direct product, metabelian, nilpotent (class 3), monomial, 2-elementary
Aliases: C5×C4⋊SD16, C20⋊15SD16, C4⋊C8⋊8C10, Q8⋊1(C5×D4), (C5×Q8)⋊12D4, (C4×Q8)⋊3C10, C4⋊3(C5×SD16), (Q8×C20)⋊23C2, C4.32(D4×C10), C20.393(C2×D4), C4⋊1D4.3C10, (C2×C20).322D4, D4⋊C4⋊10C10, C2.7(C10×SD16), (C10×SD16)⋊28C2, (C2×SD16)⋊11C10, C42.15(C2×C10), C10.87(C2×SD16), C22.84(D4×C10), C20.342(C4○D4), (C4×C20).257C22, (C2×C20).919C23, (C2×C40).299C22, C10.143(C4⋊D4), C10.135(C8⋊C22), (D4×C10).186C22, (Q8×C10).262C22, (C5×C4⋊C8)⋊27C2, C4.41(C5×C4○D4), C4⋊C4.52(C2×C10), (C2×C8).36(C2×C10), (C2×C4).128(C5×D4), C2.12(C5×C4⋊D4), C2.10(C5×C8⋊C22), (C5×D4⋊C4)⋊34C2, (C2×D4).10(C2×C10), (C5×C4⋊1D4).10C2, (C2×C10).640(C2×D4), (C2×Q8).47(C2×C10), (C5×C4⋊C4).373C22, (C2×C4).94(C22×C10), SmallGroup(320,961)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C5×C4⋊SD16
G = < a,b,c,d | a5=b4=c8=d2=1, ab=ba, ac=ca, ad=da, cbc-1=dbd=b-1, dcd=c3 >
Subgroups: 266 in 128 conjugacy classes, 58 normal (34 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, D4, Q8, Q8, C23, C10, C10, C42, C42, C4⋊C4, C4⋊C4, C2×C8, SD16, C2×D4, C2×D4, C2×Q8, C20, C20, C20, C2×C10, C2×C10, D4⋊C4, C4⋊C8, C4×Q8, C4⋊1D4, C2×SD16, C40, C2×C20, C2×C20, C5×D4, C5×Q8, C5×Q8, C22×C10, C4⋊SD16, C4×C20, C4×C20, C5×C4⋊C4, C5×C4⋊C4, C2×C40, C5×SD16, D4×C10, D4×C10, Q8×C10, C5×D4⋊C4, C5×C4⋊C8, Q8×C20, C5×C4⋊1D4, C10×SD16, C5×C4⋊SD16
Quotients: C1, C2, C22, C5, D4, C23, C10, SD16, C2×D4, C4○D4, C2×C10, C4⋊D4, C2×SD16, C8⋊C22, C5×D4, C22×C10, C4⋊SD16, C5×SD16, D4×C10, C5×C4○D4, C5×C4⋊D4, C10×SD16, C5×C8⋊C22, C5×C4⋊SD16
►On 160 points
Generators in S160
(1 12 59 18 51)(2 13 60 19 52)(3 14 61 20 53)(4 15 62 21 54)(5 16 63 22 55)(6 9 64 23 56)(7 10 57 24 49)(8 11 58 17 50)(25 85 74 33 66)(26 86 75 34 67)(27 87 76 35 68)(28 88 77 36 69)(29 81 78 37 70)(30 82 79 38 71)(31 83 80 39 72)(32 84 73 40 65)(41 157 120 149 112)(42 158 113 150 105)(43 159 114 151 106)(44 160 115 152 107)(45 153 116 145 108)(46 154 117 146 109)(47 155 118 147 110)(48 156 119 148 111)(89 126 142 97 134)(90 127 143 98 135)(91 128 144 99 136)(92 121 137 100 129)(93 122 138 101 130)(94 123 139 102 131)(95 124 140 103 132)(96 125 141 104 133)
(1 122 30 106)(2 107 31 123)(3 124 32 108)(4 109 25 125)(5 126 26 110)(6 111 27 127)(7 128 28 112)(8 105 29 121)(9 48 87 143)(10 144 88 41)(11 42 81 137)(12 138 82 43)(13 44 83 139)(14 140 84 45)(15 46 85 141)(16 142 86 47)(17 113 37 129)(18 130 38 114)(19 115 39 131)(20 132 40 116)(21 117 33 133)(22 134 34 118)(23 119 35 135)(24 136 36 120)(49 91 69 149)(50 150 70 92)(51 93 71 151)(52 152 72 94)(53 95 65 145)(54 146 66 96)(55 89 67 147)(56 148 68 90)(57 99 77 157)(58 158 78 100)(59 101 79 159)(60 160 80 102)(61 103 73 153)(62 154 74 104)(63 97 75 155)(64 156 76 98)
(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)
(2 4)(3 7)(6 8)(9 11)(10 14)(13 15)(17 23)(19 21)(20 24)(25 31)(27 29)(28 32)(33 39)(35 37)(36 40)(41 140)(42 143)(43 138)(44 141)(45 144)(46 139)(47 142)(48 137)(49 53)(50 56)(52 54)(57 61)(58 64)(60 62)(65 69)(66 72)(68 70)(73 77)(74 80)(76 78)(81 87)(83 85)(84 88)(89 147)(90 150)(91 145)(92 148)(93 151)(94 146)(95 149)(96 152)(97 155)(98 158)(99 153)(100 156)(101 159)(102 154)(103 157)(104 160)(105 127)(106 122)(107 125)(108 128)(109 123)(110 126)(111 121)(112 124)(113 135)(114 130)(115 133)(116 136)(117 131)(118 134)(119 129)(120 132)
G:=sub<Sym(160)| (1,12,59,18,51)(2,13,60,19,52)(3,14,61,20,53)(4,15,62,21,54)(5,16,63,22,55)(6,9,64,23,56)(7,10,57,24,49)(8,11,58,17,50)(25,85,74,33,66)(26,86,75,34,67)(27,87,76,35,68)(28,88,77,36,69)(29,81,78,37,70)(30,82,79,38,71)(31,83,80,39,72)(32,84,73,40,65)(41,157,120,149,112)(42,158,113,150,105)(43,159,114,151,106)(44,160,115,152,107)(45,153,116,145,108)(46,154,117,146,109)(47,155,118,147,110)(48,156,119,148,111)(89,126,142,97,134)(90,127,143,98,135)(91,128,144,99,136)(92,121,137,100,129)(93,122,138,101,130)(94,123,139,102,131)(95,124,140,103,132)(96,125,141,104,133), (1,122,30,106)(2,107,31,123)(3,124,32,108)(4,109,25,125)(5,126,26,110)(6,111,27,127)(7,128,28,112)(8,105,29,121)(9,48,87,143)(10,144,88,41)(11,42,81,137)(12,138,82,43)(13,44,83,139)(14,140,84,45)(15,46,85,141)(16,142,86,47)(17,113,37,129)(18,130,38,114)(19,115,39,131)(20,132,40,116)(21,117,33,133)(22,134,34,118)(23,119,35,135)(24,136,36,120)(49,91,69,149)(50,150,70,92)(51,93,71,151)(52,152,72,94)(53,95,65,145)(54,146,66,96)(55,89,67,147)(56,148,68,90)(57,99,77,157)(58,158,78,100)(59,101,79,159)(60,160,80,102)(61,103,73,153)(62,154,74,104)(63,97,75,155)(64,156,76,98), (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), (2,4)(3,7)(6,8)(9,11)(10,14)(13,15)(17,23)(19,21)(20,24)(25,31)(27,29)(28,32)(33,39)(35,37)(36,40)(41,140)(42,143)(43,138)(44,141)(45,144)(46,139)(47,142)(48,137)(49,53)(50,56)(52,54)(57,61)(58,64)(60,62)(65,69)(66,72)(68,70)(73,77)(74,80)(76,78)(81,87)(83,85)(84,88)(89,147)(90,150)(91,145)(92,148)(93,151)(94,146)(95,149)(96,152)(97,155)(98,158)(99,153)(100,156)(101,159)(102,154)(103,157)(104,160)(105,127)(106,122)(107,125)(108,128)(109,123)(110,126)(111,121)(112,124)(113,135)(114,130)(115,133)(116,136)(117,131)(118,134)(119,129)(120,132)>;
G:=Group( (1,12,59,18,51)(2,13,60,19,52)(3,14,61,20,53)(4,15,62,21,54)(5,16,63,22,55)(6,9,64,23,56)(7,10,57,24,49)(8,11,58,17,50)(25,85,74,33,66)(26,86,75,34,67)(27,87,76,35,68)(28,88,77,36,69)(29,81,78,37,70)(30,82,79,38,71)(31,83,80,39,72)(32,84,73,40,65)(41,157,120,149,112)(42,158,113,150,105)(43,159,114,151,106)(44,160,115,152,107)(45,153,116,145,108)(46,154,117,146,109)(47,155,118,147,110)(48,156,119,148,111)(89,126,142,97,134)(90,127,143,98,135)(91,128,144,99,136)(92,121,137,100,129)(93,122,138,101,130)(94,123,139,102,131)(95,124,140,103,132)(96,125,141,104,133), (1,122,30,106)(2,107,31,123)(3,124,32,108)(4,109,25,125)(5,126,26,110)(6,111,27,127)(7,128,28,112)(8,105,29,121)(9,48,87,143)(10,144,88,41)(11,42,81,137)(12,138,82,43)(13,44,83,139)(14,140,84,45)(15,46,85,141)(16,142,86,47)(17,113,37,129)(18,130,38,114)(19,115,39,131)(20,132,40,116)(21,117,33,133)(22,134,34,118)(23,119,35,135)(24,136,36,120)(49,91,69,149)(50,150,70,92)(51,93,71,151)(52,152,72,94)(53,95,65,145)(54,146,66,96)(55,89,67,147)(56,148,68,90)(57,99,77,157)(58,158,78,100)(59,101,79,159)(60,160,80,102)(61,103,73,153)(62,154,74,104)(63,97,75,155)(64,156,76,98), (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), (2,4)(3,7)(6,8)(9,11)(10,14)(13,15)(17,23)(19,21)(20,24)(25,31)(27,29)(28,32)(33,39)(35,37)(36,40)(41,140)(42,143)(43,138)(44,141)(45,144)(46,139)(47,142)(48,137)(49,53)(50,56)(52,54)(57,61)(58,64)(60,62)(65,69)(66,72)(68,70)(73,77)(74,80)(76,78)(81,87)(83,85)(84,88)(89,147)(90,150)(91,145)(92,148)(93,151)(94,146)(95,149)(96,152)(97,155)(98,158)(99,153)(100,156)(101,159)(102,154)(103,157)(104,160)(105,127)(106,122)(107,125)(108,128)(109,123)(110,126)(111,121)(112,124)(113,135)(114,130)(115,133)(116,136)(117,131)(118,134)(119,129)(120,132) );
G=PermutationGroup([[(1,12,59,18,51),(2,13,60,19,52),(3,14,61,20,53),(4,15,62,21,54),(5,16,63,22,55),(6,9,64,23,56),(7,10,57,24,49),(8,11,58,17,50),(25,85,74,33,66),(26,86,75,34,67),(27,87,76,35,68),(28,88,77,36,69),(29,81,78,37,70),(30,82,79,38,71),(31,83,80,39,72),(32,84,73,40,65),(41,157,120,149,112),(42,158,113,150,105),(43,159,114,151,106),(44,160,115,152,107),(45,153,116,145,108),(46,154,117,146,109),(47,155,118,147,110),(48,156,119,148,111),(89,126,142,97,134),(90,127,143,98,135),(91,128,144,99,136),(92,121,137,100,129),(93,122,138,101,130),(94,123,139,102,131),(95,124,140,103,132),(96,125,141,104,133)], [(1,122,30,106),(2,107,31,123),(3,124,32,108),(4,109,25,125),(5,126,26,110),(6,111,27,127),(7,128,28,112),(8,105,29,121),(9,48,87,143),(10,144,88,41),(11,42,81,137),(12,138,82,43),(13,44,83,139),(14,140,84,45),(15,46,85,141),(16,142,86,47),(17,113,37,129),(18,130,38,114),(19,115,39,131),(20,132,40,116),(21,117,33,133),(22,134,34,118),(23,119,35,135),(24,136,36,120),(49,91,69,149),(50,150,70,92),(51,93,71,151),(52,152,72,94),(53,95,65,145),(54,146,66,96),(55,89,67,147),(56,148,68,90),(57,99,77,157),(58,158,78,100),(59,101,79,159),(60,160,80,102),(61,103,73,153),(62,154,74,104),(63,97,75,155),(64,156,76,98)], [(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)], [(2,4),(3,7),(6,8),(9,11),(10,14),(13,15),(17,23),(19,21),(20,24),(25,31),(27,29),(28,32),(33,39),(35,37),(36,40),(41,140),(42,143),(43,138),(44,141),(45,144),(46,139),(47,142),(48,137),(49,53),(50,56),(52,54),(57,61),(58,64),(60,62),(65,69),(66,72),(68,70),(73,77),(74,80),(76,78),(81,87),(83,85),(84,88),(89,147),(90,150),(91,145),(92,148),(93,151),(94,146),(95,149),(96,152),(97,155),(98,158),(99,153),(100,156),(101,159),(102,154),(103,157),(104,160),(105,127),(106,122),(107,125),(108,128),(109,123),(110,126),(111,121),(112,124),(113,135),(114,130),(115,133),(116,136),(117,131),(118,134),(119,129),(120,132)]])
95 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | ··· | 4I | 5A | 5B | 5C | 5D | 8A | 8B | 8C | 8D | 10A | ··· | 10L | 10M | ··· | 10T | 20A | ··· | 20P | 20Q | ··· | 20AJ | 40A | ··· | 40P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 5 | 5 | 5 | 5 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 10 | ··· | 10 | 20 | ··· | 20 | 20 | ··· | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 1 | 1 | 8 | 8 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 4 | 1 | ··· | 1 | 8 | ··· | 8 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
95 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | |||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C5 | C10 | C10 | C10 | C10 | C10 | D4 | D4 | SD16 | C4○D4 | C5×D4 | C5×D4 | C5×SD16 | C5×C4○D4 | C8⋊C22 | C5×C8⋊C22 |
| kernel | C5×C4⋊SD16 | C5×D4⋊C4 | C5×C4⋊C8 | Q8×C20 | C5×C4⋊1D4 | C10×SD16 | C4⋊SD16 | D4⋊C4 | C4⋊C8 | C4×Q8 | C4⋊1D4 | C2×SD16 | C2×C20 | C5×Q8 | C20 | C20 | C2×C4 | Q8 | C4 | C4 | C10 | C2 |
| # reps | 1 | 2 | 1 | 1 | 1 | 2 | 4 | 8 | 4 | 4 | 4 | 8 | 2 | 2 | 4 | 2 | 8 | 8 | 16 | 8 | 1 | 4 |
Matrix representation of C5×C4⋊SD16 ►in GL4(𝔽41) generated by
| 10 | 0 | 0 | 0 |
| 0 | 10 | 0 | 0 |
| 0 | 0 | 37 | 0 |
| 0 | 0 | 0 | 37 |
| 9 | 18 | 0 | 0 |
| 0 | 32 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 40 | 39 | 0 | 0 |
| 1 | 1 | 0 | 0 |
| 0 | 0 | 26 | 26 |
| 0 | 0 | 15 | 26 |
| 1 | 0 | 0 | 0 |
| 40 | 40 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 40 |
G:=sub<GL(4,GF(41))| [10,0,0,0,0,10,0,0,0,0,37,0,0,0,0,37],[9,0,0,0,18,32,0,0,0,0,1,0,0,0,0,1],[40,1,0,0,39,1,0,0,0,0,26,15,0,0,26,26],[1,40,0,0,0,40,0,0,0,0,1,0,0,0,0,40] >;
C5×C4⋊SD16 in GAP, Magma, Sage, TeX
C_5\times C_4\rtimes {\rm SD}_{16} % in TeX
G:=Group("C5xC4:SD16"); // GroupNames label
G:=SmallGroup(320,961);
// by ID
G=gap.SmallGroup(320,961);
# by ID
G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-2,589,288,1766,856,10085,2539,124]);
// Polycyclic
G:=Group<a,b,c,d|a^5=b^4=c^8=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d=b^-1,d*c*d=c^3>;
// generators/relations