metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.117D10, C10.1062+ 1+4, (C4×D4)⋊25D5, (D4×C20)⋊27C2, (C4×D20)⋊34C2, C20⋊7D4⋊20C2, C20⋊2D4⋊11C2, C4⋊C4.320D10, C20⋊2Q8⋊26C2, (C2×D4).224D10, C4.66(C4○D20), C20.114(C4○D4), (C2×C20).165C23, (C4×C20).161C22, (C2×C10).107C24, C22⋊C4.119D10, (C22×C4).215D10, C2.19(D4⋊8D10), C4.118(D4⋊2D5), Dic5.5D4⋊11C2, (D4×C10).266C22, (C2×D20).222C22, C23.21D10⋊9C2, C4⋊Dic5.302C22, (C22×D5).41C23, C22.132(C23×D5), C23.104(C22×D5), D10⋊C4.55C22, (C22×C20).111C22, (C22×C10).177C23, C5⋊2(C22.49C24), (C2×Dic10).30C22, (C4×Dic5).226C22, (C2×Dic5).219C23, C23.D5.108C22, C4⋊C4⋊7D5⋊16C2, C2.56(C2×C4○D20), C10.49(C2×C4○D4), (C2×C4×D5).77C22, C2.24(C2×D4⋊2D5), (C5×C4⋊C4).335C22, (C2×C4).163(C22×D5), (C2×C5⋊D4).20C22, (C5×C22⋊C4).130C22, SmallGroup(320,1235)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42.117D10
G = < a,b,c,d | a4=b4=c10=1, d2=a2b2, ab=ba, cac-1=dad-1=a-1, bc=cb, dbd-1=b-1, dcd-1=a2b2c-1 >
Subgroups: 838 in 236 conjugacy classes, 99 normal (29 characteristic)
C1, C2 [×3], C2 [×4], C4 [×4], C4 [×9], C22, C22 [×12], C5, C2×C4 [×3], C2×C4 [×2], C2×C4 [×14], D4 [×8], Q8 [×2], C23 [×2], C23 [×2], D5 [×2], C10 [×3], C10 [×2], C42, C42 [×4], C22⋊C4 [×2], C22⋊C4 [×10], C4⋊C4, C4⋊C4 [×5], C22×C4 [×2], C22×C4 [×2], C2×D4, C2×D4 [×5], C2×Q8 [×2], Dic5 [×6], C20 [×4], C20 [×3], D10 [×6], C2×C10, C2×C10 [×6], C42⋊C2 [×4], C4×D4, C4×D4, C4⋊D4 [×4], C4.4D4 [×4], C4⋊Q8, Dic10 [×2], C4×D5 [×4], D20 [×2], C2×Dic5 [×6], C5⋊D4 [×4], C2×C20 [×3], C2×C20 [×2], C2×C20 [×4], C5×D4 [×2], C22×D5 [×2], C22×C10 [×2], C22.49C24, C4×Dic5 [×4], C4⋊Dic5, C4⋊Dic5 [×4], D10⋊C4 [×6], C23.D5 [×4], C4×C20, C5×C22⋊C4 [×2], C5×C4⋊C4, C2×Dic10 [×2], C2×C4×D5 [×2], C2×D20, C2×C5⋊D4 [×4], C22×C20 [×2], D4×C10, C20⋊2Q8, C4×D20, Dic5.5D4 [×4], C4⋊C4⋊7D5 [×2], C23.21D10 [×2], C20⋊7D4 [×2], C20⋊2D4 [×2], D4×C20, C42.117D10
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], C22.49C24, C4○D20 [×2], D4⋊2D5 [×2], C23×D5, C2×C4○D20, C2×D4⋊2D5, D4⋊8D10, C42.117D10
►On 160 points
Generators in S160
(1 116 91 33)(2 34 92 117)(3 118 93 35)(4 36 94 119)(5 120 95 37)(6 38 96 111)(7 112 97 39)(8 40 98 113)(9 114 99 31)(10 32 100 115)(11 27 42 110)(12 101 43 28)(13 29 44 102)(14 103 45 30)(15 21 46 104)(16 105 47 22)(17 23 48 106)(18 107 49 24)(19 25 50 108)(20 109 41 26)(51 76 124 149)(52 150 125 77)(53 78 126 141)(54 142 127 79)(55 80 128 143)(56 144 129 71)(57 72 130 145)(58 146 121 73)(59 74 122 147)(60 148 123 75)(61 82 134 159)(62 160 135 83)(63 84 136 151)(64 152 137 85)(65 86 138 153)(66 154 139 87)(67 88 140 155)(68 156 131 89)(69 90 132 157)(70 158 133 81)
(1 70 60 28)(2 61 51 29)(3 62 52 30)(4 63 53 21)(5 64 54 22)(6 65 55 23)(7 66 56 24)(8 67 57 25)(9 68 58 26)(10 69 59 27)(11 115 157 147)(12 116 158 148)(13 117 159 149)(14 118 160 150)(15 119 151 141)(16 120 152 142)(17 111 153 143)(18 112 154 144)(19 113 155 145)(20 114 156 146)(31 89 73 41)(32 90 74 42)(33 81 75 43)(34 82 76 44)(35 83 77 45)(36 84 78 46)(37 85 79 47)(38 86 80 48)(39 87 71 49)(40 88 72 50)(91 133 123 101)(92 134 124 102)(93 135 125 103)(94 136 126 104)(95 137 127 105)(96 138 128 106)(97 139 129 107)(98 140 130 108)(99 131 121 109)(100 132 122 110)
(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 10 123 122)(2 121 124 9)(3 8 125 130)(4 129 126 7)(5 6 127 128)(11 43 90 158)(12 157 81 42)(13 41 82 156)(14 155 83 50)(15 49 84 154)(16 153 85 48)(17 47 86 152)(18 151 87 46)(19 45 88 160)(20 159 89 44)(21 107 136 66)(22 65 137 106)(23 105 138 64)(24 63 139 104)(25 103 140 62)(26 61 131 102)(27 101 132 70)(28 69 133 110)(29 109 134 68)(30 67 135 108)(31 34 146 149)(32 148 147 33)(35 40 150 145)(36 144 141 39)(37 38 142 143)(51 99 92 58)(52 57 93 98)(53 97 94 56)(54 55 95 96)(59 91 100 60)(71 78 112 119)(72 118 113 77)(73 76 114 117)(74 116 115 75)(79 80 120 111)
G:=sub<Sym(160)| (1,116,91,33)(2,34,92,117)(3,118,93,35)(4,36,94,119)(5,120,95,37)(6,38,96,111)(7,112,97,39)(8,40,98,113)(9,114,99,31)(10,32,100,115)(11,27,42,110)(12,101,43,28)(13,29,44,102)(14,103,45,30)(15,21,46,104)(16,105,47,22)(17,23,48,106)(18,107,49,24)(19,25,50,108)(20,109,41,26)(51,76,124,149)(52,150,125,77)(53,78,126,141)(54,142,127,79)(55,80,128,143)(56,144,129,71)(57,72,130,145)(58,146,121,73)(59,74,122,147)(60,148,123,75)(61,82,134,159)(62,160,135,83)(63,84,136,151)(64,152,137,85)(65,86,138,153)(66,154,139,87)(67,88,140,155)(68,156,131,89)(69,90,132,157)(70,158,133,81), (1,70,60,28)(2,61,51,29)(3,62,52,30)(4,63,53,21)(5,64,54,22)(6,65,55,23)(7,66,56,24)(8,67,57,25)(9,68,58,26)(10,69,59,27)(11,115,157,147)(12,116,158,148)(13,117,159,149)(14,118,160,150)(15,119,151,141)(16,120,152,142)(17,111,153,143)(18,112,154,144)(19,113,155,145)(20,114,156,146)(31,89,73,41)(32,90,74,42)(33,81,75,43)(34,82,76,44)(35,83,77,45)(36,84,78,46)(37,85,79,47)(38,86,80,48)(39,87,71,49)(40,88,72,50)(91,133,123,101)(92,134,124,102)(93,135,125,103)(94,136,126,104)(95,137,127,105)(96,138,128,106)(97,139,129,107)(98,140,130,108)(99,131,121,109)(100,132,122,110), (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), (1,10,123,122)(2,121,124,9)(3,8,125,130)(4,129,126,7)(5,6,127,128)(11,43,90,158)(12,157,81,42)(13,41,82,156)(14,155,83,50)(15,49,84,154)(16,153,85,48)(17,47,86,152)(18,151,87,46)(19,45,88,160)(20,159,89,44)(21,107,136,66)(22,65,137,106)(23,105,138,64)(24,63,139,104)(25,103,140,62)(26,61,131,102)(27,101,132,70)(28,69,133,110)(29,109,134,68)(30,67,135,108)(31,34,146,149)(32,148,147,33)(35,40,150,145)(36,144,141,39)(37,38,142,143)(51,99,92,58)(52,57,93,98)(53,97,94,56)(54,55,95,96)(59,91,100,60)(71,78,112,119)(72,118,113,77)(73,76,114,117)(74,116,115,75)(79,80,120,111)>;
G:=Group( (1,116,91,33)(2,34,92,117)(3,118,93,35)(4,36,94,119)(5,120,95,37)(6,38,96,111)(7,112,97,39)(8,40,98,113)(9,114,99,31)(10,32,100,115)(11,27,42,110)(12,101,43,28)(13,29,44,102)(14,103,45,30)(15,21,46,104)(16,105,47,22)(17,23,48,106)(18,107,49,24)(19,25,50,108)(20,109,41,26)(51,76,124,149)(52,150,125,77)(53,78,126,141)(54,142,127,79)(55,80,128,143)(56,144,129,71)(57,72,130,145)(58,146,121,73)(59,74,122,147)(60,148,123,75)(61,82,134,159)(62,160,135,83)(63,84,136,151)(64,152,137,85)(65,86,138,153)(66,154,139,87)(67,88,140,155)(68,156,131,89)(69,90,132,157)(70,158,133,81), (1,70,60,28)(2,61,51,29)(3,62,52,30)(4,63,53,21)(5,64,54,22)(6,65,55,23)(7,66,56,24)(8,67,57,25)(9,68,58,26)(10,69,59,27)(11,115,157,147)(12,116,158,148)(13,117,159,149)(14,118,160,150)(15,119,151,141)(16,120,152,142)(17,111,153,143)(18,112,154,144)(19,113,155,145)(20,114,156,146)(31,89,73,41)(32,90,74,42)(33,81,75,43)(34,82,76,44)(35,83,77,45)(36,84,78,46)(37,85,79,47)(38,86,80,48)(39,87,71,49)(40,88,72,50)(91,133,123,101)(92,134,124,102)(93,135,125,103)(94,136,126,104)(95,137,127,105)(96,138,128,106)(97,139,129,107)(98,140,130,108)(99,131,121,109)(100,132,122,110), (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), (1,10,123,122)(2,121,124,9)(3,8,125,130)(4,129,126,7)(5,6,127,128)(11,43,90,158)(12,157,81,42)(13,41,82,156)(14,155,83,50)(15,49,84,154)(16,153,85,48)(17,47,86,152)(18,151,87,46)(19,45,88,160)(20,159,89,44)(21,107,136,66)(22,65,137,106)(23,105,138,64)(24,63,139,104)(25,103,140,62)(26,61,131,102)(27,101,132,70)(28,69,133,110)(29,109,134,68)(30,67,135,108)(31,34,146,149)(32,148,147,33)(35,40,150,145)(36,144,141,39)(37,38,142,143)(51,99,92,58)(52,57,93,98)(53,97,94,56)(54,55,95,96)(59,91,100,60)(71,78,112,119)(72,118,113,77)(73,76,114,117)(74,116,115,75)(79,80,120,111) );
G=PermutationGroup([(1,116,91,33),(2,34,92,117),(3,118,93,35),(4,36,94,119),(5,120,95,37),(6,38,96,111),(7,112,97,39),(8,40,98,113),(9,114,99,31),(10,32,100,115),(11,27,42,110),(12,101,43,28),(13,29,44,102),(14,103,45,30),(15,21,46,104),(16,105,47,22),(17,23,48,106),(18,107,49,24),(19,25,50,108),(20,109,41,26),(51,76,124,149),(52,150,125,77),(53,78,126,141),(54,142,127,79),(55,80,128,143),(56,144,129,71),(57,72,130,145),(58,146,121,73),(59,74,122,147),(60,148,123,75),(61,82,134,159),(62,160,135,83),(63,84,136,151),(64,152,137,85),(65,86,138,153),(66,154,139,87),(67,88,140,155),(68,156,131,89),(69,90,132,157),(70,158,133,81)], [(1,70,60,28),(2,61,51,29),(3,62,52,30),(4,63,53,21),(5,64,54,22),(6,65,55,23),(7,66,56,24),(8,67,57,25),(9,68,58,26),(10,69,59,27),(11,115,157,147),(12,116,158,148),(13,117,159,149),(14,118,160,150),(15,119,151,141),(16,120,152,142),(17,111,153,143),(18,112,154,144),(19,113,155,145),(20,114,156,146),(31,89,73,41),(32,90,74,42),(33,81,75,43),(34,82,76,44),(35,83,77,45),(36,84,78,46),(37,85,79,47),(38,86,80,48),(39,87,71,49),(40,88,72,50),(91,133,123,101),(92,134,124,102),(93,135,125,103),(94,136,126,104),(95,137,127,105),(96,138,128,106),(97,139,129,107),(98,140,130,108),(99,131,121,109),(100,132,122,110)], [(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)], [(1,10,123,122),(2,121,124,9),(3,8,125,130),(4,129,126,7),(5,6,127,128),(11,43,90,158),(12,157,81,42),(13,41,82,156),(14,155,83,50),(15,49,84,154),(16,153,85,48),(17,47,86,152),(18,151,87,46),(19,45,88,160),(20,159,89,44),(21,107,136,66),(22,65,137,106),(23,105,138,64),(24,63,139,104),(25,103,140,62),(26,61,131,102),(27,101,132,70),(28,69,133,110),(29,109,134,68),(30,67,135,108),(31,34,146,149),(32,148,147,33),(35,40,150,145),(36,144,141,39),(37,38,142,143),(51,99,92,58),(52,57,93,98),(53,97,94,56),(54,55,95,96),(59,91,100,60),(71,78,112,119),(72,118,113,77),(73,76,114,117),(74,116,115,75),(79,80,120,111)])
65 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | ··· | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 4O | 4P | 4Q | 5A | 5B | 10A | ··· | 10F | 10G | ··· | 10N | 20A | ··· | 20H | 20I | ··· | 20X |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 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 | 20 | 20 | 2 | ··· | 2 | 4 | 10 | 10 | 10 | 10 | 20 | 20 | 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⋊8D10 |
| kernel | C42.117D10 | C20⋊2Q8 | C4×D20 | Dic5.5D4 | C4⋊C4⋊7D5 | C23.21D10 | C20⋊7D4 | C20⋊2D4 | 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 |
Matrix representation of C42.117D10 ►in GL4(𝔽41) generated by
| 40 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 0 | 0 | 32 | 0 |
| 0 | 0 | 29 | 9 |
| 39 | 28 | 0 | 0 |
| 13 | 2 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 20 | 21 | 0 | 0 |
| 20 | 23 | 0 | 0 |
| 0 | 0 | 15 | 39 |
| 0 | 0 | 30 | 26 |
| 20 | 21 | 0 | 0 |
| 18 | 21 | 0 | 0 |
| 0 | 0 | 26 | 2 |
| 0 | 0 | 10 | 15 |
G:=sub<GL(4,GF(41))| [40,0,0,0,0,40,0,0,0,0,32,29,0,0,0,9],[39,13,0,0,28,2,0,0,0,0,1,0,0,0,0,1],[20,20,0,0,21,23,0,0,0,0,15,30,0,0,39,26],[20,18,0,0,21,21,0,0,0,0,26,10,0,0,2,15] >;
C42.117D10 in GAP, Magma, Sage, TeX
C_4^2._{117}D_{10} % in TeX
G:=Group("C4^2.117D10"); // GroupNames label
G:=SmallGroup(320,1235);
// by ID
G=gap.SmallGroup(320,1235);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,477,758,219,1571,192,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^10=1,d^2=a^2*b^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=a^2*b^2*c^-1>;
// generators/relations