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, Dic5.14(C22×C4), (C4×Dic5).75C22, C23.149(C22×D5), C23.D5.90C22, C2.1(D4.10D10), (C22×C20).221C22, (C22×C10).130C23, 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), C2.17(D5×C22×C4), C22.25(C2×C4×D5), (C2×C20).302(C2×C4), (C5×C4⋊C4).301C22, (C2×Dic5).38(C2×C4), (C2×C4).268(C22×D5), (C5×C42⋊C2).11C2, (C2×C10).120(C22×C4), (C5×C22⋊C4).133C22, SmallGroup(320,1188)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42.87D10
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 >
Subgroups: 686 in 266 conjugacy classes, 151 normal (19 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C5, C2×C4, C2×C4, C2×C4, Q8, C23, C10, C10, C10, C42, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C2×Q8, Dic5, Dic5, C20, C20, C2×C10, C2×C10, C2×C10, C42⋊C2, C42⋊C2, C4×Q8, C22×Q8, Dic10, C2×Dic5, C2×C20, C2×C20, C22×C10, C23.32C23, C4×Dic5, C10.D4, C4⋊Dic5, C23.D5, C4×C20, C5×C22⋊C4, C5×C4⋊C4, C2×Dic10, C22×Dic5, C22×C20, C4×Dic10, C23.11D10, Dic5⋊3Q8, C23.21D10, C5×C42⋊C2, C22×Dic10, C42.87D10
Quotients: C1, C2, C4, C22, C2×C4, C23, D5, C22×C4, C24, D10, C23×C4, 2- 1+4, C4×D5, C22×D5, C23.32C23, C2×C4×D5, C23×D5, D5×C22×C4, D4.10D10, C42.87D10
►On 160 points
Generators in S160
(1 50 8 45)(2 46 9 41)(3 47 10 42)(4 48 6 43)(5 49 7 44)(11 56 61 53)(12 57 62 54)(13 58 63 55)(14 59 64 51)(15 60 65 52)(16 77 31 75)(17 78 32 71)(18 79 33 72)(19 80 34 73)(20 76 35 74)(21 70 28 37)(22 66 29 38)(23 67 30 39)(24 68 26 40)(25 69 27 36)(81 137 86 132)(82 138 87 133)(83 139 88 134)(84 140 89 135)(85 131 90 136)(91 113 96 118)(92 114 97 119)(93 115 98 120)(94 116 99 111)(95 117 100 112)(101 148 106 143)(102 149 107 144)(103 150 108 145)(104 141 109 146)(105 142 110 147)(121 151 126 156)(122 152 127 157)(123 153 128 158)(124 154 129 159)(125 155 130 160)
(1 83 13 112)(2 89 14 118)(3 85 15 114)(4 81 11 120)(5 87 12 116)(6 86 61 115)(7 82 62 111)(8 88 63 117)(9 84 64 113)(10 90 65 119)(16 129 39 102)(17 125 40 108)(18 121 36 104)(19 127 37 110)(20 123 38 106)(21 147 80 157)(22 143 76 153)(23 149 77 159)(24 145 78 155)(25 141 79 151)(26 150 71 160)(27 146 72 156)(28 142 73 152)(29 148 74 158)(30 144 75 154)(31 124 67 107)(32 130 68 103)(33 126 69 109)(34 122 70 105)(35 128 66 101)(41 140 51 96)(42 136 52 92)(43 132 53 98)(44 138 54 94)(45 134 55 100)(46 135 59 91)(47 131 60 97)(48 137 56 93)(49 133 57 99)(50 139 58 95)
(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 33 8 18)(2 32 9 17)(3 31 10 16)(4 35 6 20)(5 34 7 19)(11 66 61 38)(12 70 62 37)(13 69 63 36)(14 68 64 40)(15 67 65 39)(21 54 28 57)(22 53 29 56)(23 52 30 60)(24 51 26 59)(25 55 27 58)(41 71 46 78)(42 75 47 77)(43 74 48 76)(44 73 49 80)(45 72 50 79)(81 123 86 128)(82 122 87 127)(83 121 88 126)(84 130 89 125)(85 129 90 124)(91 150 96 145)(92 149 97 144)(93 148 98 143)(94 147 99 142)(95 146 100 141)(101 120 106 115)(102 119 107 114)(103 118 108 113)(104 117 109 112)(105 116 110 111)(131 154 136 159)(132 153 137 158)(133 152 138 157)(134 151 139 156)(135 160 140 155)
G:=sub<Sym(160)| (1,50,8,45)(2,46,9,41)(3,47,10,42)(4,48,6,43)(5,49,7,44)(11,56,61,53)(12,57,62,54)(13,58,63,55)(14,59,64,51)(15,60,65,52)(16,77,31,75)(17,78,32,71)(18,79,33,72)(19,80,34,73)(20,76,35,74)(21,70,28,37)(22,66,29,38)(23,67,30,39)(24,68,26,40)(25,69,27,36)(81,137,86,132)(82,138,87,133)(83,139,88,134)(84,140,89,135)(85,131,90,136)(91,113,96,118)(92,114,97,119)(93,115,98,120)(94,116,99,111)(95,117,100,112)(101,148,106,143)(102,149,107,144)(103,150,108,145)(104,141,109,146)(105,142,110,147)(121,151,126,156)(122,152,127,157)(123,153,128,158)(124,154,129,159)(125,155,130,160), (1,83,13,112)(2,89,14,118)(3,85,15,114)(4,81,11,120)(5,87,12,116)(6,86,61,115)(7,82,62,111)(8,88,63,117)(9,84,64,113)(10,90,65,119)(16,129,39,102)(17,125,40,108)(18,121,36,104)(19,127,37,110)(20,123,38,106)(21,147,80,157)(22,143,76,153)(23,149,77,159)(24,145,78,155)(25,141,79,151)(26,150,71,160)(27,146,72,156)(28,142,73,152)(29,148,74,158)(30,144,75,154)(31,124,67,107)(32,130,68,103)(33,126,69,109)(34,122,70,105)(35,128,66,101)(41,140,51,96)(42,136,52,92)(43,132,53,98)(44,138,54,94)(45,134,55,100)(46,135,59,91)(47,131,60,97)(48,137,56,93)(49,133,57,99)(50,139,58,95), (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), (1,33,8,18)(2,32,9,17)(3,31,10,16)(4,35,6,20)(5,34,7,19)(11,66,61,38)(12,70,62,37)(13,69,63,36)(14,68,64,40)(15,67,65,39)(21,54,28,57)(22,53,29,56)(23,52,30,60)(24,51,26,59)(25,55,27,58)(41,71,46,78)(42,75,47,77)(43,74,48,76)(44,73,49,80)(45,72,50,79)(81,123,86,128)(82,122,87,127)(83,121,88,126)(84,130,89,125)(85,129,90,124)(91,150,96,145)(92,149,97,144)(93,148,98,143)(94,147,99,142)(95,146,100,141)(101,120,106,115)(102,119,107,114)(103,118,108,113)(104,117,109,112)(105,116,110,111)(131,154,136,159)(132,153,137,158)(133,152,138,157)(134,151,139,156)(135,160,140,155)>;
G:=Group( (1,50,8,45)(2,46,9,41)(3,47,10,42)(4,48,6,43)(5,49,7,44)(11,56,61,53)(12,57,62,54)(13,58,63,55)(14,59,64,51)(15,60,65,52)(16,77,31,75)(17,78,32,71)(18,79,33,72)(19,80,34,73)(20,76,35,74)(21,70,28,37)(22,66,29,38)(23,67,30,39)(24,68,26,40)(25,69,27,36)(81,137,86,132)(82,138,87,133)(83,139,88,134)(84,140,89,135)(85,131,90,136)(91,113,96,118)(92,114,97,119)(93,115,98,120)(94,116,99,111)(95,117,100,112)(101,148,106,143)(102,149,107,144)(103,150,108,145)(104,141,109,146)(105,142,110,147)(121,151,126,156)(122,152,127,157)(123,153,128,158)(124,154,129,159)(125,155,130,160), (1,83,13,112)(2,89,14,118)(3,85,15,114)(4,81,11,120)(5,87,12,116)(6,86,61,115)(7,82,62,111)(8,88,63,117)(9,84,64,113)(10,90,65,119)(16,129,39,102)(17,125,40,108)(18,121,36,104)(19,127,37,110)(20,123,38,106)(21,147,80,157)(22,143,76,153)(23,149,77,159)(24,145,78,155)(25,141,79,151)(26,150,71,160)(27,146,72,156)(28,142,73,152)(29,148,74,158)(30,144,75,154)(31,124,67,107)(32,130,68,103)(33,126,69,109)(34,122,70,105)(35,128,66,101)(41,140,51,96)(42,136,52,92)(43,132,53,98)(44,138,54,94)(45,134,55,100)(46,135,59,91)(47,131,60,97)(48,137,56,93)(49,133,57,99)(50,139,58,95), (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), (1,33,8,18)(2,32,9,17)(3,31,10,16)(4,35,6,20)(5,34,7,19)(11,66,61,38)(12,70,62,37)(13,69,63,36)(14,68,64,40)(15,67,65,39)(21,54,28,57)(22,53,29,56)(23,52,30,60)(24,51,26,59)(25,55,27,58)(41,71,46,78)(42,75,47,77)(43,74,48,76)(44,73,49,80)(45,72,50,79)(81,123,86,128)(82,122,87,127)(83,121,88,126)(84,130,89,125)(85,129,90,124)(91,150,96,145)(92,149,97,144)(93,148,98,143)(94,147,99,142)(95,146,100,141)(101,120,106,115)(102,119,107,114)(103,118,108,113)(104,117,109,112)(105,116,110,111)(131,154,136,159)(132,153,137,158)(133,152,138,157)(134,151,139,156)(135,160,140,155) );
G=PermutationGroup([[(1,50,8,45),(2,46,9,41),(3,47,10,42),(4,48,6,43),(5,49,7,44),(11,56,61,53),(12,57,62,54),(13,58,63,55),(14,59,64,51),(15,60,65,52),(16,77,31,75),(17,78,32,71),(18,79,33,72),(19,80,34,73),(20,76,35,74),(21,70,28,37),(22,66,29,38),(23,67,30,39),(24,68,26,40),(25,69,27,36),(81,137,86,132),(82,138,87,133),(83,139,88,134),(84,140,89,135),(85,131,90,136),(91,113,96,118),(92,114,97,119),(93,115,98,120),(94,116,99,111),(95,117,100,112),(101,148,106,143),(102,149,107,144),(103,150,108,145),(104,141,109,146),(105,142,110,147),(121,151,126,156),(122,152,127,157),(123,153,128,158),(124,154,129,159),(125,155,130,160)], [(1,83,13,112),(2,89,14,118),(3,85,15,114),(4,81,11,120),(5,87,12,116),(6,86,61,115),(7,82,62,111),(8,88,63,117),(9,84,64,113),(10,90,65,119),(16,129,39,102),(17,125,40,108),(18,121,36,104),(19,127,37,110),(20,123,38,106),(21,147,80,157),(22,143,76,153),(23,149,77,159),(24,145,78,155),(25,141,79,151),(26,150,71,160),(27,146,72,156),(28,142,73,152),(29,148,74,158),(30,144,75,154),(31,124,67,107),(32,130,68,103),(33,126,69,109),(34,122,70,105),(35,128,66,101),(41,140,51,96),(42,136,52,92),(43,132,53,98),(44,138,54,94),(45,134,55,100),(46,135,59,91),(47,131,60,97),(48,137,56,93),(49,133,57,99),(50,139,58,95)], [(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)], [(1,33,8,18),(2,32,9,17),(3,31,10,16),(4,35,6,20),(5,34,7,19),(11,66,61,38),(12,70,62,37),(13,69,63,36),(14,68,64,40),(15,67,65,39),(21,54,28,57),(22,53,29,56),(23,52,30,60),(24,51,26,59),(25,55,27,58),(41,71,46,78),(42,75,47,77),(43,74,48,76),(44,73,49,80),(45,72,50,79),(81,123,86,128),(82,122,87,127),(83,121,88,126),(84,130,89,125),(85,129,90,124),(91,150,96,145),(92,149,97,144),(93,148,98,143),(94,147,99,142),(95,146,100,141),(101,120,106,115),(102,119,107,114),(103,118,108,113),(104,117,109,112),(105,116,110,111),(131,154,136,159),(132,153,137,158),(133,152,138,157),(134,151,139,156),(135,160,140,155)]])
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 |
Matrix representation of C42.87D10 ►in 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] >;
C42.87D10 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