metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C10.10C42, C23.27D10, C22.11D20, C22.3Dic10, (C2×C20)⋊8C4, (C2×C4)⋊2Dic5, (C2×C10).4Q8, (C2×Dic5)⋊3C4, (C2×C10).32D4, C10.16(C4⋊C4), C2.5(C4×Dic5), (C22×C4).2D5, (C22×C20).1C2, C2.2(C4⋊Dic5), C22.12(C4×D5), C5⋊2(C2.C42), C2.2(C23.D5), C2.2(D10⋊C4), C10.22(C22⋊C4), C2.2(C10.D4), C22.16(C5⋊D4), (C22×Dic5).1C2, C22.10(C2×Dic5), (C22×C10).31C22, (C2×C10).33(C2×C4), SmallGroup(160,38)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C10.10C42
G = < a,b,c | a10=b4=c4=1, bab-1=a-1, ac=ca, cbc-1=a5b >
Subgroups: 192 in 76 conjugacy classes, 45 normal (19 characteristic)
C1, C2, C2, C4, C22, C22, C5, C2×C4, C2×C4, C23, C10, C10, C22×C4, C22×C4, Dic5, C20, C2×C10, C2×C10, C2.C42, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C22×C10, C22×Dic5, C22×C20, C10.10C42
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, D5, C42, C22⋊C4, C4⋊C4, Dic5, D10, C2.C42, Dic10, C4×D5, D20, C2×Dic5, C5⋊D4, C4×Dic5, C10.D4, C4⋊Dic5, D10⋊C4, C23.D5, C10.10C42
►Regular action on 160 points
(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)
(1 107 23 87)(2 106 24 86)(3 105 25 85)(4 104 26 84)(5 103 27 83)(6 102 28 82)(7 101 29 81)(8 110 30 90)(9 109 21 89)(10 108 22 88)(11 119 31 99)(12 118 32 98)(13 117 33 97)(14 116 34 96)(15 115 35 95)(16 114 36 94)(17 113 37 93)(18 112 38 92)(19 111 39 91)(20 120 40 100)(41 149 61 129)(42 148 62 128)(43 147 63 127)(44 146 64 126)(45 145 65 125)(46 144 66 124)(47 143 67 123)(48 142 68 122)(49 141 69 121)(50 150 70 130)(51 159 71 139)(52 158 72 138)(53 157 73 137)(54 156 74 136)(55 155 75 135)(56 154 76 134)(57 153 77 133)(58 152 78 132)(59 151 79 131)(60 160 80 140)
(1 53 13 43)(2 54 14 44)(3 55 15 45)(4 56 16 46)(5 57 17 47)(6 58 18 48)(7 59 19 49)(8 60 20 50)(9 51 11 41)(10 52 12 42)(21 71 31 61)(22 72 32 62)(23 73 33 63)(24 74 34 64)(25 75 35 65)(26 76 36 66)(27 77 37 67)(28 78 38 68)(29 79 39 69)(30 80 40 70)(81 136 91 126)(82 137 92 127)(83 138 93 128)(84 139 94 129)(85 140 95 130)(86 131 96 121)(87 132 97 122)(88 133 98 123)(89 134 99 124)(90 135 100 125)(101 156 111 146)(102 157 112 147)(103 158 113 148)(104 159 114 149)(105 160 115 150)(106 151 116 141)(107 152 117 142)(108 153 118 143)(109 154 119 144)(110 155 120 145)
G:=sub<Sym(160)| (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), (1,107,23,87)(2,106,24,86)(3,105,25,85)(4,104,26,84)(5,103,27,83)(6,102,28,82)(7,101,29,81)(8,110,30,90)(9,109,21,89)(10,108,22,88)(11,119,31,99)(12,118,32,98)(13,117,33,97)(14,116,34,96)(15,115,35,95)(16,114,36,94)(17,113,37,93)(18,112,38,92)(19,111,39,91)(20,120,40,100)(41,149,61,129)(42,148,62,128)(43,147,63,127)(44,146,64,126)(45,145,65,125)(46,144,66,124)(47,143,67,123)(48,142,68,122)(49,141,69,121)(50,150,70,130)(51,159,71,139)(52,158,72,138)(53,157,73,137)(54,156,74,136)(55,155,75,135)(56,154,76,134)(57,153,77,133)(58,152,78,132)(59,151,79,131)(60,160,80,140), (1,53,13,43)(2,54,14,44)(3,55,15,45)(4,56,16,46)(5,57,17,47)(6,58,18,48)(7,59,19,49)(8,60,20,50)(9,51,11,41)(10,52,12,42)(21,71,31,61)(22,72,32,62)(23,73,33,63)(24,74,34,64)(25,75,35,65)(26,76,36,66)(27,77,37,67)(28,78,38,68)(29,79,39,69)(30,80,40,70)(81,136,91,126)(82,137,92,127)(83,138,93,128)(84,139,94,129)(85,140,95,130)(86,131,96,121)(87,132,97,122)(88,133,98,123)(89,134,99,124)(90,135,100,125)(101,156,111,146)(102,157,112,147)(103,158,113,148)(104,159,114,149)(105,160,115,150)(106,151,116,141)(107,152,117,142)(108,153,118,143)(109,154,119,144)(110,155,120,145)>;
G:=Group( (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), (1,107,23,87)(2,106,24,86)(3,105,25,85)(4,104,26,84)(5,103,27,83)(6,102,28,82)(7,101,29,81)(8,110,30,90)(9,109,21,89)(10,108,22,88)(11,119,31,99)(12,118,32,98)(13,117,33,97)(14,116,34,96)(15,115,35,95)(16,114,36,94)(17,113,37,93)(18,112,38,92)(19,111,39,91)(20,120,40,100)(41,149,61,129)(42,148,62,128)(43,147,63,127)(44,146,64,126)(45,145,65,125)(46,144,66,124)(47,143,67,123)(48,142,68,122)(49,141,69,121)(50,150,70,130)(51,159,71,139)(52,158,72,138)(53,157,73,137)(54,156,74,136)(55,155,75,135)(56,154,76,134)(57,153,77,133)(58,152,78,132)(59,151,79,131)(60,160,80,140), (1,53,13,43)(2,54,14,44)(3,55,15,45)(4,56,16,46)(5,57,17,47)(6,58,18,48)(7,59,19,49)(8,60,20,50)(9,51,11,41)(10,52,12,42)(21,71,31,61)(22,72,32,62)(23,73,33,63)(24,74,34,64)(25,75,35,65)(26,76,36,66)(27,77,37,67)(28,78,38,68)(29,79,39,69)(30,80,40,70)(81,136,91,126)(82,137,92,127)(83,138,93,128)(84,139,94,129)(85,140,95,130)(86,131,96,121)(87,132,97,122)(88,133,98,123)(89,134,99,124)(90,135,100,125)(101,156,111,146)(102,157,112,147)(103,158,113,148)(104,159,114,149)(105,160,115,150)(106,151,116,141)(107,152,117,142)(108,153,118,143)(109,154,119,144)(110,155,120,145) );
G=PermutationGroup([[(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)], [(1,107,23,87),(2,106,24,86),(3,105,25,85),(4,104,26,84),(5,103,27,83),(6,102,28,82),(7,101,29,81),(8,110,30,90),(9,109,21,89),(10,108,22,88),(11,119,31,99),(12,118,32,98),(13,117,33,97),(14,116,34,96),(15,115,35,95),(16,114,36,94),(17,113,37,93),(18,112,38,92),(19,111,39,91),(20,120,40,100),(41,149,61,129),(42,148,62,128),(43,147,63,127),(44,146,64,126),(45,145,65,125),(46,144,66,124),(47,143,67,123),(48,142,68,122),(49,141,69,121),(50,150,70,130),(51,159,71,139),(52,158,72,138),(53,157,73,137),(54,156,74,136),(55,155,75,135),(56,154,76,134),(57,153,77,133),(58,152,78,132),(59,151,79,131),(60,160,80,140)], [(1,53,13,43),(2,54,14,44),(3,55,15,45),(4,56,16,46),(5,57,17,47),(6,58,18,48),(7,59,19,49),(8,60,20,50),(9,51,11,41),(10,52,12,42),(21,71,31,61),(22,72,32,62),(23,73,33,63),(24,74,34,64),(25,75,35,65),(26,76,36,66),(27,77,37,67),(28,78,38,68),(29,79,39,69),(30,80,40,70),(81,136,91,126),(82,137,92,127),(83,138,93,128),(84,139,94,129),(85,140,95,130),(86,131,96,121),(87,132,97,122),(88,133,98,123),(89,134,99,124),(90,135,100,125),(101,156,111,146),(102,157,112,147),(103,158,113,148),(104,159,114,149),(105,160,115,150),(106,151,116,141),(107,152,117,142),(108,153,118,143),(109,154,119,144),(110,155,120,145)]])
C10.10C42 is a maximal subgroup of
C23.30D20 C22.2D40 C4⋊C4⋊Dic5 C10.29C4≀C2 (C2×C20)⋊Q8 C10.49(C4×D4) Dic5.15C42 Dic5⋊2C42 C5⋊2(C42⋊8C4) C5⋊2(C42⋊5C4) C10.51(C4×D4) C2.(C4×D20) C4⋊Dic5⋊15C4 C10.52(C4×D4) (C2×Dic5)⋊Q8 C2.(C20⋊Q8) (C2×Dic5).Q8 (C2×C20).28D4 (C2×C4).Dic10 C10.(C4⋊Q8) (C22×C4).D10 D5×C2.C42 C22.58(D4×D5) (C2×C4)⋊9D20 D10⋊2C42 D10⋊2(C4⋊C4) D10⋊3(C4⋊C4) C10.54(C4×D4) C10.55(C4×D4) (C2×C4).20D20 (C2×C4).21D20 C10.(C4⋊D4) C20⋊7(C4⋊C4) (C2×C20)⋊10Q8 C4×C10.D4 C42⋊4Dic5 C10.92(C4×D4) C4×C4⋊Dic5 C42⋊9Dic5 C42⋊5Dic5 C4×D10⋊C4 (C2×C42)⋊D5 C22⋊C4×Dic5 C24.44D10 C23.42D20 C24.3D10 C24.4D10 C24.46D10 C23⋊Dic10 C24.6D10 C24.7D10 C24.47D10 C24.8D10 C24.9D10 C23.14D20 C24.12D10 C23.45D20 C24.14D10 C23⋊2D20 C24.16D10 C10.96(C4×D4) C20⋊4(C4⋊C4) (C2×Dic5)⋊6Q8 C4⋊C4×Dic5 C20.48(C4⋊C4) C10.97(C4×D4) (C2×C4)⋊Dic10 (C2×C20).287D4 C4⋊C4⋊5Dic5 (C2×C20).288D4 (C2×C20).53D4 (C2×C20).54D4 C20⋊6(C4⋊C4) (C2×C20).55D4 D10⋊4(C4⋊C4) D10⋊5(C4⋊C4) C10.90(C4×D4) (C2×C4)⋊3D20 (C2×C20).289D4 (C2×C20).290D4 (C2×C20).56D4 C4×C23.D5 C24.62D10 C24.63D10 C24.64D10 C24.65D10 C24.18D10 C24.20D10 C24.21D10 C10.C22≀C2 (Q8×C10)⋊17C4 (C22×D5)⋊Q8 C30.24C42 C30.29C42
C10.10C42 is a maximal quotient of
C42⋊6Dic5 (C2×C20)⋊8C8 C24.D10 C24.2D10 C20.31C42 (C2×C20).Q8 C42⋊1Dic5 C20.32C42 C20.60(C4⋊C4) (C2×C40)⋊15C4 C20.39C42 C20.40C42 M4(2)⋊Dic5 C20.33C42 (C2×C40)⋊C4 C23.9D20 C20.34C42 M4(2)⋊4Dic5 C20.51C42 C30.24C42 C30.29C42
52 conjugacy classes
| class | 1 | 2A | ··· | 2G | 4A | 4B | 4C | 4D | 4E | ··· | 4L | 5A | 5B | 10A | ··· | 10N | 20A | ··· | 20P |
| order | 1 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 5 | 5 | 10 | ··· | 10 | 20 | ··· | 20 |
| size | 1 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 10 | ··· | 10 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
52 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | - | + | - | + | - | + | ||||
| image | C1 | C2 | C2 | C4 | C4 | D4 | Q8 | D5 | Dic5 | D10 | Dic10 | C4×D5 | D20 | C5⋊D4 |
| kernel | C10.10C42 | C22×Dic5 | C22×C20 | C2×Dic5 | C2×C20 | C2×C10 | C2×C10 | C22×C4 | C2×C4 | C23 | C22 | C22 | C22 | C22 |
| # reps | 1 | 2 | 1 | 8 | 4 | 3 | 1 | 2 | 4 | 2 | 4 | 8 | 4 | 8 |
Matrix representation of C10.10C42 ►in GL4(𝔽41) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 34 |
| 0 | 0 | 6 | 35 |
| 9 | 0 | 0 | 0 |
| 0 | 32 | 0 | 0 |
| 0 | 0 | 23 | 17 |
| 0 | 0 | 5 | 18 |
| 9 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 0 | 0 | 39 | 32 |
| 0 | 0 | 37 | 2 |
G:=sub<GL(4,GF(41))| [1,0,0,0,0,1,0,0,0,0,0,6,0,0,34,35],[9,0,0,0,0,32,0,0,0,0,23,5,0,0,17,18],[9,0,0,0,0,40,0,0,0,0,39,37,0,0,32,2] >;
C10.10C42 in GAP, Magma, Sage, TeX
C_{10}._{10}C_4^2 % in TeX
G:=Group("C10.10C4^2"); // GroupNames label
G:=SmallGroup(160,38);
// by ID
G=gap.SmallGroup(160,38);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-5,24,217,55,4613]);
// Polycyclic
G:=Group<a,b,c|a^10=b^4=c^4=1,b*a*b^-1=a^-1,a*c=c*a,c*b*c^-1=a^5*b>;
// generators/relations