metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.12F5, C20.14M4(2), (C4×D5)⋊5C8, C20.14(C2×C8), (C4×C20).13C4, C4.9(D5⋊C8), C20⋊C8⋊18C2, D10.15(C2×C8), C10.4(C22×C8), C4.12(C4.F5), D10⋊C8.7C2, Dic5.15(C2×C8), (C4×Dic5).26C4, (D5×C42).26C2, C10.5(C2×M4(2)), C5⋊2(C42.12C4), C10.2(C42⋊C2), Dic5.24(C4○D4), C22.29(C22×F5), (C4×Dic5).344C22, (C2×Dic5).316C23, C2.2(D10.C23), (C4×C5⋊C8)⋊9C2, C2.6(C2×D5⋊C8), (C2×C4×D5).44C4, C2.2(C2×C4.F5), (C2×C4).97(C2×F5), (C2×C5⋊C8).18C22, (C2×C20).100(C2×C4), (C2×C4×D5).357C22, (C2×C10).18(C22×C4), (C2×Dic5).166(C2×C4), (C22×D5).118(C2×C4), SmallGroup(320,1018)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 378 in 118 conjugacy classes, 58 normal (30 characteristic)
C1, C2 [×3], C2 [×2], C4 [×4], C4 [×6], C22, C22 [×4], C5, C8 [×4], C2×C4 [×3], C2×C4 [×11], C23, D5 [×2], C10 [×3], C42, C42 [×3], C2×C8 [×4], C22×C4 [×3], Dic5 [×4], Dic5, C20 [×4], C20, D10 [×2], D10 [×2], C2×C10, C4×C8 [×2], C22⋊C8 [×2], C4⋊C8 [×2], C2×C42, C5⋊C8 [×4], C4×D5 [×4], C4×D5 [×4], C2×Dic5 [×3], C2×C20 [×3], C22×D5, C42.12C4, C4×Dic5 [×3], C4×C20, C2×C5⋊C8 [×4], C2×C4×D5 [×3], C4×C5⋊C8 [×2], C20⋊C8 [×2], D10⋊C8 [×2], D5×C42, C42.12F5
Quotients:
C1, C2 [×7], C4 [×4], C22 [×7], C8 [×4], C2×C4 [×6], C23, C2×C8 [×6], M4(2) [×2], C22×C4, C4○D4 [×2], F5, C42⋊C2, C22×C8, C2×M4(2), C2×F5 [×3], C42.12C4, D5⋊C8 [×2], C4.F5 [×2], C22×F5, C2×D5⋊C8, C2×C4.F5, D10.C23, C42.12F5
Generators and relations
G = < a,b,c,d | a4=b4=c5=1, d4=a2b2, ab=ba, ac=ca, dad-1=a-1, bc=cb, bd=db, dcd-1=c3 >
►On 160 points
Generators in S160
(1 93 132 24)(2 17 133 94)(3 95 134 18)(4 19 135 96)(5 89 136 20)(6 21 129 90)(7 91 130 22)(8 23 131 92)(9 108 80 51)(10 52 73 109)(11 110 74 53)(12 54 75 111)(13 112 76 55)(14 56 77 105)(15 106 78 49)(16 50 79 107)(25 71 33 63)(26 64 34 72)(27 65 35 57)(28 58 36 66)(29 67 37 59)(30 60 38 68)(31 69 39 61)(32 62 40 70)(41 147 141 87)(42 88 142 148)(43 149 143 81)(44 82 144 150)(45 151 137 83)(46 84 138 152)(47 145 139 85)(48 86 140 146)(97 154 122 116)(98 117 123 155)(99 156 124 118)(100 119 125 157)(101 158 126 120)(102 113 127 159)(103 160 128 114)(104 115 121 153)
(1 22 136 95)(2 23 129 96)(3 24 130 89)(4 17 131 90)(5 18 132 91)(6 19 133 92)(7 20 134 93)(8 21 135 94)(9 97 76 126)(10 98 77 127)(11 99 78 128)(12 100 79 121)(13 101 80 122)(14 102 73 123)(15 103 74 124)(16 104 75 125)(25 139 37 43)(26 140 38 44)(27 141 39 45)(28 142 40 46)(29 143 33 47)(30 144 34 48)(31 137 35 41)(32 138 36 42)(49 114 110 156)(50 115 111 157)(51 116 112 158)(52 117 105 159)(53 118 106 160)(54 119 107 153)(55 120 108 154)(56 113 109 155)(57 147 69 83)(58 148 70 84)(59 149 71 85)(60 150 72 86)(61 151 65 87)(62 152 66 88)(63 145 67 81)(64 146 68 82)
(1 12 57 143 113)(2 144 13 114 58)(3 115 137 59 14)(4 60 116 15 138)(5 16 61 139 117)(6 140 9 118 62)(7 119 141 63 10)(8 64 120 11 142)(17 150 112 103 36)(18 104 151 37 105)(19 38 97 106 152)(20 107 39 145 98)(21 146 108 99 40)(22 100 147 33 109)(23 34 101 110 148)(24 111 35 149 102)(25 52 91 125 87)(26 126 53 88 92)(27 81 127 93 54)(28 94 82 55 128)(29 56 95 121 83)(30 122 49 84 96)(31 85 123 89 50)(32 90 86 51 124)(41 71 73 130 157)(42 131 72 158 74)(43 159 132 75 65)(44 76 160 66 133)(45 67 77 134 153)(46 135 68 154 78)(47 155 136 79 69)(48 80 156 70 129)
(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)| (1,93,132,24)(2,17,133,94)(3,95,134,18)(4,19,135,96)(5,89,136,20)(6,21,129,90)(7,91,130,22)(8,23,131,92)(9,108,80,51)(10,52,73,109)(11,110,74,53)(12,54,75,111)(13,112,76,55)(14,56,77,105)(15,106,78,49)(16,50,79,107)(25,71,33,63)(26,64,34,72)(27,65,35,57)(28,58,36,66)(29,67,37,59)(30,60,38,68)(31,69,39,61)(32,62,40,70)(41,147,141,87)(42,88,142,148)(43,149,143,81)(44,82,144,150)(45,151,137,83)(46,84,138,152)(47,145,139,85)(48,86,140,146)(97,154,122,116)(98,117,123,155)(99,156,124,118)(100,119,125,157)(101,158,126,120)(102,113,127,159)(103,160,128,114)(104,115,121,153), (1,22,136,95)(2,23,129,96)(3,24,130,89)(4,17,131,90)(5,18,132,91)(6,19,133,92)(7,20,134,93)(8,21,135,94)(9,97,76,126)(10,98,77,127)(11,99,78,128)(12,100,79,121)(13,101,80,122)(14,102,73,123)(15,103,74,124)(16,104,75,125)(25,139,37,43)(26,140,38,44)(27,141,39,45)(28,142,40,46)(29,143,33,47)(30,144,34,48)(31,137,35,41)(32,138,36,42)(49,114,110,156)(50,115,111,157)(51,116,112,158)(52,117,105,159)(53,118,106,160)(54,119,107,153)(55,120,108,154)(56,113,109,155)(57,147,69,83)(58,148,70,84)(59,149,71,85)(60,150,72,86)(61,151,65,87)(62,152,66,88)(63,145,67,81)(64,146,68,82), (1,12,57,143,113)(2,144,13,114,58)(3,115,137,59,14)(4,60,116,15,138)(5,16,61,139,117)(6,140,9,118,62)(7,119,141,63,10)(8,64,120,11,142)(17,150,112,103,36)(18,104,151,37,105)(19,38,97,106,152)(20,107,39,145,98)(21,146,108,99,40)(22,100,147,33,109)(23,34,101,110,148)(24,111,35,149,102)(25,52,91,125,87)(26,126,53,88,92)(27,81,127,93,54)(28,94,82,55,128)(29,56,95,121,83)(30,122,49,84,96)(31,85,123,89,50)(32,90,86,51,124)(41,71,73,130,157)(42,131,72,158,74)(43,159,132,75,65)(44,76,160,66,133)(45,67,77,134,153)(46,135,68,154,78)(47,155,136,79,69)(48,80,156,70,129), (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( (1,93,132,24)(2,17,133,94)(3,95,134,18)(4,19,135,96)(5,89,136,20)(6,21,129,90)(7,91,130,22)(8,23,131,92)(9,108,80,51)(10,52,73,109)(11,110,74,53)(12,54,75,111)(13,112,76,55)(14,56,77,105)(15,106,78,49)(16,50,79,107)(25,71,33,63)(26,64,34,72)(27,65,35,57)(28,58,36,66)(29,67,37,59)(30,60,38,68)(31,69,39,61)(32,62,40,70)(41,147,141,87)(42,88,142,148)(43,149,143,81)(44,82,144,150)(45,151,137,83)(46,84,138,152)(47,145,139,85)(48,86,140,146)(97,154,122,116)(98,117,123,155)(99,156,124,118)(100,119,125,157)(101,158,126,120)(102,113,127,159)(103,160,128,114)(104,115,121,153), (1,22,136,95)(2,23,129,96)(3,24,130,89)(4,17,131,90)(5,18,132,91)(6,19,133,92)(7,20,134,93)(8,21,135,94)(9,97,76,126)(10,98,77,127)(11,99,78,128)(12,100,79,121)(13,101,80,122)(14,102,73,123)(15,103,74,124)(16,104,75,125)(25,139,37,43)(26,140,38,44)(27,141,39,45)(28,142,40,46)(29,143,33,47)(30,144,34,48)(31,137,35,41)(32,138,36,42)(49,114,110,156)(50,115,111,157)(51,116,112,158)(52,117,105,159)(53,118,106,160)(54,119,107,153)(55,120,108,154)(56,113,109,155)(57,147,69,83)(58,148,70,84)(59,149,71,85)(60,150,72,86)(61,151,65,87)(62,152,66,88)(63,145,67,81)(64,146,68,82), (1,12,57,143,113)(2,144,13,114,58)(3,115,137,59,14)(4,60,116,15,138)(5,16,61,139,117)(6,140,9,118,62)(7,119,141,63,10)(8,64,120,11,142)(17,150,112,103,36)(18,104,151,37,105)(19,38,97,106,152)(20,107,39,145,98)(21,146,108,99,40)(22,100,147,33,109)(23,34,101,110,148)(24,111,35,149,102)(25,52,91,125,87)(26,126,53,88,92)(27,81,127,93,54)(28,94,82,55,128)(29,56,95,121,83)(30,122,49,84,96)(31,85,123,89,50)(32,90,86,51,124)(41,71,73,130,157)(42,131,72,158,74)(43,159,132,75,65)(44,76,160,66,133)(45,67,77,134,153)(46,135,68,154,78)(47,155,136,79,69)(48,80,156,70,129), (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([(1,93,132,24),(2,17,133,94),(3,95,134,18),(4,19,135,96),(5,89,136,20),(6,21,129,90),(7,91,130,22),(8,23,131,92),(9,108,80,51),(10,52,73,109),(11,110,74,53),(12,54,75,111),(13,112,76,55),(14,56,77,105),(15,106,78,49),(16,50,79,107),(25,71,33,63),(26,64,34,72),(27,65,35,57),(28,58,36,66),(29,67,37,59),(30,60,38,68),(31,69,39,61),(32,62,40,70),(41,147,141,87),(42,88,142,148),(43,149,143,81),(44,82,144,150),(45,151,137,83),(46,84,138,152),(47,145,139,85),(48,86,140,146),(97,154,122,116),(98,117,123,155),(99,156,124,118),(100,119,125,157),(101,158,126,120),(102,113,127,159),(103,160,128,114),(104,115,121,153)], [(1,22,136,95),(2,23,129,96),(3,24,130,89),(4,17,131,90),(5,18,132,91),(6,19,133,92),(7,20,134,93),(8,21,135,94),(9,97,76,126),(10,98,77,127),(11,99,78,128),(12,100,79,121),(13,101,80,122),(14,102,73,123),(15,103,74,124),(16,104,75,125),(25,139,37,43),(26,140,38,44),(27,141,39,45),(28,142,40,46),(29,143,33,47),(30,144,34,48),(31,137,35,41),(32,138,36,42),(49,114,110,156),(50,115,111,157),(51,116,112,158),(52,117,105,159),(53,118,106,160),(54,119,107,153),(55,120,108,154),(56,113,109,155),(57,147,69,83),(58,148,70,84),(59,149,71,85),(60,150,72,86),(61,151,65,87),(62,152,66,88),(63,145,67,81),(64,146,68,82)], [(1,12,57,143,113),(2,144,13,114,58),(3,115,137,59,14),(4,60,116,15,138),(5,16,61,139,117),(6,140,9,118,62),(7,119,141,63,10),(8,64,120,11,142),(17,150,112,103,36),(18,104,151,37,105),(19,38,97,106,152),(20,107,39,145,98),(21,146,108,99,40),(22,100,147,33,109),(23,34,101,110,148),(24,111,35,149,102),(25,52,91,125,87),(26,126,53,88,92),(27,81,127,93,54),(28,94,82,55,128),(29,56,95,121,83),(30,122,49,84,96),(31,85,123,89,50),(32,90,86,51,124),(41,71,73,130,157),(42,131,72,158,74),(43,159,132,75,65),(44,76,160,66,133),(45,67,77,134,153),(46,135,68,154,78),(47,155,136,79,69),(48,80,156,70,129)], [(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)])
Matrix representation ►G ⊆ GL6(𝔽41)
| 32 | 0 | 0 | 0 | 0 | 0 |
| 18 | 9 | 0 | 0 | 0 | 0 |
| 0 | 0 | 34 | 0 | 27 | 27 |
| 0 | 0 | 14 | 7 | 14 | 0 |
| 0 | 0 | 0 | 14 | 7 | 14 |
| 0 | 0 | 27 | 27 | 0 | 34 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 9 | 0 | 0 | 0 |
| 0 | 0 | 0 | 9 | 0 | 0 |
| 0 | 0 | 0 | 0 | 9 | 0 |
| 0 | 0 | 0 | 0 | 0 | 9 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 40 | 40 | 40 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 5 | 5 | 0 | 0 | 0 | 0 |
| 26 | 36 | 0 | 0 | 0 | 0 |
| 0 | 0 | 6 | 12 | 36 | 17 |
| 0 | 0 | 24 | 5 | 29 | 35 |
| 0 | 0 | 24 | 30 | 36 | 19 |
| 0 | 0 | 6 | 30 | 11 | 35 |
G:=sub<GL(6,GF(41))| [32,18,0,0,0,0,0,9,0,0,0,0,0,0,34,14,0,27,0,0,0,7,14,27,0,0,27,14,7,0,0,0,27,0,14,34],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,9,0,0,0,0,0,0,9,0,0,0,0,0,0,9,0,0,0,0,0,0,9],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,1,0,0,0,0,40,0,1,0,0,0,40,0,0,1,0,0,40,0,0,0],[5,26,0,0,0,0,5,36,0,0,0,0,0,0,6,24,24,6,0,0,12,5,30,30,0,0,36,29,36,11,0,0,17,35,19,35] >;
56 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | ··· | 4P | 4Q | 4R | 5 | 8A | ··· | 8P | 10A | 10B | 10C | 20A | ··· | 20L |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | 4 | 5 | 8 | ··· | 8 | 10 | 10 | 10 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 10 | 10 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 5 | ··· | 5 | 10 | 10 | 4 | 10 | ··· | 10 | 4 | 4 | 4 | 4 | ··· | 4 |
56 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | |||||||||
| image | C1 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | C8 | C4○D4 | M4(2) | F5 | C2×F5 | D5⋊C8 | C4.F5 | D10.C23 |
| kernel | C42.12F5 | C4×C5⋊C8 | C20⋊C8 | D10⋊C8 | D5×C42 | C4×Dic5 | C4×C20 | C2×C4×D5 | C4×D5 | Dic5 | C20 | C42 | C2×C4 | C4 | C4 | C2 |
| # reps | 1 | 2 | 2 | 2 | 1 | 2 | 2 | 4 | 16 | 4 | 4 | 1 | 3 | 4 | 4 | 4 |
In GAP, Magma, Sage, TeX
C_4^2._{12}F_5 % in TeX
G:=Group("C4^2.12F5"); // GroupNames label
G:=SmallGroup(320,1018);
// by ID
G=gap.SmallGroup(320,1018);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,253,120,268,136,6278,1595]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^5=1,d^4=a^2*b^2,a*b=b*a,a*c=c*a,d*a*d^-1=a^-1,b*c=c*b,b*d=d*b,d*c*d^-1=c^3>;
// generators/relations
×
⋊
ℤ
𝔽
○
≀