metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.87D10, C10.462- (1+4), C4⋊C4.308D10, (C4×Dic10)⋊5C2, (C2×Dic10)⋊29C4, (C4×C20).20C22, (C2×C10).60C24, C10.36(C23×C4), Dic5⋊3Q8⋊11C2, (C2×C20).581C23, C20.178(C22×C4), C22⋊C4.123D10, Dic10.46(C2×C4), C42⋊C2.10D5, (C22×C4).185D10, C22.25(C23×D5), C4⋊Dic5.396C22, (C4×Dic5).75C22, Dic5.14(C22×C4), C23.149(C22×D5), C23.D5.90C22, C2.1(D4.10D10), (C22×C10).130C23, (C22×C20).221C22, C5⋊2(C23.32C23), (C22×Dic10).18C2, (C2×Dic5).202C23, C23.11D10.5C2, (C2×Dic10).291C22, C10.D4.130C22, C23.21D10.21C2, (C22×Dic5).84C22, C4.57(C2×C4×D5), (C2×C4).57(C4×D5), C22.25(C2×C4×D5), C2.17(D5×C22×C4), (C2×C20).302(C2×C4), (C5×C4⋊C4).301C22, (C2×Dic5).38(C2×C4), (C2×C4).268(C22×D5), (C2×C10).120(C22×C4), (C5×C42⋊C2).11C2, (C5×C22⋊C4).133C22, SmallGroup(320,1188)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 686 in 266 conjugacy classes, 151 normal (19 characteristic)
C1, C2, C2 [×2], C2 [×2], C4 [×4], C4 [×16], C22, C22 [×2], C22 [×2], C5, C2×C4 [×2], C2×C4 [×8], C2×C4 [×16], Q8 [×16], C23, C10, C10 [×2], C10 [×2], C42 [×2], C42 [×10], C22⋊C4 [×2], C22⋊C4 [×2], C4⋊C4 [×2], C4⋊C4 [×10], C22×C4, C22×C4 [×2], C2×Q8 [×12], Dic5 [×8], Dic5 [×4], C20 [×4], C20 [×4], C2×C10, C2×C10 [×2], C2×C10 [×2], C42⋊C2, C42⋊C2 [×5], C4×Q8 [×8], C22×Q8, Dic10 [×16], C2×Dic5 [×16], C2×C20 [×2], C2×C20 [×8], C22×C10, C23.32C23, C4×Dic5 [×10], C10.D4 [×8], C4⋊Dic5 [×2], C23.D5 [×2], C4×C20 [×2], C5×C22⋊C4 [×2], C5×C4⋊C4 [×2], C2×Dic10 [×12], C22×Dic5 [×2], C22×C20, C4×Dic10 [×4], C23.11D10 [×4], Dic5⋊3Q8 [×4], C23.21D10, C5×C42⋊C2, C22×Dic10, C42.87D10
Quotients:
C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], C23 [×15], D5, C22×C4 [×14], C24, D10 [×7], C23×C4, 2- (1+4) [×2], C4×D5 [×4], C22×D5 [×7], C23.32C23, C2×C4×D5 [×6], C23×D5, D5×C22×C4, D4.10D10 [×2], C42.87D10
Generators and relations
G = < a,b,c,d | a4=b4=c10=1, d2=a2, ab=ba, ac=ca, dad-1=a-1, cbc-1=dbd-1=a2b, dcd-1=c-1 >
►On 160 points
Generators in S160
(1 48 8 43)(2 49 9 44)(3 50 10 45)(4 46 6 41)(5 47 7 42)(11 56 16 51)(12 57 17 52)(13 58 18 53)(14 59 19 54)(15 60 20 55)(21 28 72 62)(22 29 73 63)(23 30 74 64)(24 26 75 65)(25 27 71 61)(31 70 39 77)(32 66 40 78)(33 67 36 79)(34 68 37 80)(35 69 38 76)(81 109 86 104)(82 110 87 105)(83 101 88 106)(84 102 89 107)(85 103 90 108)(91 153 96 158)(92 154 97 159)(93 155 98 160)(94 156 99 151)(95 157 100 152)(111 125 116 130)(112 126 117 121)(113 127 118 122)(114 128 119 123)(115 129 120 124)(131 142 136 147)(132 143 137 148)(133 144 138 149)(134 145 139 150)(135 146 140 141)
(1 89 13 151)(2 85 14 157)(3 81 15 153)(4 87 11 159)(5 83 12 155)(6 82 16 154)(7 88 17 160)(8 84 18 156)(9 90 19 152)(10 86 20 158)(21 118 79 148)(22 114 80 144)(23 120 76 150)(24 116 77 146)(25 112 78 142)(26 130 31 140)(27 126 32 136)(28 122 33 132)(29 128 34 138)(30 124 35 134)(36 137 62 127)(37 133 63 123)(38 139 64 129)(39 135 65 125)(40 131 61 121)(41 110 51 97)(42 106 52 93)(43 102 53 99)(44 108 54 95)(45 104 55 91)(46 105 56 92)(47 101 57 98)(48 107 58 94)(49 103 59 100)(50 109 60 96)(66 147 71 117)(67 143 72 113)(68 149 73 119)(69 145 74 115)(70 141 75 111)
(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 39 8 31)(2 38 9 35)(3 37 10 34)(4 36 6 33)(5 40 7 32)(11 62 16 28)(12 61 17 27)(13 65 18 26)(14 64 19 30)(15 63 20 29)(21 56 72 51)(22 60 73 55)(23 59 74 54)(24 58 75 53)(25 57 71 52)(41 79 46 67)(42 78 47 66)(43 77 48 70)(44 76 49 69)(45 80 50 68)(81 138 86 133)(82 137 87 132)(83 136 88 131)(84 135 89 140)(85 134 90 139)(91 119 96 114)(92 118 97 113)(93 117 98 112)(94 116 99 111)(95 115 100 120)(101 142 106 147)(102 141 107 146)(103 150 108 145)(104 149 109 144)(105 148 110 143)(121 155 126 160)(122 154 127 159)(123 153 128 158)(124 152 129 157)(125 151 130 156)
G:=sub<Sym(160)| (1,48,8,43)(2,49,9,44)(3,50,10,45)(4,46,6,41)(5,47,7,42)(11,56,16,51)(12,57,17,52)(13,58,18,53)(14,59,19,54)(15,60,20,55)(21,28,72,62)(22,29,73,63)(23,30,74,64)(24,26,75,65)(25,27,71,61)(31,70,39,77)(32,66,40,78)(33,67,36,79)(34,68,37,80)(35,69,38,76)(81,109,86,104)(82,110,87,105)(83,101,88,106)(84,102,89,107)(85,103,90,108)(91,153,96,158)(92,154,97,159)(93,155,98,160)(94,156,99,151)(95,157,100,152)(111,125,116,130)(112,126,117,121)(113,127,118,122)(114,128,119,123)(115,129,120,124)(131,142,136,147)(132,143,137,148)(133,144,138,149)(134,145,139,150)(135,146,140,141), (1,89,13,151)(2,85,14,157)(3,81,15,153)(4,87,11,159)(5,83,12,155)(6,82,16,154)(7,88,17,160)(8,84,18,156)(9,90,19,152)(10,86,20,158)(21,118,79,148)(22,114,80,144)(23,120,76,150)(24,116,77,146)(25,112,78,142)(26,130,31,140)(27,126,32,136)(28,122,33,132)(29,128,34,138)(30,124,35,134)(36,137,62,127)(37,133,63,123)(38,139,64,129)(39,135,65,125)(40,131,61,121)(41,110,51,97)(42,106,52,93)(43,102,53,99)(44,108,54,95)(45,104,55,91)(46,105,56,92)(47,101,57,98)(48,107,58,94)(49,103,59,100)(50,109,60,96)(66,147,71,117)(67,143,72,113)(68,149,73,119)(69,145,74,115)(70,141,75,111), (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), (1,39,8,31)(2,38,9,35)(3,37,10,34)(4,36,6,33)(5,40,7,32)(11,62,16,28)(12,61,17,27)(13,65,18,26)(14,64,19,30)(15,63,20,29)(21,56,72,51)(22,60,73,55)(23,59,74,54)(24,58,75,53)(25,57,71,52)(41,79,46,67)(42,78,47,66)(43,77,48,70)(44,76,49,69)(45,80,50,68)(81,138,86,133)(82,137,87,132)(83,136,88,131)(84,135,89,140)(85,134,90,139)(91,119,96,114)(92,118,97,113)(93,117,98,112)(94,116,99,111)(95,115,100,120)(101,142,106,147)(102,141,107,146)(103,150,108,145)(104,149,109,144)(105,148,110,143)(121,155,126,160)(122,154,127,159)(123,153,128,158)(124,152,129,157)(125,151,130,156)>;
G:=Group( (1,48,8,43)(2,49,9,44)(3,50,10,45)(4,46,6,41)(5,47,7,42)(11,56,16,51)(12,57,17,52)(13,58,18,53)(14,59,19,54)(15,60,20,55)(21,28,72,62)(22,29,73,63)(23,30,74,64)(24,26,75,65)(25,27,71,61)(31,70,39,77)(32,66,40,78)(33,67,36,79)(34,68,37,80)(35,69,38,76)(81,109,86,104)(82,110,87,105)(83,101,88,106)(84,102,89,107)(85,103,90,108)(91,153,96,158)(92,154,97,159)(93,155,98,160)(94,156,99,151)(95,157,100,152)(111,125,116,130)(112,126,117,121)(113,127,118,122)(114,128,119,123)(115,129,120,124)(131,142,136,147)(132,143,137,148)(133,144,138,149)(134,145,139,150)(135,146,140,141), (1,89,13,151)(2,85,14,157)(3,81,15,153)(4,87,11,159)(5,83,12,155)(6,82,16,154)(7,88,17,160)(8,84,18,156)(9,90,19,152)(10,86,20,158)(21,118,79,148)(22,114,80,144)(23,120,76,150)(24,116,77,146)(25,112,78,142)(26,130,31,140)(27,126,32,136)(28,122,33,132)(29,128,34,138)(30,124,35,134)(36,137,62,127)(37,133,63,123)(38,139,64,129)(39,135,65,125)(40,131,61,121)(41,110,51,97)(42,106,52,93)(43,102,53,99)(44,108,54,95)(45,104,55,91)(46,105,56,92)(47,101,57,98)(48,107,58,94)(49,103,59,100)(50,109,60,96)(66,147,71,117)(67,143,72,113)(68,149,73,119)(69,145,74,115)(70,141,75,111), (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), (1,39,8,31)(2,38,9,35)(3,37,10,34)(4,36,6,33)(5,40,7,32)(11,62,16,28)(12,61,17,27)(13,65,18,26)(14,64,19,30)(15,63,20,29)(21,56,72,51)(22,60,73,55)(23,59,74,54)(24,58,75,53)(25,57,71,52)(41,79,46,67)(42,78,47,66)(43,77,48,70)(44,76,49,69)(45,80,50,68)(81,138,86,133)(82,137,87,132)(83,136,88,131)(84,135,89,140)(85,134,90,139)(91,119,96,114)(92,118,97,113)(93,117,98,112)(94,116,99,111)(95,115,100,120)(101,142,106,147)(102,141,107,146)(103,150,108,145)(104,149,109,144)(105,148,110,143)(121,155,126,160)(122,154,127,159)(123,153,128,158)(124,152,129,157)(125,151,130,156) );
G=PermutationGroup([(1,48,8,43),(2,49,9,44),(3,50,10,45),(4,46,6,41),(5,47,7,42),(11,56,16,51),(12,57,17,52),(13,58,18,53),(14,59,19,54),(15,60,20,55),(21,28,72,62),(22,29,73,63),(23,30,74,64),(24,26,75,65),(25,27,71,61),(31,70,39,77),(32,66,40,78),(33,67,36,79),(34,68,37,80),(35,69,38,76),(81,109,86,104),(82,110,87,105),(83,101,88,106),(84,102,89,107),(85,103,90,108),(91,153,96,158),(92,154,97,159),(93,155,98,160),(94,156,99,151),(95,157,100,152),(111,125,116,130),(112,126,117,121),(113,127,118,122),(114,128,119,123),(115,129,120,124),(131,142,136,147),(132,143,137,148),(133,144,138,149),(134,145,139,150),(135,146,140,141)], [(1,89,13,151),(2,85,14,157),(3,81,15,153),(4,87,11,159),(5,83,12,155),(6,82,16,154),(7,88,17,160),(8,84,18,156),(9,90,19,152),(10,86,20,158),(21,118,79,148),(22,114,80,144),(23,120,76,150),(24,116,77,146),(25,112,78,142),(26,130,31,140),(27,126,32,136),(28,122,33,132),(29,128,34,138),(30,124,35,134),(36,137,62,127),(37,133,63,123),(38,139,64,129),(39,135,65,125),(40,131,61,121),(41,110,51,97),(42,106,52,93),(43,102,53,99),(44,108,54,95),(45,104,55,91),(46,105,56,92),(47,101,57,98),(48,107,58,94),(49,103,59,100),(50,109,60,96),(66,147,71,117),(67,143,72,113),(68,149,73,119),(69,145,74,115),(70,141,75,111)], [(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)], [(1,39,8,31),(2,38,9,35),(3,37,10,34),(4,36,6,33),(5,40,7,32),(11,62,16,28),(12,61,17,27),(13,65,18,26),(14,64,19,30),(15,63,20,29),(21,56,72,51),(22,60,73,55),(23,59,74,54),(24,58,75,53),(25,57,71,52),(41,79,46,67),(42,78,47,66),(43,77,48,70),(44,76,49,69),(45,80,50,68),(81,138,86,133),(82,137,87,132),(83,136,88,131),(84,135,89,140),(85,134,90,139),(91,119,96,114),(92,118,97,113),(93,117,98,112),(94,116,99,111),(95,115,100,120),(101,142,106,147),(102,141,107,146),(103,150,108,145),(104,149,109,144),(105,148,110,143),(121,155,126,160),(122,154,127,159),(123,153,128,158),(124,152,129,157),(125,151,130,156)])
Matrix representation ►G ⊆ GL6(𝔽41)
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 2 | 32 | 0 | 0 |
| 0 | 0 | 37 | 39 | 0 | 0 |
| 0 | 0 | 0 | 0 | 11 | 32 |
| 0 | 0 | 0 | 0 | 9 | 30 |
| 9 | 0 | 0 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 | 0 | 0 |
| 0 | 0 | 9 | 0 | 23 | 23 |
| 0 | 0 | 0 | 9 | 36 | 38 |
| 0 | 0 | 0 | 0 | 32 | 0 |
| 0 | 0 | 0 | 0 | 0 | 32 |
| 0 | 6 | 0 | 0 | 0 | 0 |
| 34 | 7 | 0 | 0 | 0 | 0 |
| 0 | 0 | 6 | 7 | 0 | 0 |
| 0 | 0 | 35 | 0 | 0 | 0 |
| 0 | 0 | 19 | 10 | 1 | 34 |
| 0 | 0 | 28 | 38 | 7 | 34 |
| 26 | 38 | 0 | 0 | 0 | 0 |
| 20 | 15 | 0 | 0 | 0 | 0 |
| 0 | 0 | 13 | 22 | 0 | 0 |
| 0 | 0 | 37 | 28 | 0 | 0 |
| 0 | 0 | 6 | 14 | 22 | 19 |
| 0 | 0 | 7 | 8 | 9 | 19 |
G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,2,37,0,0,0,0,32,39,0,0,0,0,0,0,11,9,0,0,0,0,32,30],[9,0,0,0,0,0,0,9,0,0,0,0,0,0,9,0,0,0,0,0,0,9,0,0,0,0,23,36,32,0,0,0,23,38,0,32],[0,34,0,0,0,0,6,7,0,0,0,0,0,0,6,35,19,28,0,0,7,0,10,38,0,0,0,0,1,7,0,0,0,0,34,34],[26,20,0,0,0,0,38,15,0,0,0,0,0,0,13,37,6,7,0,0,22,28,14,8,0,0,0,0,22,9,0,0,0,0,19,19] >;
74 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | ··· | 4L | 4M | ··· | 4AB | 5A | 5B | 10A | ··· | 10F | 10G | 10H | 10I | 10J | 20A | ··· | 20H | 20I | ··· | 20AB |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 5 | 5 | 10 | ··· | 10 | 10 | 10 | 10 | 10 | 20 | ··· | 20 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | ··· | 2 | 10 | ··· | 10 | 2 | 2 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
74 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | - | - | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | D5 | D10 | D10 | D10 | D10 | C4×D5 | 2- (1+4) | D4.10D10 |
| kernel | C42.87D10 | C4×Dic10 | C23.11D10 | Dic5⋊3Q8 | C23.21D10 | C5×C42⋊C2 | C22×Dic10 | C2×Dic10 | C42⋊C2 | C42 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×C4 | C10 | C2 |
| # reps | 1 | 4 | 4 | 4 | 1 | 1 | 1 | 16 | 2 | 4 | 4 | 4 | 2 | 16 | 2 | 8 |
In GAP, Magma, Sage, TeX
C_4^2._{87}D_{10} % in TeX
G:=Group("C4^2.87D10"); // GroupNames label
G:=SmallGroup(320,1188);
// by ID
G=gap.SmallGroup(320,1188);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,224,758,184,570,80,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^10=1,d^2=a^2,a*b=b*a,a*c=c*a,d*a*d^-1=a^-1,c*b*c^-1=d*b*d^-1=a^2*b,d*c*d^-1=c^-1>;
// generators/relations
×
⋊
ℤ
𝔽
○
≀