metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.274D10, (C2×C20)⋊13Q8, C20.89(C2×Q8), C20⋊2Q8⋊37C2, (C4×Dic10)⋊3C2, (C2×C4)⋊10Dic10, (C2×C42).21D5, C10.4(C22×Q8), (C2×C10).14C24, C20.6Q8⋊31C2, C4.54(C2×Dic10), C4.117(C4○D20), C20.233(C4○D4), (C2×C20).692C23, (C4×C20).314C22, (C22×C4).436D10, (C2×Dic5).3C23, C2.6(C22×Dic10), C22.61(C23×D5), C20.48D4.20C2, C4⋊Dic5.288C22, C22.10(C2×Dic10), C23.216(C22×D5), C23.D5.80C22, (C22×C20).522C22, (C22×C10).376C23, C5⋊1(C23.37C23), (C4×Dic5).210C22, C23.21D10.6C2, C10.D4.95C22, (C2×Dic10).231C22, (C2×C4×C20).23C2, C2.8(C2×C4○D20), C10.3(C2×C4○D4), (C2×C10).48(C2×Q8), (C2×C4).728(C22×D5), SmallGroup(320,1142)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42.274D10
G = < a,b,c,d | a4=b4=c10=1, d2=a2b2, ab=ba, ac=ca, dad-1=a-1, bc=cb, bd=db, dcd-1=b2c-1 >
Subgroups: 606 in 222 conjugacy classes, 119 normal (21 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C5, C2×C4, C2×C4, C2×C4, Q8, C23, C10, C10, C10, C42, C42, C42, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C2×Q8, Dic5, C20, C20, C2×C10, C2×C10, C2×C10, C2×C42, C42⋊C2, C4×Q8, C22⋊Q8, C42.C2, C4⋊Q8, Dic10, C2×Dic5, C2×C20, C2×C20, C2×C20, C22×C10, C23.37C23, C4×Dic5, C10.D4, C4⋊Dic5, C23.D5, C4×C20, C4×C20, C2×Dic10, C22×C20, C22×C20, C4×Dic10, C20⋊2Q8, C20.6Q8, C20.48D4, C23.21D10, C2×C4×C20, C42.274D10
Quotients: C1, C2, C22, Q8, C23, D5, C2×Q8, C4○D4, C24, D10, C22×Q8, C2×C4○D4, Dic10, C22×D5, C23.37C23, C2×Dic10, C4○D20, C23×D5, C22×Dic10, C2×C4○D20, C42.274D10
►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 52)(12 57 17 53)(13 58 18 54)(14 59 19 55)(15 60 20 51)(21 70 26 61)(22 66 27 62)(23 67 28 63)(24 68 29 64)(25 69 30 65)(31 79 36 74)(32 80 37 75)(33 76 38 71)(34 77 39 72)(35 78 40 73)(81 132 91 145)(82 133 92 146)(83 134 93 147)(84 135 94 148)(85 136 95 149)(86 137 96 150)(87 138 97 141)(88 139 98 142)(89 140 99 143)(90 131 100 144)(101 154 127 120)(102 155 128 111)(103 156 129 112)(104 157 130 113)(105 158 121 114)(106 159 122 115)(107 160 123 116)(108 151 124 117)(109 152 125 118)(110 153 126 119)
(1 33 13 23)(2 34 14 24)(3 35 15 25)(4 31 11 21)(5 32 12 22)(6 36 16 26)(7 37 17 27)(8 38 18 28)(9 39 19 29)(10 40 20 30)(41 74 52 61)(42 75 53 62)(43 71 54 63)(44 72 55 64)(45 73 51 65)(46 79 56 70)(47 80 57 66)(48 76 58 67)(49 77 59 68)(50 78 60 69)(81 104 86 109)(82 105 87 110)(83 106 88 101)(84 107 89 102)(85 108 90 103)(91 130 96 125)(92 121 97 126)(93 122 98 127)(94 123 99 128)(95 124 100 129)(111 148 116 143)(112 149 117 144)(113 150 118 145)(114 141 119 146)(115 142 120 147)(131 156 136 151)(132 157 137 152)(133 158 138 153)(134 159 139 154)(135 160 140 155)
(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)
(1 100 18 85)(2 94 19 89)(3 98 20 83)(4 92 16 87)(5 96 17 81)(6 82 11 97)(7 86 12 91)(8 90 13 95)(9 84 14 99)(10 88 15 93)(21 126 36 105)(22 130 37 109)(23 124 38 103)(24 128 39 107)(25 122 40 101)(26 110 31 121)(27 104 32 125)(28 108 33 129)(29 102 34 123)(30 106 35 127)(41 146 56 138)(42 150 57 132)(43 144 58 136)(44 148 59 140)(45 142 60 134)(46 133 52 141)(47 137 53 145)(48 131 54 149)(49 135 55 143)(50 139 51 147)(61 119 79 158)(62 113 80 152)(63 117 76 156)(64 111 77 160)(65 115 78 154)(66 157 75 118)(67 151 71 112)(68 155 72 116)(69 159 73 120)(70 153 74 114)
G:=sub<Sym(160)| (1,48,8,43)(2,49,9,44)(3,50,10,45)(4,46,6,41)(5,47,7,42)(11,56,16,52)(12,57,17,53)(13,58,18,54)(14,59,19,55)(15,60,20,51)(21,70,26,61)(22,66,27,62)(23,67,28,63)(24,68,29,64)(25,69,30,65)(31,79,36,74)(32,80,37,75)(33,76,38,71)(34,77,39,72)(35,78,40,73)(81,132,91,145)(82,133,92,146)(83,134,93,147)(84,135,94,148)(85,136,95,149)(86,137,96,150)(87,138,97,141)(88,139,98,142)(89,140,99,143)(90,131,100,144)(101,154,127,120)(102,155,128,111)(103,156,129,112)(104,157,130,113)(105,158,121,114)(106,159,122,115)(107,160,123,116)(108,151,124,117)(109,152,125,118)(110,153,126,119), (1,33,13,23)(2,34,14,24)(3,35,15,25)(4,31,11,21)(5,32,12,22)(6,36,16,26)(7,37,17,27)(8,38,18,28)(9,39,19,29)(10,40,20,30)(41,74,52,61)(42,75,53,62)(43,71,54,63)(44,72,55,64)(45,73,51,65)(46,79,56,70)(47,80,57,66)(48,76,58,67)(49,77,59,68)(50,78,60,69)(81,104,86,109)(82,105,87,110)(83,106,88,101)(84,107,89,102)(85,108,90,103)(91,130,96,125)(92,121,97,126)(93,122,98,127)(94,123,99,128)(95,124,100,129)(111,148,116,143)(112,149,117,144)(113,150,118,145)(114,141,119,146)(115,142,120,147)(131,156,136,151)(132,157,137,152)(133,158,138,153)(134,159,139,154)(135,160,140,155), (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), (1,100,18,85)(2,94,19,89)(3,98,20,83)(4,92,16,87)(5,96,17,81)(6,82,11,97)(7,86,12,91)(8,90,13,95)(9,84,14,99)(10,88,15,93)(21,126,36,105)(22,130,37,109)(23,124,38,103)(24,128,39,107)(25,122,40,101)(26,110,31,121)(27,104,32,125)(28,108,33,129)(29,102,34,123)(30,106,35,127)(41,146,56,138)(42,150,57,132)(43,144,58,136)(44,148,59,140)(45,142,60,134)(46,133,52,141)(47,137,53,145)(48,131,54,149)(49,135,55,143)(50,139,51,147)(61,119,79,158)(62,113,80,152)(63,117,76,156)(64,111,77,160)(65,115,78,154)(66,157,75,118)(67,151,71,112)(68,155,72,116)(69,159,73,120)(70,153,74,114)>;
G:=Group( (1,48,8,43)(2,49,9,44)(3,50,10,45)(4,46,6,41)(5,47,7,42)(11,56,16,52)(12,57,17,53)(13,58,18,54)(14,59,19,55)(15,60,20,51)(21,70,26,61)(22,66,27,62)(23,67,28,63)(24,68,29,64)(25,69,30,65)(31,79,36,74)(32,80,37,75)(33,76,38,71)(34,77,39,72)(35,78,40,73)(81,132,91,145)(82,133,92,146)(83,134,93,147)(84,135,94,148)(85,136,95,149)(86,137,96,150)(87,138,97,141)(88,139,98,142)(89,140,99,143)(90,131,100,144)(101,154,127,120)(102,155,128,111)(103,156,129,112)(104,157,130,113)(105,158,121,114)(106,159,122,115)(107,160,123,116)(108,151,124,117)(109,152,125,118)(110,153,126,119), (1,33,13,23)(2,34,14,24)(3,35,15,25)(4,31,11,21)(5,32,12,22)(6,36,16,26)(7,37,17,27)(8,38,18,28)(9,39,19,29)(10,40,20,30)(41,74,52,61)(42,75,53,62)(43,71,54,63)(44,72,55,64)(45,73,51,65)(46,79,56,70)(47,80,57,66)(48,76,58,67)(49,77,59,68)(50,78,60,69)(81,104,86,109)(82,105,87,110)(83,106,88,101)(84,107,89,102)(85,108,90,103)(91,130,96,125)(92,121,97,126)(93,122,98,127)(94,123,99,128)(95,124,100,129)(111,148,116,143)(112,149,117,144)(113,150,118,145)(114,141,119,146)(115,142,120,147)(131,156,136,151)(132,157,137,152)(133,158,138,153)(134,159,139,154)(135,160,140,155), (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), (1,100,18,85)(2,94,19,89)(3,98,20,83)(4,92,16,87)(5,96,17,81)(6,82,11,97)(7,86,12,91)(8,90,13,95)(9,84,14,99)(10,88,15,93)(21,126,36,105)(22,130,37,109)(23,124,38,103)(24,128,39,107)(25,122,40,101)(26,110,31,121)(27,104,32,125)(28,108,33,129)(29,102,34,123)(30,106,35,127)(41,146,56,138)(42,150,57,132)(43,144,58,136)(44,148,59,140)(45,142,60,134)(46,133,52,141)(47,137,53,145)(48,131,54,149)(49,135,55,143)(50,139,51,147)(61,119,79,158)(62,113,80,152)(63,117,76,156)(64,111,77,160)(65,115,78,154)(66,157,75,118)(67,151,71,112)(68,155,72,116)(69,159,73,120)(70,153,74,114) );
G=PermutationGroup([[(1,48,8,43),(2,49,9,44),(3,50,10,45),(4,46,6,41),(5,47,7,42),(11,56,16,52),(12,57,17,53),(13,58,18,54),(14,59,19,55),(15,60,20,51),(21,70,26,61),(22,66,27,62),(23,67,28,63),(24,68,29,64),(25,69,30,65),(31,79,36,74),(32,80,37,75),(33,76,38,71),(34,77,39,72),(35,78,40,73),(81,132,91,145),(82,133,92,146),(83,134,93,147),(84,135,94,148),(85,136,95,149),(86,137,96,150),(87,138,97,141),(88,139,98,142),(89,140,99,143),(90,131,100,144),(101,154,127,120),(102,155,128,111),(103,156,129,112),(104,157,130,113),(105,158,121,114),(106,159,122,115),(107,160,123,116),(108,151,124,117),(109,152,125,118),(110,153,126,119)], [(1,33,13,23),(2,34,14,24),(3,35,15,25),(4,31,11,21),(5,32,12,22),(6,36,16,26),(7,37,17,27),(8,38,18,28),(9,39,19,29),(10,40,20,30),(41,74,52,61),(42,75,53,62),(43,71,54,63),(44,72,55,64),(45,73,51,65),(46,79,56,70),(47,80,57,66),(48,76,58,67),(49,77,59,68),(50,78,60,69),(81,104,86,109),(82,105,87,110),(83,106,88,101),(84,107,89,102),(85,108,90,103),(91,130,96,125),(92,121,97,126),(93,122,98,127),(94,123,99,128),(95,124,100,129),(111,148,116,143),(112,149,117,144),(113,150,118,145),(114,141,119,146),(115,142,120,147),(131,156,136,151),(132,157,137,152),(133,158,138,153),(134,159,139,154),(135,160,140,155)], [(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)], [(1,100,18,85),(2,94,19,89),(3,98,20,83),(4,92,16,87),(5,96,17,81),(6,82,11,97),(7,86,12,91),(8,90,13,95),(9,84,14,99),(10,88,15,93),(21,126,36,105),(22,130,37,109),(23,124,38,103),(24,128,39,107),(25,122,40,101),(26,110,31,121),(27,104,32,125),(28,108,33,129),(29,102,34,123),(30,106,35,127),(41,146,56,138),(42,150,57,132),(43,144,58,136),(44,148,59,140),(45,142,60,134),(46,133,52,141),(47,137,53,145),(48,131,54,149),(49,135,55,143),(50,139,51,147),(61,119,79,158),(62,113,80,152),(63,117,76,156),(64,111,77,160),(65,115,78,154),(66,157,75,118),(67,151,71,112),(68,155,72,116),(69,159,73,120),(70,153,74,114)]])
92 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | ··· | 4N | 4O | ··· | 4V | 5A | 5B | 10A | ··· | 10N | 20A | ··· | 20AV |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | ··· | 4 | 5 | 5 | 10 | ··· | 10 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 20 | ··· | 20 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
92 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | - | + | + | + | - | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | Q8 | D5 | C4○D4 | D10 | D10 | Dic10 | C4○D20 |
| kernel | C42.274D10 | C4×Dic10 | C20⋊2Q8 | C20.6Q8 | C20.48D4 | C23.21D10 | C2×C4×C20 | C2×C20 | C2×C42 | C20 | C42 | C22×C4 | C2×C4 | C4 |
| # reps | 1 | 4 | 2 | 2 | 4 | 2 | 1 | 4 | 2 | 8 | 8 | 6 | 16 | 32 |
Matrix representation of C42.274D10 ►in GL4(𝔽41) generated by
| 32 | 0 | 0 | 0 |
| 18 | 9 | 0 | 0 |
| 0 | 0 | 2 | 28 |
| 0 | 0 | 13 | 39 |
| 9 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 |
| 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 40 |
| 1 | 0 | 0 | 0 |
| 39 | 40 | 0 | 0 |
| 0 | 0 | 40 | 35 |
| 0 | 0 | 6 | 35 |
| 9 | 9 | 0 | 0 |
| 23 | 32 | 0 | 0 |
| 0 | 0 | 4 | 14 |
| 0 | 0 | 31 | 37 |
G:=sub<GL(4,GF(41))| [32,18,0,0,0,9,0,0,0,0,2,13,0,0,28,39],[9,0,0,0,0,9,0,0,0,0,40,0,0,0,0,40],[1,39,0,0,0,40,0,0,0,0,40,6,0,0,35,35],[9,23,0,0,9,32,0,0,0,0,4,31,0,0,14,37] >;
C42.274D10 in GAP, Magma, Sage, TeX
C_4^2._{274}D_{10} % in TeX
G:=Group("C4^2.274D10"); // GroupNames label
G:=SmallGroup(320,1142);
// by ID
G=gap.SmallGroup(320,1142);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,224,477,232,100,675,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^10=1,d^2=a^2*b^2,a*b=b*a,a*c=c*a,d*a*d^-1=a^-1,b*c=c*b,b*d=d*b,d*c*d^-1=b^2*c^-1>;
// generators/relations