metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.105D10, C10.562- 1+4, (C4×D4).12D5, C4⋊C4.280D10, (D4×C20).13C2, (C4×Dic10)⋊27C2, (C2×D4).209D10, C20.48D4⋊8C2, (C2×C10).85C24, C20.6Q8⋊15C2, C20.291(C4○D4), (C4×C20).147C22, (C2×C20).585C23, C22⋊C4.130D10, Dic5.Q8⋊7C2, (C22×C4).204D10, C23.D10⋊6C2, C4.115(D4⋊2D5), C23.D5.9C22, (D4×C10).303C22, C22.11(C4○D20), C23.21D10⋊6C2, C4⋊Dic5.296C22, (C22×C20).79C22, (C2×Dic5).35C23, C22.113(C23×D5), C23.165(C22×D5), C23.11D10⋊27C2, (C22×C10).155C23, C5⋊4(C22.46C24), (C4×Dic5).222C22, C23.18D10.5C2, C2.14(D4.10D10), (C2×Dic10).244C22, C10.D4.153C22, (C22×Dic5).93C22, (C2×C4⋊Dic5)⋊23C2, C2.41(C2×C4○D20), C10.37(C2×C4○D4), C2.19(C2×D4⋊2D5), (C2×C10).15(C4○D4), (C5×C4⋊C4).321C22, (C2×C4).155(C22×D5), (C5×C22⋊C4).142C22, SmallGroup(320,1213)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42.105D10
G = < a,b,c,d | a4=b4=c10=1, d2=a2, ab=ba, cac-1=a-1, dad-1=a-1b2, bc=cb, dbd-1=b-1, dcd-1=c-1 >
Subgroups: 598 in 214 conjugacy classes, 99 normal (51 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C5, C2×C4, C2×C4, D4, Q8, C23, C10, C10, C42, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×Q8, Dic5, C20, C20, C2×C10, C2×C10, C2×C10, C2×C4⋊C4, C42⋊C2, C4×D4, C4×Q8, C22⋊Q8, C22.D4, C42.C2, C42⋊2C2, Dic10, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C5×D4, C22×C10, C22.46C24, C4×Dic5, C10.D4, C4⋊Dic5, C4⋊Dic5, C23.D5, C4×C20, C5×C22⋊C4, C5×C4⋊C4, C2×Dic10, C22×Dic5, C22×C20, D4×C10, C4×Dic10, C20.6Q8, C23.11D10, C23.D10, Dic5.Q8, C20.48D4, C2×C4⋊Dic5, C23.21D10, C23.18D10, D4×C20, C42.105D10
Quotients: C1, C2, C22, C23, D5, C4○D4, C24, D10, C2×C4○D4, 2- 1+4, C22×D5, C22.46C24, C4○D20, D4⋊2D5, C23×D5, C2×C4○D20, C2×D4⋊2D5, D4.10D10, C42.105D10
►On 160 points
Generators in S160
(1 68 63 6)(2 7 64 69)(3 70 65 8)(4 9 66 61)(5 62 67 10)(11 78 73 16)(12 17 74 79)(13 80 75 18)(14 19 76 71)(15 72 77 20)(21 26 31 36)(22 37 32 27)(23 28 33 38)(24 39 34 29)(25 30 35 40)(41 153 133 127)(42 128 134 154)(43 155 135 129)(44 130 136 156)(45 157 137 121)(46 122 138 158)(47 159 139 123)(48 124 140 160)(49 151 131 125)(50 126 132 152)(51 90 85 56)(52 57 86 81)(53 82 87 58)(54 59 88 83)(55 84 89 60)(91 146 119 104)(92 105 120 147)(93 148 111 106)(94 107 112 149)(95 150 113 108)(96 109 114 141)(97 142 115 110)(98 101 116 143)(99 144 117 102)(100 103 118 145)
(1 57 30 11)(2 58 21 12)(3 59 22 13)(4 60 23 14)(5 51 24 15)(6 52 25 16)(7 53 26 17)(8 54 27 18)(9 55 28 19)(10 56 29 20)(31 74 64 82)(32 75 65 83)(33 76 66 84)(34 77 67 85)(35 78 68 86)(36 79 69 87)(37 80 70 88)(38 71 61 89)(39 72 62 90)(40 73 63 81)(41 148 158 116)(42 149 159 117)(43 150 160 118)(44 141 151 119)(45 142 152 120)(46 143 153 111)(47 144 154 112)(48 145 155 113)(49 146 156 114)(50 147 157 115)(91 136 109 125)(92 137 110 126)(93 138 101 127)(94 139 102 128)(95 140 103 129)(96 131 104 130)(97 132 105 121)(98 133 106 122)(99 134 107 123)(100 135 108 124)
(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)
(1 146 63 104)(2 145 64 103)(3 144 65 102)(4 143 66 101)(5 142 67 110)(6 141 68 109)(7 150 69 108)(8 149 70 107)(9 148 61 106)(10 147 62 105)(11 156 73 130)(12 155 74 129)(13 154 75 128)(14 153 76 127)(15 152 77 126)(16 151 78 125)(17 160 79 124)(18 159 80 123)(19 158 71 122)(20 157 72 121)(21 113 31 95)(22 112 32 94)(23 111 33 93)(24 120 34 92)(25 119 35 91)(26 118 36 100)(27 117 37 99)(28 116 38 98)(29 115 39 97)(30 114 40 96)(41 89 133 55)(42 88 134 54)(43 87 135 53)(44 86 136 52)(45 85 137 51)(46 84 138 60)(47 83 139 59)(48 82 140 58)(49 81 131 57)(50 90 132 56)
G:=sub<Sym(160)| (1,68,63,6)(2,7,64,69)(3,70,65,8)(4,9,66,61)(5,62,67,10)(11,78,73,16)(12,17,74,79)(13,80,75,18)(14,19,76,71)(15,72,77,20)(21,26,31,36)(22,37,32,27)(23,28,33,38)(24,39,34,29)(25,30,35,40)(41,153,133,127)(42,128,134,154)(43,155,135,129)(44,130,136,156)(45,157,137,121)(46,122,138,158)(47,159,139,123)(48,124,140,160)(49,151,131,125)(50,126,132,152)(51,90,85,56)(52,57,86,81)(53,82,87,58)(54,59,88,83)(55,84,89,60)(91,146,119,104)(92,105,120,147)(93,148,111,106)(94,107,112,149)(95,150,113,108)(96,109,114,141)(97,142,115,110)(98,101,116,143)(99,144,117,102)(100,103,118,145), (1,57,30,11)(2,58,21,12)(3,59,22,13)(4,60,23,14)(5,51,24,15)(6,52,25,16)(7,53,26,17)(8,54,27,18)(9,55,28,19)(10,56,29,20)(31,74,64,82)(32,75,65,83)(33,76,66,84)(34,77,67,85)(35,78,68,86)(36,79,69,87)(37,80,70,88)(38,71,61,89)(39,72,62,90)(40,73,63,81)(41,148,158,116)(42,149,159,117)(43,150,160,118)(44,141,151,119)(45,142,152,120)(46,143,153,111)(47,144,154,112)(48,145,155,113)(49,146,156,114)(50,147,157,115)(91,136,109,125)(92,137,110,126)(93,138,101,127)(94,139,102,128)(95,140,103,129)(96,131,104,130)(97,132,105,121)(98,133,106,122)(99,134,107,123)(100,135,108,124), (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), (1,146,63,104)(2,145,64,103)(3,144,65,102)(4,143,66,101)(5,142,67,110)(6,141,68,109)(7,150,69,108)(8,149,70,107)(9,148,61,106)(10,147,62,105)(11,156,73,130)(12,155,74,129)(13,154,75,128)(14,153,76,127)(15,152,77,126)(16,151,78,125)(17,160,79,124)(18,159,80,123)(19,158,71,122)(20,157,72,121)(21,113,31,95)(22,112,32,94)(23,111,33,93)(24,120,34,92)(25,119,35,91)(26,118,36,100)(27,117,37,99)(28,116,38,98)(29,115,39,97)(30,114,40,96)(41,89,133,55)(42,88,134,54)(43,87,135,53)(44,86,136,52)(45,85,137,51)(46,84,138,60)(47,83,139,59)(48,82,140,58)(49,81,131,57)(50,90,132,56)>;
G:=Group( (1,68,63,6)(2,7,64,69)(3,70,65,8)(4,9,66,61)(5,62,67,10)(11,78,73,16)(12,17,74,79)(13,80,75,18)(14,19,76,71)(15,72,77,20)(21,26,31,36)(22,37,32,27)(23,28,33,38)(24,39,34,29)(25,30,35,40)(41,153,133,127)(42,128,134,154)(43,155,135,129)(44,130,136,156)(45,157,137,121)(46,122,138,158)(47,159,139,123)(48,124,140,160)(49,151,131,125)(50,126,132,152)(51,90,85,56)(52,57,86,81)(53,82,87,58)(54,59,88,83)(55,84,89,60)(91,146,119,104)(92,105,120,147)(93,148,111,106)(94,107,112,149)(95,150,113,108)(96,109,114,141)(97,142,115,110)(98,101,116,143)(99,144,117,102)(100,103,118,145), (1,57,30,11)(2,58,21,12)(3,59,22,13)(4,60,23,14)(5,51,24,15)(6,52,25,16)(7,53,26,17)(8,54,27,18)(9,55,28,19)(10,56,29,20)(31,74,64,82)(32,75,65,83)(33,76,66,84)(34,77,67,85)(35,78,68,86)(36,79,69,87)(37,80,70,88)(38,71,61,89)(39,72,62,90)(40,73,63,81)(41,148,158,116)(42,149,159,117)(43,150,160,118)(44,141,151,119)(45,142,152,120)(46,143,153,111)(47,144,154,112)(48,145,155,113)(49,146,156,114)(50,147,157,115)(91,136,109,125)(92,137,110,126)(93,138,101,127)(94,139,102,128)(95,140,103,129)(96,131,104,130)(97,132,105,121)(98,133,106,122)(99,134,107,123)(100,135,108,124), (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), (1,146,63,104)(2,145,64,103)(3,144,65,102)(4,143,66,101)(5,142,67,110)(6,141,68,109)(7,150,69,108)(8,149,70,107)(9,148,61,106)(10,147,62,105)(11,156,73,130)(12,155,74,129)(13,154,75,128)(14,153,76,127)(15,152,77,126)(16,151,78,125)(17,160,79,124)(18,159,80,123)(19,158,71,122)(20,157,72,121)(21,113,31,95)(22,112,32,94)(23,111,33,93)(24,120,34,92)(25,119,35,91)(26,118,36,100)(27,117,37,99)(28,116,38,98)(29,115,39,97)(30,114,40,96)(41,89,133,55)(42,88,134,54)(43,87,135,53)(44,86,136,52)(45,85,137,51)(46,84,138,60)(47,83,139,59)(48,82,140,58)(49,81,131,57)(50,90,132,56) );
G=PermutationGroup([[(1,68,63,6),(2,7,64,69),(3,70,65,8),(4,9,66,61),(5,62,67,10),(11,78,73,16),(12,17,74,79),(13,80,75,18),(14,19,76,71),(15,72,77,20),(21,26,31,36),(22,37,32,27),(23,28,33,38),(24,39,34,29),(25,30,35,40),(41,153,133,127),(42,128,134,154),(43,155,135,129),(44,130,136,156),(45,157,137,121),(46,122,138,158),(47,159,139,123),(48,124,140,160),(49,151,131,125),(50,126,132,152),(51,90,85,56),(52,57,86,81),(53,82,87,58),(54,59,88,83),(55,84,89,60),(91,146,119,104),(92,105,120,147),(93,148,111,106),(94,107,112,149),(95,150,113,108),(96,109,114,141),(97,142,115,110),(98,101,116,143),(99,144,117,102),(100,103,118,145)], [(1,57,30,11),(2,58,21,12),(3,59,22,13),(4,60,23,14),(5,51,24,15),(6,52,25,16),(7,53,26,17),(8,54,27,18),(9,55,28,19),(10,56,29,20),(31,74,64,82),(32,75,65,83),(33,76,66,84),(34,77,67,85),(35,78,68,86),(36,79,69,87),(37,80,70,88),(38,71,61,89),(39,72,62,90),(40,73,63,81),(41,148,158,116),(42,149,159,117),(43,150,160,118),(44,141,151,119),(45,142,152,120),(46,143,153,111),(47,144,154,112),(48,145,155,113),(49,146,156,114),(50,147,157,115),(91,136,109,125),(92,137,110,126),(93,138,101,127),(94,139,102,128),(95,140,103,129),(96,131,104,130),(97,132,105,121),(98,133,106,122),(99,134,107,123),(100,135,108,124)], [(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)], [(1,146,63,104),(2,145,64,103),(3,144,65,102),(4,143,66,101),(5,142,67,110),(6,141,68,109),(7,150,69,108),(8,149,70,107),(9,148,61,106),(10,147,62,105),(11,156,73,130),(12,155,74,129),(13,154,75,128),(14,153,76,127),(15,152,77,126),(16,151,78,125),(17,160,79,124),(18,159,80,123),(19,158,71,122),(20,157,72,121),(21,113,31,95),(22,112,32,94),(23,111,33,93),(24,120,34,92),(25,119,35,91),(26,118,36,100),(27,117,37,99),(28,116,38,98),(29,115,39,97),(30,114,40,96),(41,89,133,55),(42,88,134,54),(43,87,135,53),(44,86,136,52),(45,85,137,51),(46,84,138,60),(47,83,139,59),(48,82,140,58),(49,81,131,57),(50,90,132,56)]])
65 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 4A | ··· | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 4M | ··· | 4R | 5A | 5B | 10A | ··· | 10F | 10G | ··· | 10N | 20A | ··· | 20H | 20I | ··· | 20X |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 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 | 2 | 2 | 4 | 2 | ··· | 2 | 4 | 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 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | - | - | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D5 | C4○D4 | C4○D4 | D10 | D10 | D10 | D10 | D10 | C4○D20 | 2- 1+4 | D4⋊2D5 | D4.10D10 |
| kernel | C42.105D10 | C4×Dic10 | C20.6Q8 | C23.11D10 | C23.D10 | Dic5.Q8 | C20.48D4 | C2×C4⋊Dic5 | C23.21D10 | C23.18D10 | D4×C20 | C4×D4 | C20 | C2×C10 | C42 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×D4 | C22 | C10 | C4 | C2 |
| # reps | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 1 | 1 | 2 | 1 | 2 | 4 | 4 | 2 | 4 | 2 | 4 | 2 | 16 | 1 | 4 | 4 |
Matrix representation of C42.105D10 ►in GL4(𝔽41) generated by
| 40 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 4 | 12 |
| 0 | 0 | 2 | 37 |
| 32 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 |
| 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 40 |
| 25 | 0 | 0 | 0 |
| 0 | 23 | 0 | 0 |
| 0 | 0 | 37 | 28 |
| 0 | 0 | 39 | 4 |
| 0 | 23 | 0 | 0 |
| 25 | 0 | 0 | 0 |
| 0 | 0 | 5 | 6 |
| 0 | 0 | 23 | 36 |
G:=sub<GL(4,GF(41))| [40,0,0,0,0,1,0,0,0,0,4,2,0,0,12,37],[32,0,0,0,0,9,0,0,0,0,40,0,0,0,0,40],[25,0,0,0,0,23,0,0,0,0,37,39,0,0,28,4],[0,25,0,0,23,0,0,0,0,0,5,23,0,0,6,36] >;
C42.105D10 in GAP, Magma, Sage, TeX
C_4^2._{105}D_{10} % in TeX
G:=Group("C4^2.105D10"); // GroupNames label
G:=SmallGroup(320,1213);
// by ID
G=gap.SmallGroup(320,1213);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,224,387,100,675,570,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^10=1,d^2=a^2,a*b=b*a,c*a*c^-1=a^-1,d*a*d^-1=a^-1*b^2,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=c^-1>;
// generators/relations