metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.179D10, C10.852+ (1+4), C4⋊Q8⋊17D5, (C4×D20)⋊52C2, C4⋊2D20⋊41C2, C4⋊C4.222D10, (C2×Q8).89D10, C20.141(C4○D4), C20.23D4⋊28C2, (C4×C20).219C22, (C2×C10).278C24, (C2×C20).640C23, C4.42(Q8⋊2D5), C2.89(D4⋊6D10), (C2×D20).283C22, C4⋊Dic5.387C22, (Q8×C10).145C22, C22.299(C23×D5), D10⋊C4.53C22, C5⋊5(C22.49C24), (C2×Dic5).285C23, (C4×Dic5).175C22, (C22×D5).123C23, (C5×C4⋊Q8)⋊20C2, C4⋊C4⋊7D5⋊44C2, C10.125(C2×C4○D4), C2.33(C2×Q8⋊2D5), (C2×C4×D5).160C22, (C5×C4⋊C4).221C22, (C2×C4).603(C22×D5), SmallGroup(320,1406)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 918 in 236 conjugacy classes, 99 normal (13 characteristic)
C1, C2, C2 [×2], C2 [×4], C4 [×4], C4 [×9], C22, C22 [×12], C5, C2×C4, C2×C4 [×6], C2×C4 [×12], D4 [×8], Q8 [×2], C23 [×4], D5 [×4], C10, C10 [×2], C42, C42 [×4], C22⋊C4 [×12], C4⋊C4 [×4], C4⋊C4 [×2], C22×C4 [×4], C2×D4 [×6], C2×Q8 [×2], Dic5 [×4], C20 [×4], C20 [×5], D10 [×12], C2×C10, C42⋊C2 [×4], C4×D4 [×2], C4⋊D4 [×4], C4.4D4 [×4], C4⋊Q8, C4×D5 [×8], D20 [×8], C2×Dic5 [×4], C2×C20, C2×C20 [×6], C5×Q8 [×2], C22×D5 [×4], C22.49C24, C4×Dic5 [×4], C4⋊Dic5 [×2], D10⋊C4 [×12], C4×C20, C5×C4⋊C4 [×4], C2×C4×D5 [×4], C2×D20 [×6], Q8×C10 [×2], C4×D20 [×2], C4⋊C4⋊7D5 [×4], C4⋊2D20 [×4], C20.23D4 [×4], C5×C4⋊Q8, C42.179D10
Quotients:
C1, C2 [×15], C22 [×35], C23 [×15], D5, C4○D4 [×4], C24, D10 [×7], C2×C4○D4 [×2], 2+ (1+4), C22×D5 [×7], C22.49C24, Q8⋊2D5 [×4], C23×D5, D4⋊6D10, C2×Q8⋊2D5 [×2], C42.179D10
Generators and relations
G = < a,b,c,d | a4=b4=1, c10=b2, d2=a2b2, ab=ba, cac-1=dad-1=a-1, cbc-1=b-1, dbd-1=a2b, dcd-1=a2c9 >
►On 160 points
Generators in S160
(1 136 76 34)(2 35 77 137)(3 138 78 36)(4 37 79 139)(5 140 80 38)(6 39 61 121)(7 122 62 40)(8 21 63 123)(9 124 64 22)(10 23 65 125)(11 126 66 24)(12 25 67 127)(13 128 68 26)(14 27 69 129)(15 130 70 28)(16 29 71 131)(17 132 72 30)(18 31 73 133)(19 134 74 32)(20 33 75 135)(41 153 87 115)(42 116 88 154)(43 155 89 117)(44 118 90 156)(45 157 91 119)(46 120 92 158)(47 159 93 101)(48 102 94 160)(49 141 95 103)(50 104 96 142)(51 143 97 105)(52 106 98 144)(53 145 99 107)(54 108 100 146)(55 147 81 109)(56 110 82 148)(57 149 83 111)(58 112 84 150)(59 151 85 113)(60 114 86 152)
(1 143 11 153)(2 154 12 144)(3 145 13 155)(4 156 14 146)(5 147 15 157)(6 158 16 148)(7 149 17 159)(8 160 18 150)(9 151 19 141)(10 142 20 152)(21 48 31 58)(22 59 32 49)(23 50 33 60)(24 41 34 51)(25 52 35 42)(26 43 36 53)(27 54 37 44)(28 45 38 55)(29 56 39 46)(30 47 40 57)(61 120 71 110)(62 111 72 101)(63 102 73 112)(64 113 74 103)(65 104 75 114)(66 115 76 105)(67 106 77 116)(68 117 78 107)(69 108 79 118)(70 119 80 109)(81 130 91 140)(82 121 92 131)(83 132 93 122)(84 123 94 133)(85 134 95 124)(86 125 96 135)(87 136 97 126)(88 127 98 137)(89 138 99 128)(90 129 100 139)
(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 81 66 45)(2 44 67 100)(3 99 68 43)(4 42 69 98)(5 97 70 41)(6 60 71 96)(7 95 72 59)(8 58 73 94)(9 93 74 57)(10 56 75 92)(11 91 76 55)(12 54 77 90)(13 89 78 53)(14 52 79 88)(15 87 80 51)(16 50 61 86)(17 85 62 49)(18 48 63 84)(19 83 64 47)(20 46 65 82)(21 150 133 102)(22 101 134 149)(23 148 135 120)(24 119 136 147)(25 146 137 118)(26 117 138 145)(27 144 139 116)(28 115 140 143)(29 142 121 114)(30 113 122 141)(31 160 123 112)(32 111 124 159)(33 158 125 110)(34 109 126 157)(35 156 127 108)(36 107 128 155)(37 154 129 106)(38 105 130 153)(39 152 131 104)(40 103 132 151)
G:=sub<Sym(160)| (1,136,76,34)(2,35,77,137)(3,138,78,36)(4,37,79,139)(5,140,80,38)(6,39,61,121)(7,122,62,40)(8,21,63,123)(9,124,64,22)(10,23,65,125)(11,126,66,24)(12,25,67,127)(13,128,68,26)(14,27,69,129)(15,130,70,28)(16,29,71,131)(17,132,72,30)(18,31,73,133)(19,134,74,32)(20,33,75,135)(41,153,87,115)(42,116,88,154)(43,155,89,117)(44,118,90,156)(45,157,91,119)(46,120,92,158)(47,159,93,101)(48,102,94,160)(49,141,95,103)(50,104,96,142)(51,143,97,105)(52,106,98,144)(53,145,99,107)(54,108,100,146)(55,147,81,109)(56,110,82,148)(57,149,83,111)(58,112,84,150)(59,151,85,113)(60,114,86,152), (1,143,11,153)(2,154,12,144)(3,145,13,155)(4,156,14,146)(5,147,15,157)(6,158,16,148)(7,149,17,159)(8,160,18,150)(9,151,19,141)(10,142,20,152)(21,48,31,58)(22,59,32,49)(23,50,33,60)(24,41,34,51)(25,52,35,42)(26,43,36,53)(27,54,37,44)(28,45,38,55)(29,56,39,46)(30,47,40,57)(61,120,71,110)(62,111,72,101)(63,102,73,112)(64,113,74,103)(65,104,75,114)(66,115,76,105)(67,106,77,116)(68,117,78,107)(69,108,79,118)(70,119,80,109)(81,130,91,140)(82,121,92,131)(83,132,93,122)(84,123,94,133)(85,134,95,124)(86,125,96,135)(87,136,97,126)(88,127,98,137)(89,138,99,128)(90,129,100,139), (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), (1,81,66,45)(2,44,67,100)(3,99,68,43)(4,42,69,98)(5,97,70,41)(6,60,71,96)(7,95,72,59)(8,58,73,94)(9,93,74,57)(10,56,75,92)(11,91,76,55)(12,54,77,90)(13,89,78,53)(14,52,79,88)(15,87,80,51)(16,50,61,86)(17,85,62,49)(18,48,63,84)(19,83,64,47)(20,46,65,82)(21,150,133,102)(22,101,134,149)(23,148,135,120)(24,119,136,147)(25,146,137,118)(26,117,138,145)(27,144,139,116)(28,115,140,143)(29,142,121,114)(30,113,122,141)(31,160,123,112)(32,111,124,159)(33,158,125,110)(34,109,126,157)(35,156,127,108)(36,107,128,155)(37,154,129,106)(38,105,130,153)(39,152,131,104)(40,103,132,151)>;
G:=Group( (1,136,76,34)(2,35,77,137)(3,138,78,36)(4,37,79,139)(5,140,80,38)(6,39,61,121)(7,122,62,40)(8,21,63,123)(9,124,64,22)(10,23,65,125)(11,126,66,24)(12,25,67,127)(13,128,68,26)(14,27,69,129)(15,130,70,28)(16,29,71,131)(17,132,72,30)(18,31,73,133)(19,134,74,32)(20,33,75,135)(41,153,87,115)(42,116,88,154)(43,155,89,117)(44,118,90,156)(45,157,91,119)(46,120,92,158)(47,159,93,101)(48,102,94,160)(49,141,95,103)(50,104,96,142)(51,143,97,105)(52,106,98,144)(53,145,99,107)(54,108,100,146)(55,147,81,109)(56,110,82,148)(57,149,83,111)(58,112,84,150)(59,151,85,113)(60,114,86,152), (1,143,11,153)(2,154,12,144)(3,145,13,155)(4,156,14,146)(5,147,15,157)(6,158,16,148)(7,149,17,159)(8,160,18,150)(9,151,19,141)(10,142,20,152)(21,48,31,58)(22,59,32,49)(23,50,33,60)(24,41,34,51)(25,52,35,42)(26,43,36,53)(27,54,37,44)(28,45,38,55)(29,56,39,46)(30,47,40,57)(61,120,71,110)(62,111,72,101)(63,102,73,112)(64,113,74,103)(65,104,75,114)(66,115,76,105)(67,106,77,116)(68,117,78,107)(69,108,79,118)(70,119,80,109)(81,130,91,140)(82,121,92,131)(83,132,93,122)(84,123,94,133)(85,134,95,124)(86,125,96,135)(87,136,97,126)(88,127,98,137)(89,138,99,128)(90,129,100,139), (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), (1,81,66,45)(2,44,67,100)(3,99,68,43)(4,42,69,98)(5,97,70,41)(6,60,71,96)(7,95,72,59)(8,58,73,94)(9,93,74,57)(10,56,75,92)(11,91,76,55)(12,54,77,90)(13,89,78,53)(14,52,79,88)(15,87,80,51)(16,50,61,86)(17,85,62,49)(18,48,63,84)(19,83,64,47)(20,46,65,82)(21,150,133,102)(22,101,134,149)(23,148,135,120)(24,119,136,147)(25,146,137,118)(26,117,138,145)(27,144,139,116)(28,115,140,143)(29,142,121,114)(30,113,122,141)(31,160,123,112)(32,111,124,159)(33,158,125,110)(34,109,126,157)(35,156,127,108)(36,107,128,155)(37,154,129,106)(38,105,130,153)(39,152,131,104)(40,103,132,151) );
G=PermutationGroup([(1,136,76,34),(2,35,77,137),(3,138,78,36),(4,37,79,139),(5,140,80,38),(6,39,61,121),(7,122,62,40),(8,21,63,123),(9,124,64,22),(10,23,65,125),(11,126,66,24),(12,25,67,127),(13,128,68,26),(14,27,69,129),(15,130,70,28),(16,29,71,131),(17,132,72,30),(18,31,73,133),(19,134,74,32),(20,33,75,135),(41,153,87,115),(42,116,88,154),(43,155,89,117),(44,118,90,156),(45,157,91,119),(46,120,92,158),(47,159,93,101),(48,102,94,160),(49,141,95,103),(50,104,96,142),(51,143,97,105),(52,106,98,144),(53,145,99,107),(54,108,100,146),(55,147,81,109),(56,110,82,148),(57,149,83,111),(58,112,84,150),(59,151,85,113),(60,114,86,152)], [(1,143,11,153),(2,154,12,144),(3,145,13,155),(4,156,14,146),(5,147,15,157),(6,158,16,148),(7,149,17,159),(8,160,18,150),(9,151,19,141),(10,142,20,152),(21,48,31,58),(22,59,32,49),(23,50,33,60),(24,41,34,51),(25,52,35,42),(26,43,36,53),(27,54,37,44),(28,45,38,55),(29,56,39,46),(30,47,40,57),(61,120,71,110),(62,111,72,101),(63,102,73,112),(64,113,74,103),(65,104,75,114),(66,115,76,105),(67,106,77,116),(68,117,78,107),(69,108,79,118),(70,119,80,109),(81,130,91,140),(82,121,92,131),(83,132,93,122),(84,123,94,133),(85,134,95,124),(86,125,96,135),(87,136,97,126),(88,127,98,137),(89,138,99,128),(90,129,100,139)], [(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)], [(1,81,66,45),(2,44,67,100),(3,99,68,43),(4,42,69,98),(5,97,70,41),(6,60,71,96),(7,95,72,59),(8,58,73,94),(9,93,74,57),(10,56,75,92),(11,91,76,55),(12,54,77,90),(13,89,78,53),(14,52,79,88),(15,87,80,51),(16,50,61,86),(17,85,62,49),(18,48,63,84),(19,83,64,47),(20,46,65,82),(21,150,133,102),(22,101,134,149),(23,148,135,120),(24,119,136,147),(25,146,137,118),(26,117,138,145),(27,144,139,116),(28,115,140,143),(29,142,121,114),(30,113,122,141),(31,160,123,112),(32,111,124,159),(33,158,125,110),(34,109,126,157),(35,156,127,108),(36,107,128,155),(37,154,129,106),(38,105,130,153),(39,152,131,104),(40,103,132,151)])
Matrix representation ►G ⊆ GL6(𝔽41)
| 13 | 34 | 0 | 0 | 0 | 0 |
| 36 | 28 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 |
| 28 | 7 | 0 | 0 | 0 | 0 |
| 5 | 13 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 32 | 0 |
| 0 | 0 | 0 | 0 | 20 | 9 |
| 6 | 1 | 0 | 0 | 0 | 0 |
| 4 | 35 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 6 | 0 | 0 |
| 0 | 0 | 34 | 7 | 0 | 0 |
| 0 | 0 | 0 | 0 | 32 | 37 |
| 0 | 0 | 0 | 0 | 0 | 9 |
| 1 | 38 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 34 | 6 | 0 | 0 |
| 0 | 0 | 33 | 7 | 0 | 0 |
| 0 | 0 | 0 | 0 | 9 | 0 |
| 0 | 0 | 0 | 0 | 0 | 9 |
G:=sub<GL(6,GF(41))| [13,36,0,0,0,0,34,28,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[28,5,0,0,0,0,7,13,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,32,20,0,0,0,0,0,9],[6,4,0,0,0,0,1,35,0,0,0,0,0,0,0,34,0,0,0,0,6,7,0,0,0,0,0,0,32,0,0,0,0,0,37,9],[1,0,0,0,0,0,38,40,0,0,0,0,0,0,34,33,0,0,0,0,6,7,0,0,0,0,0,0,9,0,0,0,0,0,0,9] >;
53 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | ··· | 4I | 4J | ··· | 4Q | 5A | 5B | 10A | ··· | 10F | 20A | ··· | 20L | 20M | ··· | 20T |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | ··· | 4 | 5 | 5 | 10 | ··· | 10 | 20 | ··· | 20 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 20 | 20 | 20 | 20 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 10 | ··· | 10 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | ··· | 8 |
53 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | D5 | C4○D4 | D10 | D10 | D10 | 2+ (1+4) | Q8⋊2D5 | D4⋊6D10 |
| kernel | C42.179D10 | C4×D20 | C4⋊C4⋊7D5 | C4⋊2D20 | C20.23D4 | C5×C4⋊Q8 | C4⋊Q8 | C20 | C42 | C4⋊C4 | C2×Q8 | C10 | C4 | C2 |
| # reps | 1 | 2 | 4 | 4 | 4 | 1 | 2 | 8 | 2 | 8 | 4 | 1 | 8 | 4 |
In GAP, Magma, Sage, TeX
C_4^2._{179}D_{10} % in TeX
G:=Group("C4^2.179D10"); // GroupNames label
G:=SmallGroup(320,1406);
// by ID
G=gap.SmallGroup(320,1406);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,232,758,387,100,675,570,185,80,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^10=b^2,d^2=a^2*b^2,a*b=b*a,c*a*c^-1=d*a*d^-1=a^-1,c*b*c^-1=b^-1,d*b*d^-1=a^2*b,d*c*d^-1=a^2*c^9>;
// generators/relations
×
⋊
ℤ
𝔽
○
≀