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
C1 — C5 — C10 — C2×C10 — C22×D5 — C2×C5⋊D4 — C20⋊2D4 — C42.117D10 |
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
(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