direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C10×C4⋊C4, C4⋊2(C2×C20), (C2×C4)⋊3C20, C20⋊12(C2×C4), (C2×C20)⋊13C4, C2.2(D4×C10), (C2×C10).8Q8, C2.1(Q8×C10), (C2×C10).51D4, C10.65(C2×D4), C10.18(C2×Q8), C22.3(C5×Q8), (C22×C4).3C10, C2.2(C22×C20), (C22×C20).5C2, C22.13(C5×D4), C23.13(C2×C10), C10.43(C22×C4), (C2×C10).71C23, C22.11(C2×C20), (C2×C20).120C22, C22.5(C22×C10), (C22×C10).49C22, (C2×C4).13(C2×C10), (C2×C10).60(C2×C4), SmallGroup(160,177)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C10×C4⋊C4
G = < a,b,c | a10=b4=c4=1, ab=ba, ac=ca, cbc-1=b-1 >
Subgroups: 108 in 92 conjugacy classes, 76 normal (16 characteristic)
C1, C2 [×3], C2 [×4], C4 [×4], C4 [×4], C22, C22 [×6], C5, C2×C4 [×10], C2×C4 [×4], C23, C10 [×3], C10 [×4], C4⋊C4 [×4], C22×C4, C22×C4 [×2], C20 [×4], C20 [×4], C2×C10, C2×C10 [×6], C2×C4⋊C4, C2×C20 [×10], C2×C20 [×4], C22×C10, C5×C4⋊C4 [×4], C22×C20, C22×C20 [×2], C10×C4⋊C4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C5, C2×C4 [×6], D4 [×2], Q8 [×2], C23, C10 [×7], C4⋊C4 [×4], C22×C4, C2×D4, C2×Q8, C20 [×4], C2×C10 [×7], C2×C4⋊C4, C2×C20 [×6], C5×D4 [×2], C5×Q8 [×2], C22×C10, C5×C4⋊C4 [×4], C22×C20, D4×C10, Q8×C10, C10×C4⋊C4
►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 83 52 97)(2 84 53 98)(3 85 54 99)(4 86 55 100)(5 87 56 91)(6 88 57 92)(7 89 58 93)(8 90 59 94)(9 81 60 95)(10 82 51 96)(11 143 160 115)(12 144 151 116)(13 145 152 117)(14 146 153 118)(15 147 154 119)(16 148 155 120)(17 149 156 111)(18 150 157 112)(19 141 158 113)(20 142 159 114)(21 131 39 127)(22 132 40 128)(23 133 31 129)(24 134 32 130)(25 135 33 121)(26 136 34 122)(27 137 35 123)(28 138 36 124)(29 139 37 125)(30 140 38 126)(41 110 65 72)(42 101 66 73)(43 102 67 74)(44 103 68 75)(45 104 69 76)(46 105 70 77)(47 106 61 78)(48 107 62 79)(49 108 63 80)(50 109 64 71)
(1 137 62 147)(2 138 63 148)(3 139 64 149)(4 140 65 150)(5 131 66 141)(6 132 67 142)(7 133 68 143)(8 134 69 144)(9 135 70 145)(10 136 61 146)(11 89 23 75)(12 90 24 76)(13 81 25 77)(14 82 26 78)(15 83 27 79)(16 84 28 80)(17 85 29 71)(18 86 30 72)(19 87 21 73)(20 88 22 74)(31 103 160 93)(32 104 151 94)(33 105 152 95)(34 106 153 96)(35 107 154 97)(36 108 155 98)(37 109 156 99)(38 110 157 100)(39 101 158 91)(40 102 159 92)(41 112 55 126)(42 113 56 127)(43 114 57 128)(44 115 58 129)(45 116 59 130)(46 117 60 121)(47 118 51 122)(48 119 52 123)(49 120 53 124)(50 111 54 125)
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), (1,83,52,97)(2,84,53,98)(3,85,54,99)(4,86,55,100)(5,87,56,91)(6,88,57,92)(7,89,58,93)(8,90,59,94)(9,81,60,95)(10,82,51,96)(11,143,160,115)(12,144,151,116)(13,145,152,117)(14,146,153,118)(15,147,154,119)(16,148,155,120)(17,149,156,111)(18,150,157,112)(19,141,158,113)(20,142,159,114)(21,131,39,127)(22,132,40,128)(23,133,31,129)(24,134,32,130)(25,135,33,121)(26,136,34,122)(27,137,35,123)(28,138,36,124)(29,139,37,125)(30,140,38,126)(41,110,65,72)(42,101,66,73)(43,102,67,74)(44,103,68,75)(45,104,69,76)(46,105,70,77)(47,106,61,78)(48,107,62,79)(49,108,63,80)(50,109,64,71), (1,137,62,147)(2,138,63,148)(3,139,64,149)(4,140,65,150)(5,131,66,141)(6,132,67,142)(7,133,68,143)(8,134,69,144)(9,135,70,145)(10,136,61,146)(11,89,23,75)(12,90,24,76)(13,81,25,77)(14,82,26,78)(15,83,27,79)(16,84,28,80)(17,85,29,71)(18,86,30,72)(19,87,21,73)(20,88,22,74)(31,103,160,93)(32,104,151,94)(33,105,152,95)(34,106,153,96)(35,107,154,97)(36,108,155,98)(37,109,156,99)(38,110,157,100)(39,101,158,91)(40,102,159,92)(41,112,55,126)(42,113,56,127)(43,114,57,128)(44,115,58,129)(45,116,59,130)(46,117,60,121)(47,118,51,122)(48,119,52,123)(49,120,53,124)(50,111,54,125)>;
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), (1,83,52,97)(2,84,53,98)(3,85,54,99)(4,86,55,100)(5,87,56,91)(6,88,57,92)(7,89,58,93)(8,90,59,94)(9,81,60,95)(10,82,51,96)(11,143,160,115)(12,144,151,116)(13,145,152,117)(14,146,153,118)(15,147,154,119)(16,148,155,120)(17,149,156,111)(18,150,157,112)(19,141,158,113)(20,142,159,114)(21,131,39,127)(22,132,40,128)(23,133,31,129)(24,134,32,130)(25,135,33,121)(26,136,34,122)(27,137,35,123)(28,138,36,124)(29,139,37,125)(30,140,38,126)(41,110,65,72)(42,101,66,73)(43,102,67,74)(44,103,68,75)(45,104,69,76)(46,105,70,77)(47,106,61,78)(48,107,62,79)(49,108,63,80)(50,109,64,71), (1,137,62,147)(2,138,63,148)(3,139,64,149)(4,140,65,150)(5,131,66,141)(6,132,67,142)(7,133,68,143)(8,134,69,144)(9,135,70,145)(10,136,61,146)(11,89,23,75)(12,90,24,76)(13,81,25,77)(14,82,26,78)(15,83,27,79)(16,84,28,80)(17,85,29,71)(18,86,30,72)(19,87,21,73)(20,88,22,74)(31,103,160,93)(32,104,151,94)(33,105,152,95)(34,106,153,96)(35,107,154,97)(36,108,155,98)(37,109,156,99)(38,110,157,100)(39,101,158,91)(40,102,159,92)(41,112,55,126)(42,113,56,127)(43,114,57,128)(44,115,58,129)(45,116,59,130)(46,117,60,121)(47,118,51,122)(48,119,52,123)(49,120,53,124)(50,111,54,125) );
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)], [(1,83,52,97),(2,84,53,98),(3,85,54,99),(4,86,55,100),(5,87,56,91),(6,88,57,92),(7,89,58,93),(8,90,59,94),(9,81,60,95),(10,82,51,96),(11,143,160,115),(12,144,151,116),(13,145,152,117),(14,146,153,118),(15,147,154,119),(16,148,155,120),(17,149,156,111),(18,150,157,112),(19,141,158,113),(20,142,159,114),(21,131,39,127),(22,132,40,128),(23,133,31,129),(24,134,32,130),(25,135,33,121),(26,136,34,122),(27,137,35,123),(28,138,36,124),(29,139,37,125),(30,140,38,126),(41,110,65,72),(42,101,66,73),(43,102,67,74),(44,103,68,75),(45,104,69,76),(46,105,70,77),(47,106,61,78),(48,107,62,79),(49,108,63,80),(50,109,64,71)], [(1,137,62,147),(2,138,63,148),(3,139,64,149),(4,140,65,150),(5,131,66,141),(6,132,67,142),(7,133,68,143),(8,134,69,144),(9,135,70,145),(10,136,61,146),(11,89,23,75),(12,90,24,76),(13,81,25,77),(14,82,26,78),(15,83,27,79),(16,84,28,80),(17,85,29,71),(18,86,30,72),(19,87,21,73),(20,88,22,74),(31,103,160,93),(32,104,151,94),(33,105,152,95),(34,106,153,96),(35,107,154,97),(36,108,155,98),(37,109,156,99),(38,110,157,100),(39,101,158,91),(40,102,159,92),(41,112,55,126),(42,113,56,127),(43,114,57,128),(44,115,58,129),(45,116,59,130),(46,117,60,121),(47,118,51,122),(48,119,52,123),(49,120,53,124),(50,111,54,125)])
C10×C4⋊C4 is a maximal subgroup of
(C2×C20)⋊C8 C20.31C42 (C2×C20).Q8 C20.47(C4⋊C4) C4○D20⋊9C4 (C2×C10).40D8 C4⋊C4.228D10 C4⋊C4.230D10 C4⋊C4.231D10 C10.96(C4×D4) C20⋊4(C4⋊C4) (C2×Dic5)⋊6Q8 C20⋊5(C4⋊C4) 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) (C2×D20)⋊22C4 D10⋊5(C4⋊C4) C10.90(C4×D4) (C2×C4)⋊3D20 (C2×C20).289D4 (C2×C20).290D4 (C2×C20).56D4 C10.12- 1+4 C10.82+ 1+4 C10.2- 1+4 C10.2+ 1+4 C10.102+ 1+4 C10.52- 1+4 C10.112+ 1+4 C10.62- 1+4 D4×C2×C20 Q8×C2×C20
100 conjugacy classes
| class | 1 | 2A | ··· | 2G | 4A | ··· | 4L | 5A | 5B | 5C | 5D | 10A | ··· | 10AB | 20A | ··· | 20AV |
| order | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 5 | 5 | 5 | 5 | 10 | ··· | 10 | 20 | ··· | 20 |
| size | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 2 | ··· | 2 |
100 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | - | |||||||
| image | C1 | C2 | C2 | C4 | C5 | C10 | C10 | C20 | D4 | Q8 | C5×D4 | C5×Q8 |
| kernel | C10×C4⋊C4 | C5×C4⋊C4 | C22×C20 | C2×C20 | C2×C4⋊C4 | C4⋊C4 | C22×C4 | C2×C4 | C2×C10 | C2×C10 | C22 | C22 |
| # reps | 1 | 4 | 3 | 8 | 4 | 16 | 12 | 32 | 2 | 2 | 8 | 8 |
Matrix representation of C10×C4⋊C4 ►in GL4(𝔽41) generated by
| 40 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 0 | 0 | 23 | 0 |
| 0 | 0 | 0 | 23 |
| 40 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 0 | 0 | 1 | 39 |
| 0 | 0 | 1 | 40 |
| 40 | 0 | 0 | 0 |
| 0 | 32 | 0 | 0 |
| 0 | 0 | 33 | 23 |
| 0 | 0 | 24 | 8 |
G:=sub<GL(4,GF(41))| [40,0,0,0,0,40,0,0,0,0,23,0,0,0,0,23],[40,0,0,0,0,40,0,0,0,0,1,1,0,0,39,40],[40,0,0,0,0,32,0,0,0,0,33,24,0,0,23,8] >;
C10×C4⋊C4 in GAP, Magma, Sage, TeX
C_{10}\times C_4\rtimes C_4 % in TeX
G:=Group("C10xC4:C4"); // GroupNames label
G:=SmallGroup(160,177);
// by ID
G=gap.SmallGroup(160,177);
# by ID
G:=PCGroup([6,-2,-2,-2,-5,-2,-2,480,505,247]);
// Polycyclic
G:=Group<a,b,c|a^10=b^4=c^4=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations