direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C10×C22⋊Q8, C23⋊3(C5×Q8), C4.63(D4×C10), (C22×C10)⋊6Q8, C22⋊1(Q8×C10), (C2×C20).524D4, C20.470(C2×D4), (C22×Q8)⋊3C10, C24.31(C2×C10), (C23×C20).25C2, (C23×C4).10C10, (Q8×C10)⋊48C22, C22.60(D4×C10), C10.57(C22×Q8), (C2×C10).343C24, (C2×C20).656C23, C10.182(C22×D4), (C23×C10).91C22, C23.70(C22×C10), C22.17(C23×C10), (C22×C20).444C22, (C22×C10).258C23, C2.6(D4×C2×C10), C2.3(Q8×C2×C10), (C10×C4⋊C4)⋊42C2, (C2×C4⋊C4)⋊15C10, (Q8×C2×C10)⋊15C2, (C2×C10)⋊5(C2×Q8), C4⋊C4⋊10(C2×C10), (C2×Q8)⋊8(C2×C10), C2.6(C10×C4○D4), (C5×C4⋊C4)⋊66C22, (C2×C4).135(C5×D4), C10.225(C2×C4○D4), (C2×C10).682(C2×D4), C22.30(C5×C4○D4), (C10×C22⋊C4).31C2, (C2×C22⋊C4).11C10, C22⋊C4.10(C2×C10), (C2×C4).12(C22×C10), (C22×C4).53(C2×C10), (C2×C10).230(C4○D4), (C5×C22⋊C4).144C22, SmallGroup(320,1525)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C10×C22⋊Q8
G = < a,b,c,d,e | a10=b2=c2=d4=1, e2=d2, ab=ba, ac=ca, ad=da, ae=ea, ebe-1=bc=cb, bd=db, cd=dc, ce=ec, ede-1=d-1 >
Subgroups: 450 in 322 conjugacy classes, 194 normal (26 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C5, C2×C4, C2×C4, Q8, C23, C23, C23, C10, C10, C10, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C22×C4, C2×Q8, C2×Q8, C24, C20, C20, C2×C10, C2×C10, C2×C10, C2×C22⋊C4, C2×C4⋊C4, C2×C4⋊C4, C22⋊Q8, C23×C4, C22×Q8, C2×C20, C2×C20, C5×Q8, C22×C10, C22×C10, C22×C10, C2×C22⋊Q8, C5×C22⋊C4, C5×C4⋊C4, C22×C20, C22×C20, C22×C20, Q8×C10, Q8×C10, C23×C10, C10×C22⋊C4, C10×C4⋊C4, C10×C4⋊C4, C5×C22⋊Q8, C23×C20, Q8×C2×C10, C10×C22⋊Q8
Quotients: C1, C2, C22, C5, D4, Q8, C23, C10, C2×D4, C2×Q8, C4○D4, C24, C2×C10, C22⋊Q8, C22×D4, C22×Q8, C2×C4○D4, C5×D4, C5×Q8, C22×C10, C2×C22⋊Q8, D4×C10, Q8×C10, C5×C4○D4, C23×C10, C5×C22⋊Q8, D4×C2×C10, Q8×C2×C10, C10×C4○D4, C10×C22⋊Q8
►On 160 points
Generators in S160
(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 21)(12 22)(13 23)(14 24)(15 25)(16 26)(17 27)(18 28)(19 29)(20 30)(31 160)(32 151)(33 152)(34 153)(35 154)(36 155)(37 156)(38 157)(39 158)(40 159)(111 125)(112 126)(113 127)(114 128)(115 129)(116 130)(117 121)(118 122)(119 123)(120 124)(131 147)(132 148)(133 149)(134 150)(135 141)(136 142)(137 143)(138 144)(139 145)(140 146)
(1 48)(2 49)(3 50)(4 41)(5 42)(6 43)(7 44)(8 45)(9 46)(10 47)(11 21)(12 22)(13 23)(14 24)(15 25)(16 26)(17 27)(18 28)(19 29)(20 30)(31 160)(32 151)(33 152)(34 153)(35 154)(36 155)(37 156)(38 157)(39 158)(40 159)(51 67)(52 68)(53 69)(54 70)(55 61)(56 62)(57 63)(58 64)(59 65)(60 66)(71 85)(72 86)(73 87)(74 88)(75 89)(76 90)(77 81)(78 82)(79 83)(80 84)(91 107)(92 108)(93 109)(94 110)(95 101)(96 102)(97 103)(98 104)(99 105)(100 106)(111 125)(112 126)(113 127)(114 128)(115 129)(116 130)(117 121)(118 122)(119 123)(120 124)(131 147)(132 148)(133 149)(134 150)(135 141)(136 142)(137 143)(138 144)(139 145)(140 146)
(1 79 67 96)(2 80 68 97)(3 71 69 98)(4 72 70 99)(5 73 61 100)(6 74 62 91)(7 75 63 92)(8 76 64 93)(9 77 65 94)(10 78 66 95)(11 131 40 114)(12 132 31 115)(13 133 32 116)(14 134 33 117)(15 135 34 118)(16 136 35 119)(17 137 36 120)(18 138 37 111)(19 139 38 112)(20 140 39 113)(21 147 159 128)(22 148 160 129)(23 149 151 130)(24 150 152 121)(25 141 153 122)(26 142 154 123)(27 143 155 124)(28 144 156 125)(29 145 157 126)(30 146 158 127)(41 86 54 105)(42 87 55 106)(43 88 56 107)(44 89 57 108)(45 90 58 109)(46 81 59 110)(47 82 60 101)(48 83 51 102)(49 84 52 103)(50 85 53 104)
(1 119 67 136)(2 120 68 137)(3 111 69 138)(4 112 70 139)(5 113 61 140)(6 114 62 131)(7 115 63 132)(8 116 64 133)(9 117 65 134)(10 118 66 135)(11 74 40 91)(12 75 31 92)(13 76 32 93)(14 77 33 94)(15 78 34 95)(16 79 35 96)(17 80 36 97)(18 71 37 98)(19 72 38 99)(20 73 39 100)(21 88 159 107)(22 89 160 108)(23 90 151 109)(24 81 152 110)(25 82 153 101)(26 83 154 102)(27 84 155 103)(28 85 156 104)(29 86 157 105)(30 87 158 106)(41 126 54 145)(42 127 55 146)(43 128 56 147)(44 129 57 148)(45 130 58 149)(46 121 59 150)(47 122 60 141)(48 123 51 142)(49 124 52 143)(50 125 53 144)
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), (11,21)(12,22)(13,23)(14,24)(15,25)(16,26)(17,27)(18,28)(19,29)(20,30)(31,160)(32,151)(33,152)(34,153)(35,154)(36,155)(37,156)(38,157)(39,158)(40,159)(111,125)(112,126)(113,127)(114,128)(115,129)(116,130)(117,121)(118,122)(119,123)(120,124)(131,147)(132,148)(133,149)(134,150)(135,141)(136,142)(137,143)(138,144)(139,145)(140,146), (1,48)(2,49)(3,50)(4,41)(5,42)(6,43)(7,44)(8,45)(9,46)(10,47)(11,21)(12,22)(13,23)(14,24)(15,25)(16,26)(17,27)(18,28)(19,29)(20,30)(31,160)(32,151)(33,152)(34,153)(35,154)(36,155)(37,156)(38,157)(39,158)(40,159)(51,67)(52,68)(53,69)(54,70)(55,61)(56,62)(57,63)(58,64)(59,65)(60,66)(71,85)(72,86)(73,87)(74,88)(75,89)(76,90)(77,81)(78,82)(79,83)(80,84)(91,107)(92,108)(93,109)(94,110)(95,101)(96,102)(97,103)(98,104)(99,105)(100,106)(111,125)(112,126)(113,127)(114,128)(115,129)(116,130)(117,121)(118,122)(119,123)(120,124)(131,147)(132,148)(133,149)(134,150)(135,141)(136,142)(137,143)(138,144)(139,145)(140,146), (1,79,67,96)(2,80,68,97)(3,71,69,98)(4,72,70,99)(5,73,61,100)(6,74,62,91)(7,75,63,92)(8,76,64,93)(9,77,65,94)(10,78,66,95)(11,131,40,114)(12,132,31,115)(13,133,32,116)(14,134,33,117)(15,135,34,118)(16,136,35,119)(17,137,36,120)(18,138,37,111)(19,139,38,112)(20,140,39,113)(21,147,159,128)(22,148,160,129)(23,149,151,130)(24,150,152,121)(25,141,153,122)(26,142,154,123)(27,143,155,124)(28,144,156,125)(29,145,157,126)(30,146,158,127)(41,86,54,105)(42,87,55,106)(43,88,56,107)(44,89,57,108)(45,90,58,109)(46,81,59,110)(47,82,60,101)(48,83,51,102)(49,84,52,103)(50,85,53,104), (1,119,67,136)(2,120,68,137)(3,111,69,138)(4,112,70,139)(5,113,61,140)(6,114,62,131)(7,115,63,132)(8,116,64,133)(9,117,65,134)(10,118,66,135)(11,74,40,91)(12,75,31,92)(13,76,32,93)(14,77,33,94)(15,78,34,95)(16,79,35,96)(17,80,36,97)(18,71,37,98)(19,72,38,99)(20,73,39,100)(21,88,159,107)(22,89,160,108)(23,90,151,109)(24,81,152,110)(25,82,153,101)(26,83,154,102)(27,84,155,103)(28,85,156,104)(29,86,157,105)(30,87,158,106)(41,126,54,145)(42,127,55,146)(43,128,56,147)(44,129,57,148)(45,130,58,149)(46,121,59,150)(47,122,60,141)(48,123,51,142)(49,124,52,143)(50,125,53,144)>;
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), (11,21)(12,22)(13,23)(14,24)(15,25)(16,26)(17,27)(18,28)(19,29)(20,30)(31,160)(32,151)(33,152)(34,153)(35,154)(36,155)(37,156)(38,157)(39,158)(40,159)(111,125)(112,126)(113,127)(114,128)(115,129)(116,130)(117,121)(118,122)(119,123)(120,124)(131,147)(132,148)(133,149)(134,150)(135,141)(136,142)(137,143)(138,144)(139,145)(140,146), (1,48)(2,49)(3,50)(4,41)(5,42)(6,43)(7,44)(8,45)(9,46)(10,47)(11,21)(12,22)(13,23)(14,24)(15,25)(16,26)(17,27)(18,28)(19,29)(20,30)(31,160)(32,151)(33,152)(34,153)(35,154)(36,155)(37,156)(38,157)(39,158)(40,159)(51,67)(52,68)(53,69)(54,70)(55,61)(56,62)(57,63)(58,64)(59,65)(60,66)(71,85)(72,86)(73,87)(74,88)(75,89)(76,90)(77,81)(78,82)(79,83)(80,84)(91,107)(92,108)(93,109)(94,110)(95,101)(96,102)(97,103)(98,104)(99,105)(100,106)(111,125)(112,126)(113,127)(114,128)(115,129)(116,130)(117,121)(118,122)(119,123)(120,124)(131,147)(132,148)(133,149)(134,150)(135,141)(136,142)(137,143)(138,144)(139,145)(140,146), (1,79,67,96)(2,80,68,97)(3,71,69,98)(4,72,70,99)(5,73,61,100)(6,74,62,91)(7,75,63,92)(8,76,64,93)(9,77,65,94)(10,78,66,95)(11,131,40,114)(12,132,31,115)(13,133,32,116)(14,134,33,117)(15,135,34,118)(16,136,35,119)(17,137,36,120)(18,138,37,111)(19,139,38,112)(20,140,39,113)(21,147,159,128)(22,148,160,129)(23,149,151,130)(24,150,152,121)(25,141,153,122)(26,142,154,123)(27,143,155,124)(28,144,156,125)(29,145,157,126)(30,146,158,127)(41,86,54,105)(42,87,55,106)(43,88,56,107)(44,89,57,108)(45,90,58,109)(46,81,59,110)(47,82,60,101)(48,83,51,102)(49,84,52,103)(50,85,53,104), (1,119,67,136)(2,120,68,137)(3,111,69,138)(4,112,70,139)(5,113,61,140)(6,114,62,131)(7,115,63,132)(8,116,64,133)(9,117,65,134)(10,118,66,135)(11,74,40,91)(12,75,31,92)(13,76,32,93)(14,77,33,94)(15,78,34,95)(16,79,35,96)(17,80,36,97)(18,71,37,98)(19,72,38,99)(20,73,39,100)(21,88,159,107)(22,89,160,108)(23,90,151,109)(24,81,152,110)(25,82,153,101)(26,83,154,102)(27,84,155,103)(28,85,156,104)(29,86,157,105)(30,87,158,106)(41,126,54,145)(42,127,55,146)(43,128,56,147)(44,129,57,148)(45,130,58,149)(46,121,59,150)(47,122,60,141)(48,123,51,142)(49,124,52,143)(50,125,53,144) );
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)], [(11,21),(12,22),(13,23),(14,24),(15,25),(16,26),(17,27),(18,28),(19,29),(20,30),(31,160),(32,151),(33,152),(34,153),(35,154),(36,155),(37,156),(38,157),(39,158),(40,159),(111,125),(112,126),(113,127),(114,128),(115,129),(116,130),(117,121),(118,122),(119,123),(120,124),(131,147),(132,148),(133,149),(134,150),(135,141),(136,142),(137,143),(138,144),(139,145),(140,146)], [(1,48),(2,49),(3,50),(4,41),(5,42),(6,43),(7,44),(8,45),(9,46),(10,47),(11,21),(12,22),(13,23),(14,24),(15,25),(16,26),(17,27),(18,28),(19,29),(20,30),(31,160),(32,151),(33,152),(34,153),(35,154),(36,155),(37,156),(38,157),(39,158),(40,159),(51,67),(52,68),(53,69),(54,70),(55,61),(56,62),(57,63),(58,64),(59,65),(60,66),(71,85),(72,86),(73,87),(74,88),(75,89),(76,90),(77,81),(78,82),(79,83),(80,84),(91,107),(92,108),(93,109),(94,110),(95,101),(96,102),(97,103),(98,104),(99,105),(100,106),(111,125),(112,126),(113,127),(114,128),(115,129),(116,130),(117,121),(118,122),(119,123),(120,124),(131,147),(132,148),(133,149),(134,150),(135,141),(136,142),(137,143),(138,144),(139,145),(140,146)], [(1,79,67,96),(2,80,68,97),(3,71,69,98),(4,72,70,99),(5,73,61,100),(6,74,62,91),(7,75,63,92),(8,76,64,93),(9,77,65,94),(10,78,66,95),(11,131,40,114),(12,132,31,115),(13,133,32,116),(14,134,33,117),(15,135,34,118),(16,136,35,119),(17,137,36,120),(18,138,37,111),(19,139,38,112),(20,140,39,113),(21,147,159,128),(22,148,160,129),(23,149,151,130),(24,150,152,121),(25,141,153,122),(26,142,154,123),(27,143,155,124),(28,144,156,125),(29,145,157,126),(30,146,158,127),(41,86,54,105),(42,87,55,106),(43,88,56,107),(44,89,57,108),(45,90,58,109),(46,81,59,110),(47,82,60,101),(48,83,51,102),(49,84,52,103),(50,85,53,104)], [(1,119,67,136),(2,120,68,137),(3,111,69,138),(4,112,70,139),(5,113,61,140),(6,114,62,131),(7,115,63,132),(8,116,64,133),(9,117,65,134),(10,118,66,135),(11,74,40,91),(12,75,31,92),(13,76,32,93),(14,77,33,94),(15,78,34,95),(16,79,35,96),(17,80,36,97),(18,71,37,98),(19,72,38,99),(20,73,39,100),(21,88,159,107),(22,89,160,108),(23,90,151,109),(24,81,152,110),(25,82,153,101),(26,83,154,102),(27,84,155,103),(28,85,156,104),(29,86,157,105),(30,87,158,106),(41,126,54,145),(42,127,55,146),(43,128,56,147),(44,129,57,148),(45,130,58,149),(46,121,59,150),(47,122,60,141),(48,123,51,142),(49,124,52,143),(50,125,53,144)]])
140 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 4A | ··· | 4H | 4I | ··· | 4P | 5A | 5B | 5C | 5D | 10A | ··· | 10AB | 10AC | ··· | 10AR | 20A | ··· | 20AF | 20AG | ··· | 20BL |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 5 | 5 | 5 | 5 | 10 | ··· | 10 | 10 | ··· | 10 | 20 | ··· | 20 | 20 | ··· | 20 |
| size | 1 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 |
140 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | - | ||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C5 | C10 | C10 | C10 | C10 | C10 | D4 | Q8 | C4○D4 | C5×D4 | C5×Q8 | C5×C4○D4 |
| kernel | C10×C22⋊Q8 | C10×C22⋊C4 | C10×C4⋊C4 | C5×C22⋊Q8 | C23×C20 | Q8×C2×C10 | C2×C22⋊Q8 | C2×C22⋊C4 | C2×C4⋊C4 | C22⋊Q8 | C23×C4 | C22×Q8 | C2×C20 | C22×C10 | C2×C10 | C2×C4 | C23 | C22 |
| # reps | 1 | 2 | 3 | 8 | 1 | 1 | 4 | 8 | 12 | 32 | 4 | 4 | 4 | 4 | 4 | 16 | 16 | 16 |
Matrix representation of C10×C22⋊Q8 ►in GL5(𝔽41)
| 40 | 0 | 0 | 0 | 0 |
| 0 | 18 | 0 | 0 | 0 |
| 0 | 0 | 18 | 0 | 0 |
| 0 | 0 | 0 | 25 | 0 |
| 0 | 0 | 0 | 0 | 25 |
| 40 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 40 | 39 |
| 0 | 0 | 0 | 1 | 1 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 34 | 14 |
| 0 | 0 | 0 | 14 | 7 |
G:=sub<GL(5,GF(41))| [40,0,0,0,0,0,18,0,0,0,0,0,18,0,0,0,0,0,25,0,0,0,0,0,25],[40,0,0,0,0,0,1,0,0,0,0,0,40,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,40,0,0,0,0,0,40,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,40,1,0,0,0,39,1],[1,0,0,0,0,0,0,40,0,0,0,40,0,0,0,0,0,0,34,14,0,0,0,14,7] >;
C10×C22⋊Q8 in GAP, Magma, Sage, TeX
C_{10}\times C_2^2\rtimes Q_8 % in TeX
G:=Group("C10xC2^2:Q8"); // GroupNames label
G:=SmallGroup(320,1525);
// by ID
G=gap.SmallGroup(320,1525);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-5,-2,-2,560,1149,568,3446]);
// Polycyclic
G:=Group<a,b,c,d,e|a^10=b^2=c^2=d^4=1,e^2=d^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,e*b*e^-1=b*c=c*b,b*d=d*b,c*d=d*c,c*e=e*c,e*d*e^-1=d^-1>;
// generators/relations