C20 [×4], C22×C20 [×4], C23×C10, C5×C22⋊C8 [×4], C22×C40 [×2], C23×C20, C10×C22⋊C8
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C5, C8 [×4], C2×C4 [×6], D4 [×4], C23, C10 [×7], C22⋊C4 [×4], C2×C8 [×6], M4(2) [×2], C22×C4, C2×D4 [×2], C20 [×4], C2×C10 [×7], C22⋊C8 [×4], C2×C22⋊C4, C22×C8, C2×M4(2), C40 [×4], C2×C20 [×6], C5×D4 [×4], C22×C10, C2×C22⋊C8, C5×C22⋊C4 [×4], C2×C40 [×6], C5×M4(2) [×2], C22×C20, D4×C10 [×2], C5×C22⋊C8 [×4], C10×C22⋊C4, C22×C40, C10×M4(2), C10×C22⋊C8
(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)
(11 70)(12 61)(13 62)(14 63)(15 64)(16 65)(17 66)(18 67)(19 68)(20 69)(21 152)(22 153)(23 154)(24 155)(25 156)(26 157)(27 158)(28 159)(29 160)(30 151)(31 58)(32 59)(33 60)(34 51)(35 52)(36 53)(37 54)(38 55)(39 56)(40 57)(41 77)(42 78)(43 79)(44 80)(45 71)(46 72)(47 73)(48 74)(49 75)(50 76)
(1 104)(2 105)(3 106)(4 107)(5 108)(6 109)(7 110)(8 101)(9 102)(10 103)(11 70)(12 61)(13 62)(14 63)(15 64)(16 65)(17 66)(18 67)(19 68)(20 69)(21 152)(22 153)(23 154)(24 155)(25 156)(26 157)(27 158)(28 159)(29 160)(30 151)(31 58)(32 59)(33 60)(34 51)(35 52)(36 53)(37 54)(38 55)(39 56)(40 57)(41 77)(42 78)(43 79)(44 80)(45 71)(46 72)(47 73)(48 74)(49 75)(50 76)(81 93)(82 94)(83 95)(84 96)(85 97)(86 98)(87 99)(88 100)(89 91)(90 92)(111 139)(112 140)(113 131)(114 132)(115 133)(116 134)(117 135)(118 136)(119 137)(120 138)(121 148)(122 149)(123 150)(124 141)(125 142)(126 143)(127 144)(128 145)(129 146)(130 147)
(1 11 125 56 111 71 94 26)(2 12 126 57 112 72 95 27)(3 13 127 58 113 73 96 28)(4 14 128 59 114 74 97 29)(5 15 129 60 115 75 98 30)(6 16 130 51 116 76 99 21)(7 17 121 52 117 77 100 22)(8 18 122 53 118 78 91 23)(9 19 123 54 119 79 92 24)(10 20 124 55 120 80 93 25)(31 131 47 84 159 106 62 144)(32 132 48 85 160 107 63 145)(33 133 49 86 151 108 64 146)(34 134 50 87 152 109 65 147)(35 135 41 88 153 110 66 148)(36 136 42 89 154 101 67 149)(37 137 43 90 155 102 68 150)(38 138 44 81 156 103 69 141)(39 139 45 82 157 104 70 142)(40 140 46 83 158 105 61 143)
G:=sub<Sym(160)| (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), (11,70)(12,61)(13,62)(14,63)(15,64)(16,65)(17,66)(18,67)(19,68)(20,69)(21,152)(22,153)(23,154)(24,155)(25,156)(26,157)(27,158)(28,159)(29,160)(30,151)(31,58)(32,59)(33,60)(34,51)(35,52)(36,53)(37,54)(38,55)(39,56)(40,57)(41,77)(42,78)(43,79)(44,80)(45,71)(46,72)(47,73)(48,74)(49,75)(50,76), (1,104)(2,105)(3,106)(4,107)(5,108)(6,109)(7,110)(8,101)(9,102)(10,103)(11,70)(12,61)(13,62)(14,63)(15,64)(16,65)(17,66)(18,67)(19,68)(20,69)(21,152)(22,153)(23,154)(24,155)(25,156)(26,157)(27,158)(28,159)(29,160)(30,151)(31,58)(32,59)(33,60)(34,51)(35,52)(36,53)(37,54)(38,55)(39,56)(40,57)(41,77)(42,78)(43,79)(44,80)(45,71)(46,72)(47,73)(48,74)(49,75)(50,76)(81,93)(82,94)(83,95)(84,96)(85,97)(86,98)(87,99)(88,100)(89,91)(90,92)(111,139)(112,140)(113,131)(114,132)(115,133)(116,134)(117,135)(118,136)(119,137)(120,138)(121,148)(122,149)(123,150)(124,141)(125,142)(126,143)(127,144)(128,145)(129,146)(130,147), (1,11,125,56,111,71,94,26)(2,12,126,57,112,72,95,27)(3,13,127,58,113,73,96,28)(4,14,128,59,114,74,97,29)(5,15,129,60,115,75,98,30)(6,16,130,51,116,76,99,21)(7,17,121,52,117,77,100,22)(8,18,122,53,118,78,91,23)(9,19,123,54,119,79,92,24)(10,20,124,55,120,80,93,25)(31,131,47,84,159,106,62,144)(32,132,48,85,160,107,63,145)(33,133,49,86,151,108,64,146)(34,134,50,87,152,109,65,147)(35,135,41,88,153,110,66,148)(36,136,42,89,154,101,67,149)(37,137,43,90,155,102,68,150)(38,138,44,81,156,103,69,141)(39,139,45,82,157,104,70,142)(40,140,46,83,158,105,61,143)>;
G:=Group( (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), (11,70)(12,61)(13,62)(14,63)(15,64)(16,65)(17,66)(18,67)(19,68)(20,69)(21,152)(22,153)(23,154)(24,155)(25,156)(26,157)(27,158)(28,159)(29,160)(30,151)(31,58)(32,59)(33,60)(34,51)(35,52)(36,53)(37,54)(38,55)(39,56)(40,57)(41,77)(42,78)(43,79)(44,80)(45,71)(46,72)(47,73)(48,74)(49,75)(50,76), (1,104)(2,105)(3,106)(4,107)(5,108)(6,109)(7,110)(8,101)(9,102)(10,103)(11,70)(12,61)(13,62)(14,63)(15,64)(16,65)(17,66)(18,67)(19,68)(20,69)(21,152)(22,153)(23,154)(24,155)(25,156)(26,157)(27,158)(28,159)(29,160)(30,151)(31,58)(32,59)(33,60)(34,51)(35,52)(36,53)(37,54)(38,55)(39,56)(40,57)(41,77)(42,78)(43,79)(44,80)(45,71)(46,72)(47,73)(48,74)(49,75)(50,76)(81,93)(82,94)(83,95)(84,96)(85,97)(86,98)(87,99)(88,100)(89,91)(90,92)(111,139)(112,140)(113,131)(114,132)(115,133)(116,134)(117,135)(118,136)(119,137)(120,138)(121,148)(122,149)(123,150)(124,141)(125,142)(126,143)(127,144)(128,145)(129,146)(130,147), (1,11,125,56,111,71,94,26)(2,12,126,57,112,72,95,27)(3,13,127,58,113,73,96,28)(4,14,128,59,114,74,97,29)(5,15,129,60,115,75,98,30)(6,16,130,51,116,76,99,21)(7,17,121,52,117,77,100,22)(8,18,122,53,118,78,91,23)(9,19,123,54,119,79,92,24)(10,20,124,55,120,80,93,25)(31,131,47,84,159,106,62,144)(32,132,48,85,160,107,63,145)(33,133,49,86,151,108,64,146)(34,134,50,87,152,109,65,147)(35,135,41,88,153,110,66,148)(36,136,42,89,154,101,67,149)(37,137,43,90,155,102,68,150)(38,138,44,81,156,103,69,141)(39,139,45,82,157,104,70,142)(40,140,46,83,158,105,61,143) );
G=PermutationGroup([(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)], [(11,70),(12,61),(13,62),(14,63),(15,64),(16,65),(17,66),(18,67),(19,68),(20,69),(21,152),(22,153),(23,154),(24,155),(25,156),(26,157),(27,158),(28,159),(29,160),(30,151),(31,58),(32,59),(33,60),(34,51),(35,52),(36,53),(37,54),(38,55),(39,56),(40,57),(41,77),(42,78),(43,79),(44,80),(45,71),(46,72),(47,73),(48,74),(49,75),(50,76)], [(1,104),(2,105),(3,106),(4,107),(5,108),(6,109),(7,110),(8,101),(9,102),(10,103),(11,70),(12,61),(13,62),(14,63),(15,64),(16,65),(17,66),(18,67),(19,68),(20,69),(21,152),(22,153),(23,154),(24,155),(25,156),(26,157),(27,158),(28,159),(29,160),(30,151),(31,58),(32,59),(33,60),(34,51),(35,52),(36,53),(37,54),(38,55),(39,56),(40,57),(41,77),(42,78),(43,79),(44,80),(45,71),(46,72),(47,73),(48,74),(49,75),(50,76),(81,93),(82,94),(83,95),(84,96),(85,97),(86,98),(87,99),(88,100),(89,91),(90,92),(111,139),(112,140),(113,131),(114,132),(115,133),(116,134),(117,135),(118,136),(119,137),(120,138),(121,148),(122,149),(123,150),(124,141),(125,142),(126,143),(127,144),(128,145),(129,146),(130,147)], [(1,11,125,56,111,71,94,26),(2,12,126,57,112,72,95,27),(3,13,127,58,113,73,96,28),(4,14,128,59,114,74,97,29),(5,15,129,60,115,75,98,30),(6,16,130,51,116,76,99,21),(7,17,121,52,117,77,100,22),(8,18,122,53,118,78,91,23),(9,19,123,54,119,79,92,24),(10,20,124,55,120,80,93,25),(31,131,47,84,159,106,62,144),(32,132,48,85,160,107,63,145),(33,133,49,86,151,108,64,146),(34,134,50,87,152,109,65,147),(35,135,41,88,153,110,66,148),(36,136,42,89,154,101,67,149),(37,137,43,90,155,102,68,150),(38,138,44,81,156,103,69,141),(39,139,45,82,157,104,70,142),(40,140,46,83,158,105,61,143)])
200 conjugacy classes
class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 4A | ··· | 4H | 4I | 4J | 4K | 4L | 5A | 5B | 5C | 5D | 8A | ··· | 8P | 10A | ··· | 10AB | 10AC | ··· | 10AR | 20A | ··· | 20AF | 20AG | ··· | 20AV | 40A | ··· | 40BL |
order | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 8 | ··· | 8 | 10 | ··· | 10 | 10 | ··· | 10 | 20 | ··· | 20 | 20 | ··· | 20 | 40 | ··· | 40 |
size | 1 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | ··· | 2 |
200 irreducible representations
dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
type | + | + | + | + | + | |||||||||||||
image | C1 | C2 | C2 | C2 | C4 | C4 | C5 | C8 | C10 | C10 | C10 | C20 | C20 | C40 | D4 | M4(2) | C5×D4 | C5×M4(2) |
kernel | C10×C22⋊C8 | C5×C22⋊C8 | C22×C40 | C23×C20 | C22×C20 | C23×C10 | C2×C22⋊C8 | C22×C10 | C22⋊C8 | C22×C8 | C23×C4 | C22×C4 | C24 | C23 | C2×C20 | C2×C10 | C2×C4 | C22 |
# reps | 1 | 4 | 2 | 1 | 6 | 2 | 4 | 16 | 16 | 8 | 4 | 24 | 8 | 64 | 4 | 4 | 16 | 16 |
Matrix representation of C10×C22⋊C8 ►in GL4(𝔽41) generated by
25 | 0 | 0 | 0 |
0 | 40 | 0 | 0 |
0 | 0 | 1 | 0 |
0 | 0 | 0 | 1 |
40 | 0 | 0 | 0 |
0 | 40 | 0 | 0 |
0 | 0 | 1 | 0 |
0 | 0 | 27 | 40 |
1 | 0 | 0 | 0 |
0 | 1 | 0 | 0 |
0 | 0 | 40 | 0 |
0 | 0 | 0 | 40 |
40 | 0 | 0 | 0 |
0 | 3 | 0 | 0 |
0 | 0 | 27 | 39 |
0 | 0 | 37 | 14 |
G:=sub<GL(4,GF(41))| [25,0,0,0,0,40,0,0,0,0,1,0,0,0,0,1],[40,0,0,0,0,40,0,0,0,0,1,27,0,0,0,40],[1,0,0,0,0,1,0,0,0,0,40,0,0,0,0,40],[40,0,0,0,0,3,0,0,0,0,27,37,0,0,39,14] >;
C10×C22⋊C8 in GAP, Magma, Sage, TeX
C_{10}\times C_2^2\rtimes C_8
% in TeX
G:=Group("C10xC2^2:C8");
// GroupNames label
G:=SmallGroup(320,907);
// by ID
G=gap.SmallGroup(320,907);
# by ID
G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-2,560,589,124]);
// Polycyclic
G:=Group<a,b,c,d|a^10=b^2=c^2=d^8=1,a*b=b*a,a*c=c*a,a*d=d*a,d*b*d^-1=b*c=c*b,c*d=d*c>;
// generators/relations