direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C10×C4.4D4, (C2×C42)⋊9C10, C4.13(D4×C10), C42⋊17(C2×C10), (C4×C20)⋊58C22, C20.320(C2×D4), (C2×C20).430D4, (C22×Q8)⋊4C10, C24.13(C2×C10), (Q8×C10)⋊50C22, C22.62(D4×C10), (C2×C10).346C24, (C2×C20).659C23, (C22×D4).11C10, C10.185(C22×D4), C23.6(C22×C10), (D4×C10).317C22, C22.20(C23×C10), (C23×C10).13C22, (C22×C10).85C23, (C22×C20).508C22, (C2×C4×C20)⋊22C2, C2.9(D4×C2×C10), (Q8×C2×C10)⋊16C2, (D4×C2×C10).24C2, C2.9(C10×C4○D4), (C2×C4).86(C5×D4), (C2×Q8)⋊10(C2×C10), C22⋊C4⋊14(C2×C10), (C10×C22⋊C4)⋊31C2, (C2×C22⋊C4)⋊11C10, (C2×D4).62(C2×C10), C10.228(C2×C4○D4), (C2×C10).683(C2×D4), C22.32(C5×C4○D4), (C5×C22⋊C4)⋊68C22, (C2×C4).58(C22×C10), (C2×C10).232(C4○D4), (C22×C4).100(C2×C10), SmallGroup(320,1528)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 530 in 330 conjugacy classes, 178 normal (18 characteristic)
C1, C2, C2 [×6], C2 [×4], C4 [×4], C4 [×8], C22, C22 [×6], C22 [×20], C5, C2×C4 [×14], C2×C4 [×8], D4 [×8], Q8 [×8], C23, C23 [×4], C23 [×12], C10, C10 [×6], C10 [×4], C42 [×4], C22⋊C4 [×16], C22×C4, C22×C4 [×4], C2×D4 [×4], C2×D4 [×4], C2×Q8 [×4], C2×Q8 [×4], C24 [×2], C20 [×4], C20 [×8], C2×C10, C2×C10 [×6], C2×C10 [×20], C2×C42, C2×C22⋊C4 [×4], C4.4D4 [×8], C22×D4, C22×Q8, C2×C20 [×14], C2×C20 [×8], C5×D4 [×8], C5×Q8 [×8], C22×C10, C22×C10 [×4], C22×C10 [×12], C2×C4.4D4, C4×C20 [×4], C5×C22⋊C4 [×16], C22×C20, C22×C20 [×4], D4×C10 [×4], D4×C10 [×4], Q8×C10 [×4], Q8×C10 [×4], C23×C10 [×2], C2×C4×C20, C10×C22⋊C4 [×4], C5×C4.4D4 [×8], D4×C2×C10, Q8×C2×C10, C10×C4.4D4
Quotients:
C1, C2 [×15], C22 [×35], C5, D4 [×4], C23 [×15], C10 [×15], C2×D4 [×6], C4○D4 [×4], C24, C2×C10 [×35], C4.4D4 [×4], C22×D4, C2×C4○D4 [×2], C5×D4 [×4], C22×C10 [×15], C2×C4.4D4, D4×C10 [×6], C5×C4○D4 [×4], C23×C10, C5×C4.4D4 [×4], D4×C2×C10, C10×C4○D4 [×2], C10×C4.4D4
Generators and relations
G = < a,b,c,d | a10=b4=c4=1, d2=b2, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1, dcd-1=b2c-1 >
►On 160 points
Generators in S160
(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 26 55 35)(2 27 56 36)(3 28 57 37)(4 29 58 38)(5 30 59 39)(6 21 60 40)(7 22 51 31)(8 23 52 32)(9 24 53 33)(10 25 54 34)(11 68 159 43)(12 69 160 44)(13 70 151 45)(14 61 152 46)(15 62 153 47)(16 63 154 48)(17 64 155 49)(18 65 156 50)(19 66 157 41)(20 67 158 42)(71 125 110 132)(72 126 101 133)(73 127 102 134)(74 128 103 135)(75 129 104 136)(76 130 105 137)(77 121 106 138)(78 122 107 139)(79 123 108 140)(80 124 109 131)(81 117 98 146)(82 118 99 147)(83 119 100 148)(84 120 91 149)(85 111 92 150)(86 112 93 141)(87 113 94 142)(88 114 95 143)(89 115 96 144)(90 116 97 145)
(1 140 48 148)(2 131 49 149)(3 132 50 150)(4 133 41 141)(5 134 42 142)(6 135 43 143)(7 136 44 144)(8 137 45 145)(9 138 46 146)(10 139 47 147)(11 88 21 74)(12 89 22 75)(13 90 23 76)(14 81 24 77)(15 82 25 78)(16 83 26 79)(17 84 27 80)(18 85 28 71)(19 86 29 72)(20 87 30 73)(31 104 160 96)(32 105 151 97)(33 106 152 98)(34 107 153 99)(35 108 154 100)(36 109 155 91)(37 110 156 92)(38 101 157 93)(39 102 158 94)(40 103 159 95)(51 129 69 115)(52 130 70 116)(53 121 61 117)(54 122 62 118)(55 123 63 119)(56 124 64 120)(57 125 65 111)(58 126 66 112)(59 127 67 113)(60 128 68 114)
(1 83 55 100)(2 84 56 91)(3 85 57 92)(4 86 58 93)(5 87 59 94)(6 88 60 95)(7 89 51 96)(8 90 52 97)(9 81 53 98)(10 82 54 99)(11 135 159 128)(12 136 160 129)(13 137 151 130)(14 138 152 121)(15 139 153 122)(16 140 154 123)(17 131 155 124)(18 132 156 125)(19 133 157 126)(20 134 158 127)(21 143 40 114)(22 144 31 115)(23 145 32 116)(24 146 33 117)(25 147 34 118)(26 148 35 119)(27 149 36 120)(28 150 37 111)(29 141 38 112)(30 142 39 113)(41 72 66 101)(42 73 67 102)(43 74 68 103)(44 75 69 104)(45 76 70 105)(46 77 61 106)(47 78 62 107)(48 79 63 108)(49 80 64 109)(50 71 65 110)
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), (1,26,55,35)(2,27,56,36)(3,28,57,37)(4,29,58,38)(5,30,59,39)(6,21,60,40)(7,22,51,31)(8,23,52,32)(9,24,53,33)(10,25,54,34)(11,68,159,43)(12,69,160,44)(13,70,151,45)(14,61,152,46)(15,62,153,47)(16,63,154,48)(17,64,155,49)(18,65,156,50)(19,66,157,41)(20,67,158,42)(71,125,110,132)(72,126,101,133)(73,127,102,134)(74,128,103,135)(75,129,104,136)(76,130,105,137)(77,121,106,138)(78,122,107,139)(79,123,108,140)(80,124,109,131)(81,117,98,146)(82,118,99,147)(83,119,100,148)(84,120,91,149)(85,111,92,150)(86,112,93,141)(87,113,94,142)(88,114,95,143)(89,115,96,144)(90,116,97,145), (1,140,48,148)(2,131,49,149)(3,132,50,150)(4,133,41,141)(5,134,42,142)(6,135,43,143)(7,136,44,144)(8,137,45,145)(9,138,46,146)(10,139,47,147)(11,88,21,74)(12,89,22,75)(13,90,23,76)(14,81,24,77)(15,82,25,78)(16,83,26,79)(17,84,27,80)(18,85,28,71)(19,86,29,72)(20,87,30,73)(31,104,160,96)(32,105,151,97)(33,106,152,98)(34,107,153,99)(35,108,154,100)(36,109,155,91)(37,110,156,92)(38,101,157,93)(39,102,158,94)(40,103,159,95)(51,129,69,115)(52,130,70,116)(53,121,61,117)(54,122,62,118)(55,123,63,119)(56,124,64,120)(57,125,65,111)(58,126,66,112)(59,127,67,113)(60,128,68,114), (1,83,55,100)(2,84,56,91)(3,85,57,92)(4,86,58,93)(5,87,59,94)(6,88,60,95)(7,89,51,96)(8,90,52,97)(9,81,53,98)(10,82,54,99)(11,135,159,128)(12,136,160,129)(13,137,151,130)(14,138,152,121)(15,139,153,122)(16,140,154,123)(17,131,155,124)(18,132,156,125)(19,133,157,126)(20,134,158,127)(21,143,40,114)(22,144,31,115)(23,145,32,116)(24,146,33,117)(25,147,34,118)(26,148,35,119)(27,149,36,120)(28,150,37,111)(29,141,38,112)(30,142,39,113)(41,72,66,101)(42,73,67,102)(43,74,68,103)(44,75,69,104)(45,76,70,105)(46,77,61,106)(47,78,62,107)(48,79,63,108)(49,80,64,109)(50,71,65,110)>;
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), (1,26,55,35)(2,27,56,36)(3,28,57,37)(4,29,58,38)(5,30,59,39)(6,21,60,40)(7,22,51,31)(8,23,52,32)(9,24,53,33)(10,25,54,34)(11,68,159,43)(12,69,160,44)(13,70,151,45)(14,61,152,46)(15,62,153,47)(16,63,154,48)(17,64,155,49)(18,65,156,50)(19,66,157,41)(20,67,158,42)(71,125,110,132)(72,126,101,133)(73,127,102,134)(74,128,103,135)(75,129,104,136)(76,130,105,137)(77,121,106,138)(78,122,107,139)(79,123,108,140)(80,124,109,131)(81,117,98,146)(82,118,99,147)(83,119,100,148)(84,120,91,149)(85,111,92,150)(86,112,93,141)(87,113,94,142)(88,114,95,143)(89,115,96,144)(90,116,97,145), (1,140,48,148)(2,131,49,149)(3,132,50,150)(4,133,41,141)(5,134,42,142)(6,135,43,143)(7,136,44,144)(8,137,45,145)(9,138,46,146)(10,139,47,147)(11,88,21,74)(12,89,22,75)(13,90,23,76)(14,81,24,77)(15,82,25,78)(16,83,26,79)(17,84,27,80)(18,85,28,71)(19,86,29,72)(20,87,30,73)(31,104,160,96)(32,105,151,97)(33,106,152,98)(34,107,153,99)(35,108,154,100)(36,109,155,91)(37,110,156,92)(38,101,157,93)(39,102,158,94)(40,103,159,95)(51,129,69,115)(52,130,70,116)(53,121,61,117)(54,122,62,118)(55,123,63,119)(56,124,64,120)(57,125,65,111)(58,126,66,112)(59,127,67,113)(60,128,68,114), (1,83,55,100)(2,84,56,91)(3,85,57,92)(4,86,58,93)(5,87,59,94)(6,88,60,95)(7,89,51,96)(8,90,52,97)(9,81,53,98)(10,82,54,99)(11,135,159,128)(12,136,160,129)(13,137,151,130)(14,138,152,121)(15,139,153,122)(16,140,154,123)(17,131,155,124)(18,132,156,125)(19,133,157,126)(20,134,158,127)(21,143,40,114)(22,144,31,115)(23,145,32,116)(24,146,33,117)(25,147,34,118)(26,148,35,119)(27,149,36,120)(28,150,37,111)(29,141,38,112)(30,142,39,113)(41,72,66,101)(42,73,67,102)(43,74,68,103)(44,75,69,104)(45,76,70,105)(46,77,61,106)(47,78,62,107)(48,79,63,108)(49,80,64,109)(50,71,65,110) );
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)], [(1,26,55,35),(2,27,56,36),(3,28,57,37),(4,29,58,38),(5,30,59,39),(6,21,60,40),(7,22,51,31),(8,23,52,32),(9,24,53,33),(10,25,54,34),(11,68,159,43),(12,69,160,44),(13,70,151,45),(14,61,152,46),(15,62,153,47),(16,63,154,48),(17,64,155,49),(18,65,156,50),(19,66,157,41),(20,67,158,42),(71,125,110,132),(72,126,101,133),(73,127,102,134),(74,128,103,135),(75,129,104,136),(76,130,105,137),(77,121,106,138),(78,122,107,139),(79,123,108,140),(80,124,109,131),(81,117,98,146),(82,118,99,147),(83,119,100,148),(84,120,91,149),(85,111,92,150),(86,112,93,141),(87,113,94,142),(88,114,95,143),(89,115,96,144),(90,116,97,145)], [(1,140,48,148),(2,131,49,149),(3,132,50,150),(4,133,41,141),(5,134,42,142),(6,135,43,143),(7,136,44,144),(8,137,45,145),(9,138,46,146),(10,139,47,147),(11,88,21,74),(12,89,22,75),(13,90,23,76),(14,81,24,77),(15,82,25,78),(16,83,26,79),(17,84,27,80),(18,85,28,71),(19,86,29,72),(20,87,30,73),(31,104,160,96),(32,105,151,97),(33,106,152,98),(34,107,153,99),(35,108,154,100),(36,109,155,91),(37,110,156,92),(38,101,157,93),(39,102,158,94),(40,103,159,95),(51,129,69,115),(52,130,70,116),(53,121,61,117),(54,122,62,118),(55,123,63,119),(56,124,64,120),(57,125,65,111),(58,126,66,112),(59,127,67,113),(60,128,68,114)], [(1,83,55,100),(2,84,56,91),(3,85,57,92),(4,86,58,93),(5,87,59,94),(6,88,60,95),(7,89,51,96),(8,90,52,97),(9,81,53,98),(10,82,54,99),(11,135,159,128),(12,136,160,129),(13,137,151,130),(14,138,152,121),(15,139,153,122),(16,140,154,123),(17,131,155,124),(18,132,156,125),(19,133,157,126),(20,134,158,127),(21,143,40,114),(22,144,31,115),(23,145,32,116),(24,146,33,117),(25,147,34,118),(26,148,35,119),(27,149,36,120),(28,150,37,111),(29,141,38,112),(30,142,39,113),(41,72,66,101),(42,73,67,102),(43,74,68,103),(44,75,69,104),(45,76,70,105),(46,77,61,106),(47,78,62,107),(48,79,63,108),(49,80,64,109),(50,71,65,110)])
Matrix representation ►G ⊆ GL5(𝔽41)
| 40 | 0 | 0 | 0 | 0 |
| 0 | 25 | 0 | 0 | 0 |
| 0 | 0 | 25 | 0 | 0 |
| 0 | 0 | 0 | 23 | 0 |
| 0 | 0 | 0 | 0 | 23 |
| 40 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 1 | 23 |
| 0 | 0 | 0 | 32 | 40 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 40 | 39 | 0 | 0 |
| 0 | 1 | 1 | 0 | 0 |
| 0 | 0 | 0 | 32 | 39 |
| 0 | 0 | 0 | 40 | 9 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 2 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 9 | 0 |
| 0 | 0 | 0 | 1 | 32 |
G:=sub<GL(5,GF(41))| [40,0,0,0,0,0,25,0,0,0,0,0,25,0,0,0,0,0,23,0,0,0,0,0,23],[40,0,0,0,0,0,40,0,0,0,0,0,40,0,0,0,0,0,1,32,0,0,0,23,40],[1,0,0,0,0,0,40,1,0,0,0,39,1,0,0,0,0,0,32,40,0,0,0,39,9],[1,0,0,0,0,0,1,0,0,0,0,2,40,0,0,0,0,0,9,1,0,0,0,0,32] >;
140 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 4A | ··· | 4L | 4M | 4N | 4O | 4P | 5A | 5B | 5C | 5D | 10A | ··· | 10AB | 10AC | ··· | 10AR | 20A | ··· | 20AV | 20AW | ··· | 20BL |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 10 | ··· | 10 | 10 | ··· | 10 | 20 | ··· | 20 | 20 | ··· | 20 |
| size | 1 | 1 | ··· | 1 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
140 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | |||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C5 | C10 | C10 | C10 | C10 | C10 | D4 | C4○D4 | C5×D4 | C5×C4○D4 |
| kernel | C10×C4.4D4 | C2×C4×C20 | C10×C22⋊C4 | C5×C4.4D4 | D4×C2×C10 | Q8×C2×C10 | C2×C4.4D4 | C2×C42 | C2×C22⋊C4 | C4.4D4 | C22×D4 | C22×Q8 | C2×C20 | C2×C10 | C2×C4 | C22 |
| # reps | 1 | 1 | 4 | 8 | 1 | 1 | 4 | 4 | 16 | 32 | 4 | 4 | 4 | 8 | 16 | 32 |
In GAP, Magma, Sage, TeX
C_{10}\times C_4._4D_4 % in TeX
G:=Group("C10xC4.4D4"); // GroupNames label
G:=SmallGroup(320,1528);
// by ID
G=gap.SmallGroup(320,1528);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-5,-2,-2,1149,1128,3446,436]);
// Polycyclic
G:=Group<a,b,c,d|a^10=b^4=c^4=1,d^2=b^2,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=b^2*c^-1>;
// generators/relations
×
⋊
ℤ
𝔽
○
≀