metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.106D10, C10.572- (1+4), (C4×D4).13D5, C4⋊C4.314D10, C20⋊2Q8⋊22C2, (D4×C20).14C2, (C4×Dic10)⋊28C2, (C2×D4).210D10, C4.15(C4○D20), (C2×C10).86C24, Dic5⋊3Q8⋊14C2, C20.109(C4○D4), C20.48D4⋊19C2, (C2×C20).155C23, (C4×C20).148C22, C22⋊C4.107D10, C20.17D4.9C2, C23.D10⋊7C2, (C22×C4).205D10, C4.116(D4⋊2D5), C23.92(C22×D5), (D4×C10).250C22, C23.21D10⋊7C2, C4⋊Dic5.297C22, (C2×Dic5).36C23, (C4×Dic5).81C22, C22.114(C23×D5), (C22×C10).156C23, (C22×C20).105C22, C5⋊2(C22.50C24), C23.D5.103C22, C2.15(D4.10D10), (C2×Dic10).245C22, C10.D4.109C22, C10.38(C2×C4○D4), C2.42(C2×C4○D20), C2.20(C2×D4⋊2D5), (C5×C4⋊C4).322C22, (C2×C4).281(C22×D5), (C5×C22⋊C4).120C22, SmallGroup(320,1214)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 598 in 212 conjugacy classes, 99 normal (29 characteristic)
C1, C2 [×3], C2 [×2], C4 [×4], C4 [×11], C22, C22 [×6], C5, C2×C4 [×3], C2×C4 [×2], C2×C4 [×12], D4 [×2], Q8 [×6], C23 [×2], C10 [×3], C10 [×2], C42, C42 [×6], C22⋊C4 [×2], C22⋊C4 [×8], C4⋊C4, C4⋊C4 [×11], C22×C4 [×2], C2×D4, C2×Q8 [×3], Dic5 [×8], C20 [×4], C20 [×3], C2×C10, C2×C10 [×6], C42⋊C2 [×2], C4×D4, C4×Q8 [×3], C22⋊Q8 [×2], C4.4D4 [×2], C42⋊2C2 [×4], C4⋊Q8, Dic10 [×6], C2×Dic5 [×8], C2×C20 [×3], C2×C20 [×2], C2×C20 [×4], C5×D4 [×2], C22×C10 [×2], C22.50C24, C4×Dic5 [×6], C10.D4 [×6], C4⋊Dic5, C4⋊Dic5 [×4], C23.D5 [×8], C4×C20, C5×C22⋊C4 [×2], C5×C4⋊C4, C2×Dic10, C2×Dic10 [×2], C22×C20 [×2], D4×C10, C4×Dic10, C20⋊2Q8, C23.D10 [×4], Dic5⋊3Q8 [×2], C20.48D4 [×2], C23.21D10 [×2], C20.17D4 [×2], D4×C20, C42.106D10
Quotients:
C1, C2 [×15], C22 [×35], C23 [×15], D5, C4○D4 [×4], C24, D10 [×7], C2×C4○D4 [×2], 2- (1+4), C22×D5 [×7], C22.50C24, C4○D20 [×2], D4⋊2D5 [×2], C23×D5, C2×C4○D20, C2×D4⋊2D5, D4.10D10, C42.106D10
Generators and relations
G = < a,b,c,d | a4=b4=1, c10=b2, d2=a2, ab=ba, cac-1=dad-1=a-1, bc=cb, dbd-1=b-1, dcd-1=c9 >
►On 160 points
Generators in S160
(1 107 27 49)(2 50 28 108)(3 109 29 51)(4 52 30 110)(5 111 31 53)(6 54 32 112)(7 113 33 55)(8 56 34 114)(9 115 35 57)(10 58 36 116)(11 117 37 59)(12 60 38 118)(13 119 39 41)(14 42 40 120)(15 101 21 43)(16 44 22 102)(17 103 23 45)(18 46 24 104)(19 105 25 47)(20 48 26 106)(61 144 131 90)(62 91 132 145)(63 146 133 92)(64 93 134 147)(65 148 135 94)(66 95 136 149)(67 150 137 96)(68 97 138 151)(69 152 139 98)(70 99 140 153)(71 154 121 100)(72 81 122 155)(73 156 123 82)(74 83 124 157)(75 158 125 84)(76 85 126 159)(77 160 127 86)(78 87 128 141)(79 142 129 88)(80 89 130 143)
(1 102 11 112)(2 103 12 113)(3 104 13 114)(4 105 14 115)(5 106 15 116)(6 107 16 117)(7 108 17 118)(8 109 18 119)(9 110 19 120)(10 111 20 101)(21 58 31 48)(22 59 32 49)(23 60 33 50)(24 41 34 51)(25 42 35 52)(26 43 36 53)(27 44 37 54)(28 45 38 55)(29 46 39 56)(30 47 40 57)(61 95 71 85)(62 96 72 86)(63 97 73 87)(64 98 74 88)(65 99 75 89)(66 100 76 90)(67 81 77 91)(68 82 78 92)(69 83 79 93)(70 84 80 94)(121 159 131 149)(122 160 132 150)(123 141 133 151)(124 142 134 152)(125 143 135 153)(126 144 136 154)(127 145 137 155)(128 146 138 156)(129 147 139 157)(130 148 140 158)
(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)
(1 79 27 129)(2 68 28 138)(3 77 29 127)(4 66 30 136)(5 75 31 125)(6 64 32 134)(7 73 33 123)(8 62 34 132)(9 71 35 121)(10 80 36 130)(11 69 37 139)(12 78 38 128)(13 67 39 137)(14 76 40 126)(15 65 21 135)(16 74 22 124)(17 63 23 133)(18 72 24 122)(19 61 25 131)(20 70 26 140)(41 150 119 96)(42 159 120 85)(43 148 101 94)(44 157 102 83)(45 146 103 92)(46 155 104 81)(47 144 105 90)(48 153 106 99)(49 142 107 88)(50 151 108 97)(51 160 109 86)(52 149 110 95)(53 158 111 84)(54 147 112 93)(55 156 113 82)(56 145 114 91)(57 154 115 100)(58 143 116 89)(59 152 117 98)(60 141 118 87)
G:=sub<Sym(160)| (1,107,27,49)(2,50,28,108)(3,109,29,51)(4,52,30,110)(5,111,31,53)(6,54,32,112)(7,113,33,55)(8,56,34,114)(9,115,35,57)(10,58,36,116)(11,117,37,59)(12,60,38,118)(13,119,39,41)(14,42,40,120)(15,101,21,43)(16,44,22,102)(17,103,23,45)(18,46,24,104)(19,105,25,47)(20,48,26,106)(61,144,131,90)(62,91,132,145)(63,146,133,92)(64,93,134,147)(65,148,135,94)(66,95,136,149)(67,150,137,96)(68,97,138,151)(69,152,139,98)(70,99,140,153)(71,154,121,100)(72,81,122,155)(73,156,123,82)(74,83,124,157)(75,158,125,84)(76,85,126,159)(77,160,127,86)(78,87,128,141)(79,142,129,88)(80,89,130,143), (1,102,11,112)(2,103,12,113)(3,104,13,114)(4,105,14,115)(5,106,15,116)(6,107,16,117)(7,108,17,118)(8,109,18,119)(9,110,19,120)(10,111,20,101)(21,58,31,48)(22,59,32,49)(23,60,33,50)(24,41,34,51)(25,42,35,52)(26,43,36,53)(27,44,37,54)(28,45,38,55)(29,46,39,56)(30,47,40,57)(61,95,71,85)(62,96,72,86)(63,97,73,87)(64,98,74,88)(65,99,75,89)(66,100,76,90)(67,81,77,91)(68,82,78,92)(69,83,79,93)(70,84,80,94)(121,159,131,149)(122,160,132,150)(123,141,133,151)(124,142,134,152)(125,143,135,153)(126,144,136,154)(127,145,137,155)(128,146,138,156)(129,147,139,157)(130,148,140,158), (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), (1,79,27,129)(2,68,28,138)(3,77,29,127)(4,66,30,136)(5,75,31,125)(6,64,32,134)(7,73,33,123)(8,62,34,132)(9,71,35,121)(10,80,36,130)(11,69,37,139)(12,78,38,128)(13,67,39,137)(14,76,40,126)(15,65,21,135)(16,74,22,124)(17,63,23,133)(18,72,24,122)(19,61,25,131)(20,70,26,140)(41,150,119,96)(42,159,120,85)(43,148,101,94)(44,157,102,83)(45,146,103,92)(46,155,104,81)(47,144,105,90)(48,153,106,99)(49,142,107,88)(50,151,108,97)(51,160,109,86)(52,149,110,95)(53,158,111,84)(54,147,112,93)(55,156,113,82)(56,145,114,91)(57,154,115,100)(58,143,116,89)(59,152,117,98)(60,141,118,87)>;
G:=Group( (1,107,27,49)(2,50,28,108)(3,109,29,51)(4,52,30,110)(5,111,31,53)(6,54,32,112)(7,113,33,55)(8,56,34,114)(9,115,35,57)(10,58,36,116)(11,117,37,59)(12,60,38,118)(13,119,39,41)(14,42,40,120)(15,101,21,43)(16,44,22,102)(17,103,23,45)(18,46,24,104)(19,105,25,47)(20,48,26,106)(61,144,131,90)(62,91,132,145)(63,146,133,92)(64,93,134,147)(65,148,135,94)(66,95,136,149)(67,150,137,96)(68,97,138,151)(69,152,139,98)(70,99,140,153)(71,154,121,100)(72,81,122,155)(73,156,123,82)(74,83,124,157)(75,158,125,84)(76,85,126,159)(77,160,127,86)(78,87,128,141)(79,142,129,88)(80,89,130,143), (1,102,11,112)(2,103,12,113)(3,104,13,114)(4,105,14,115)(5,106,15,116)(6,107,16,117)(7,108,17,118)(8,109,18,119)(9,110,19,120)(10,111,20,101)(21,58,31,48)(22,59,32,49)(23,60,33,50)(24,41,34,51)(25,42,35,52)(26,43,36,53)(27,44,37,54)(28,45,38,55)(29,46,39,56)(30,47,40,57)(61,95,71,85)(62,96,72,86)(63,97,73,87)(64,98,74,88)(65,99,75,89)(66,100,76,90)(67,81,77,91)(68,82,78,92)(69,83,79,93)(70,84,80,94)(121,159,131,149)(122,160,132,150)(123,141,133,151)(124,142,134,152)(125,143,135,153)(126,144,136,154)(127,145,137,155)(128,146,138,156)(129,147,139,157)(130,148,140,158), (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), (1,79,27,129)(2,68,28,138)(3,77,29,127)(4,66,30,136)(5,75,31,125)(6,64,32,134)(7,73,33,123)(8,62,34,132)(9,71,35,121)(10,80,36,130)(11,69,37,139)(12,78,38,128)(13,67,39,137)(14,76,40,126)(15,65,21,135)(16,74,22,124)(17,63,23,133)(18,72,24,122)(19,61,25,131)(20,70,26,140)(41,150,119,96)(42,159,120,85)(43,148,101,94)(44,157,102,83)(45,146,103,92)(46,155,104,81)(47,144,105,90)(48,153,106,99)(49,142,107,88)(50,151,108,97)(51,160,109,86)(52,149,110,95)(53,158,111,84)(54,147,112,93)(55,156,113,82)(56,145,114,91)(57,154,115,100)(58,143,116,89)(59,152,117,98)(60,141,118,87) );
G=PermutationGroup([(1,107,27,49),(2,50,28,108),(3,109,29,51),(4,52,30,110),(5,111,31,53),(6,54,32,112),(7,113,33,55),(8,56,34,114),(9,115,35,57),(10,58,36,116),(11,117,37,59),(12,60,38,118),(13,119,39,41),(14,42,40,120),(15,101,21,43),(16,44,22,102),(17,103,23,45),(18,46,24,104),(19,105,25,47),(20,48,26,106),(61,144,131,90),(62,91,132,145),(63,146,133,92),(64,93,134,147),(65,148,135,94),(66,95,136,149),(67,150,137,96),(68,97,138,151),(69,152,139,98),(70,99,140,153),(71,154,121,100),(72,81,122,155),(73,156,123,82),(74,83,124,157),(75,158,125,84),(76,85,126,159),(77,160,127,86),(78,87,128,141),(79,142,129,88),(80,89,130,143)], [(1,102,11,112),(2,103,12,113),(3,104,13,114),(4,105,14,115),(5,106,15,116),(6,107,16,117),(7,108,17,118),(8,109,18,119),(9,110,19,120),(10,111,20,101),(21,58,31,48),(22,59,32,49),(23,60,33,50),(24,41,34,51),(25,42,35,52),(26,43,36,53),(27,44,37,54),(28,45,38,55),(29,46,39,56),(30,47,40,57),(61,95,71,85),(62,96,72,86),(63,97,73,87),(64,98,74,88),(65,99,75,89),(66,100,76,90),(67,81,77,91),(68,82,78,92),(69,83,79,93),(70,84,80,94),(121,159,131,149),(122,160,132,150),(123,141,133,151),(124,142,134,152),(125,143,135,153),(126,144,136,154),(127,145,137,155),(128,146,138,156),(129,147,139,157),(130,148,140,158)], [(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)], [(1,79,27,129),(2,68,28,138),(3,77,29,127),(4,66,30,136),(5,75,31,125),(6,64,32,134),(7,73,33,123),(8,62,34,132),(9,71,35,121),(10,80,36,130),(11,69,37,139),(12,78,38,128),(13,67,39,137),(14,76,40,126),(15,65,21,135),(16,74,22,124),(17,63,23,133),(18,72,24,122),(19,61,25,131),(20,70,26,140),(41,150,119,96),(42,159,120,85),(43,148,101,94),(44,157,102,83),(45,146,103,92),(46,155,104,81),(47,144,105,90),(48,153,106,99),(49,142,107,88),(50,151,108,97),(51,160,109,86),(52,149,110,95),(53,158,111,84),(54,147,112,93),(55,156,113,82),(56,145,114,91),(57,154,115,100),(58,143,116,89),(59,152,117,98),(60,141,118,87)])
Matrix representation ►G ⊆ GL4(𝔽41) generated by
| 0 | 40 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 40 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 9 | 0 |
| 0 | 0 | 31 | 32 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 21 | 0 |
| 0 | 0 | 10 | 39 |
| 0 | 9 | 0 | 0 |
| 9 | 0 | 0 | 0 |
| 0 | 0 | 39 | 21 |
| 0 | 0 | 35 | 2 |
G:=sub<GL(4,GF(41))| [0,1,0,0,40,0,0,0,0,0,40,0,0,0,0,40],[1,0,0,0,0,1,0,0,0,0,9,31,0,0,0,32],[0,1,0,0,1,0,0,0,0,0,21,10,0,0,0,39],[0,9,0,0,9,0,0,0,0,0,39,35,0,0,21,2] >;
65 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | ··· | 4H | 4I | 4J | 4K | 4L | 4M | 4N | ··· | 4S | 5A | 5B | 10A | ··· | 10F | 10G | ··· | 10N | 20A | ··· | 20H | 20I | ··· | 20X |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 5 | 5 | 10 | ··· | 10 | 10 | ··· | 10 | 20 | ··· | 20 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 2 | ··· | 2 | 4 | 10 | 10 | 10 | 10 | 20 | ··· | 20 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
65 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | - | - | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D5 | C4○D4 | D10 | D10 | D10 | D10 | D10 | C4○D20 | 2- (1+4) | D4⋊2D5 | D4.10D10 |
| kernel | C42.106D10 | C4×Dic10 | C20⋊2Q8 | C23.D10 | Dic5⋊3Q8 | C20.48D4 | C23.21D10 | C20.17D4 | D4×C20 | C4×D4 | C20 | C42 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×D4 | C4 | C10 | C4 | C2 |
| # reps | 1 | 1 | 1 | 4 | 2 | 2 | 2 | 2 | 1 | 2 | 8 | 2 | 4 | 2 | 4 | 2 | 16 | 1 | 4 | 4 |
In GAP, Magma, Sage, TeX
C_4^2._{106}D_{10} % in TeX
G:=Group("C4^2.106D10"); // GroupNames label
G:=SmallGroup(320,1214);
// by ID
G=gap.SmallGroup(320,1214);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,224,120,219,268,1571,192,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^10=b^2,d^2=a^2,a*b=b*a,c*a*c^-1=d*a*d^-1=a^-1,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=c^9>;
// generators/relations
×
⋊
ℤ
𝔽
○
≀