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, C2, C4, C4, C22, C22, C5, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, D5, C10, C10, C42, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, Dic5, C20, C20, D10, C2×C10, C2×C10, C42⋊C2, C4×D4, C4×D4, C4⋊D4, C4.4D4, C4⋊Q8, Dic10, C4×D5, D20, C2×Dic5, C5⋊D4, C2×C20, C2×C20, C2×C20, C5×D4, C22×D5, C22×C10, C22.49C24, C4×Dic5, C4⋊Dic5, C4⋊Dic5, D10⋊C4, C23.D5, C4×C20, C5×C22⋊C4, C5×C4⋊C4, C2×Dic10, C2×C4×D5, C2×D20, C2×C5⋊D4, C22×C20, D4×C10, C20⋊2Q8, C4×D20, Dic5.5D4, C4⋊C4⋊7D5, C23.21D10, C20⋊7D4, C20⋊2D4, D4×C20, C42.117D10
Quotients: C1, C2, C22, C23, D5, C4○D4, C24, D10, C2×C4○D4, 2+ 1+4, C22×D5, C22.49C24, C4○D20, D4⋊2D5, C23×D5, C2×C4○D20, C2×D4⋊2D5, D4⋊8D10, C42.117D10
►On 160 points
Generators in S160
(1 120 95 37)(2 38 96 111)(3 112 97 39)(4 40 98 113)(5 114 99 31)(6 32 100 115)(7 116 91 33)(8 34 92 117)(9 118 93 35)(10 36 94 119)(11 109 41 26)(12 27 42 110)(13 101 43 28)(14 29 44 102)(15 103 45 30)(16 21 46 104)(17 105 47 22)(18 23 48 106)(19 107 49 24)(20 25 50 108)(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 22)(2 61 51 23)(3 62 52 24)(4 63 53 25)(5 64 54 26)(6 65 55 27)(7 66 56 28)(8 67 57 29)(9 68 58 30)(10 69 59 21)(11 114 152 142)(12 115 153 143)(13 116 154 144)(14 117 155 145)(15 118 156 146)(16 119 157 147)(17 120 158 148)(18 111 159 149)(19 112 160 150)(20 113 151 141)(31 85 79 41)(32 86 80 42)(33 87 71 43)(34 88 72 44)(35 89 73 45)(36 90 74 46)(37 81 75 47)(38 82 76 48)(39 83 77 49)(40 84 78 50)(91 139 129 101)(92 140 130 102)(93 131 121 103)(94 132 122 104)(95 133 123 105)(96 134 124 106)(97 135 125 107)(98 136 126 108)(99 137 127 109)(100 138 128 110)
(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)(11 153 85 42)(12 41 86 152)(13 151 87 50)(14 49 88 160)(15 159 89 48)(16 47 90 158)(17 157 81 46)(18 45 82 156)(19 155 83 44)(20 43 84 154)(21 105 132 70)(22 69 133 104)(23 103 134 68)(24 67 135 102)(25 101 136 66)(26 65 137 110)(27 109 138 64)(28 63 139 108)(29 107 140 62)(30 61 131 106)(31 32 142 143)(33 40 144 141)(34 150 145 39)(35 38 146 149)(36 148 147 37)(51 93 96 58)(52 57 97 92)(53 91 98 56)(54 55 99 100)(59 95 94 60)(71 78 116 113)(72 112 117 77)(73 76 118 111)(74 120 119 75)(79 80 114 115)
G:=sub<Sym(160)| (1,120,95,37)(2,38,96,111)(3,112,97,39)(4,40,98,113)(5,114,99,31)(6,32,100,115)(7,116,91,33)(8,34,92,117)(9,118,93,35)(10,36,94,119)(11,109,41,26)(12,27,42,110)(13,101,43,28)(14,29,44,102)(15,103,45,30)(16,21,46,104)(17,105,47,22)(18,23,48,106)(19,107,49,24)(20,25,50,108)(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), (1,70,60,22)(2,61,51,23)(3,62,52,24)(4,63,53,25)(5,64,54,26)(6,65,55,27)(7,66,56,28)(8,67,57,29)(9,68,58,30)(10,69,59,21)(11,114,152,142)(12,115,153,143)(13,116,154,144)(14,117,155,145)(15,118,156,146)(16,119,157,147)(17,120,158,148)(18,111,159,149)(19,112,160,150)(20,113,151,141)(31,85,79,41)(32,86,80,42)(33,87,71,43)(34,88,72,44)(35,89,73,45)(36,90,74,46)(37,81,75,47)(38,82,76,48)(39,83,77,49)(40,84,78,50)(91,139,129,101)(92,140,130,102)(93,131,121,103)(94,132,122,104)(95,133,123,105)(96,134,124,106)(97,135,125,107)(98,136,126,108)(99,137,127,109)(100,138,128,110), (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)(11,153,85,42)(12,41,86,152)(13,151,87,50)(14,49,88,160)(15,159,89,48)(16,47,90,158)(17,157,81,46)(18,45,82,156)(19,155,83,44)(20,43,84,154)(21,105,132,70)(22,69,133,104)(23,103,134,68)(24,67,135,102)(25,101,136,66)(26,65,137,110)(27,109,138,64)(28,63,139,108)(29,107,140,62)(30,61,131,106)(31,32,142,143)(33,40,144,141)(34,150,145,39)(35,38,146,149)(36,148,147,37)(51,93,96,58)(52,57,97,92)(53,91,98,56)(54,55,99,100)(59,95,94,60)(71,78,116,113)(72,112,117,77)(73,76,118,111)(74,120,119,75)(79,80,114,115)>;
G:=Group( (1,120,95,37)(2,38,96,111)(3,112,97,39)(4,40,98,113)(5,114,99,31)(6,32,100,115)(7,116,91,33)(8,34,92,117)(9,118,93,35)(10,36,94,119)(11,109,41,26)(12,27,42,110)(13,101,43,28)(14,29,44,102)(15,103,45,30)(16,21,46,104)(17,105,47,22)(18,23,48,106)(19,107,49,24)(20,25,50,108)(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), (1,70,60,22)(2,61,51,23)(3,62,52,24)(4,63,53,25)(5,64,54,26)(6,65,55,27)(7,66,56,28)(8,67,57,29)(9,68,58,30)(10,69,59,21)(11,114,152,142)(12,115,153,143)(13,116,154,144)(14,117,155,145)(15,118,156,146)(16,119,157,147)(17,120,158,148)(18,111,159,149)(19,112,160,150)(20,113,151,141)(31,85,79,41)(32,86,80,42)(33,87,71,43)(34,88,72,44)(35,89,73,45)(36,90,74,46)(37,81,75,47)(38,82,76,48)(39,83,77,49)(40,84,78,50)(91,139,129,101)(92,140,130,102)(93,131,121,103)(94,132,122,104)(95,133,123,105)(96,134,124,106)(97,135,125,107)(98,136,126,108)(99,137,127,109)(100,138,128,110), (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)(11,153,85,42)(12,41,86,152)(13,151,87,50)(14,49,88,160)(15,159,89,48)(16,47,90,158)(17,157,81,46)(18,45,82,156)(19,155,83,44)(20,43,84,154)(21,105,132,70)(22,69,133,104)(23,103,134,68)(24,67,135,102)(25,101,136,66)(26,65,137,110)(27,109,138,64)(28,63,139,108)(29,107,140,62)(30,61,131,106)(31,32,142,143)(33,40,144,141)(34,150,145,39)(35,38,146,149)(36,148,147,37)(51,93,96,58)(52,57,97,92)(53,91,98,56)(54,55,99,100)(59,95,94,60)(71,78,116,113)(72,112,117,77)(73,76,118,111)(74,120,119,75)(79,80,114,115) );
G=PermutationGroup([[(1,120,95,37),(2,38,96,111),(3,112,97,39),(4,40,98,113),(5,114,99,31),(6,32,100,115),(7,116,91,33),(8,34,92,117),(9,118,93,35),(10,36,94,119),(11,109,41,26),(12,27,42,110),(13,101,43,28),(14,29,44,102),(15,103,45,30),(16,21,46,104),(17,105,47,22),(18,23,48,106),(19,107,49,24),(20,25,50,108),(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)], [(1,70,60,22),(2,61,51,23),(3,62,52,24),(4,63,53,25),(5,64,54,26),(6,65,55,27),(7,66,56,28),(8,67,57,29),(9,68,58,30),(10,69,59,21),(11,114,152,142),(12,115,153,143),(13,116,154,144),(14,117,155,145),(15,118,156,146),(16,119,157,147),(17,120,158,148),(18,111,159,149),(19,112,160,150),(20,113,151,141),(31,85,79,41),(32,86,80,42),(33,87,71,43),(34,88,72,44),(35,89,73,45),(36,90,74,46),(37,81,75,47),(38,82,76,48),(39,83,77,49),(40,84,78,50),(91,139,129,101),(92,140,130,102),(93,131,121,103),(94,132,122,104),(95,133,123,105),(96,134,124,106),(97,135,125,107),(98,136,126,108),(99,137,127,109),(100,138,128,110)], [(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),(11,153,85,42),(12,41,86,152),(13,151,87,50),(14,49,88,160),(15,159,89,48),(16,47,90,158),(17,157,81,46),(18,45,82,156),(19,155,83,44),(20,43,84,154),(21,105,132,70),(22,69,133,104),(23,103,134,68),(24,67,135,102),(25,101,136,66),(26,65,137,110),(27,109,138,64),(28,63,139,108),(29,107,140,62),(30,61,131,106),(31,32,142,143),(33,40,144,141),(34,150,145,39),(35,38,146,149),(36,148,147,37),(51,93,96,58),(52,57,97,92),(53,91,98,56),(54,55,99,100),(59,95,94,60),(71,78,116,113),(72,112,117,77),(73,76,118,111),(74,120,119,75),(79,80,114,115)]])
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