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), (C4×C20).314C22, (C2×C20).692C23, (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, C10.D4.95C22, C23.21D10.6C2, (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
Subgroups: 606 in 222 conjugacy classes, 119 normal (21 characteristic)
C1, C2, C2 [×2], C2 [×2], C4 [×8], C4 [×10], C22, C22 [×2], C22 [×2], C5, C2×C4 [×2], C2×C4 [×8], C2×C4 [×12], Q8 [×8], C23, C10, C10 [×2], C10 [×2], C42 [×2], C42 [×2], C42 [×4], C22⋊C4 [×4], C4⋊C4 [×16], C22×C4, C22×C4 [×2], C2×Q8 [×4], Dic5 [×8], C20 [×8], C20 [×2], C2×C10, C2×C10 [×2], C2×C10 [×2], C2×C42, C42⋊C2 [×2], C4×Q8 [×4], C22⋊Q8 [×4], C42.C2 [×2], C4⋊Q8 [×2], Dic10 [×8], C2×Dic5 [×8], C2×C20 [×2], C2×C20 [×8], C2×C20 [×4], C22×C10, C23.37C23, C4×Dic5 [×4], C10.D4 [×8], C4⋊Dic5 [×8], C23.D5 [×4], C4×C20 [×2], C4×C20 [×2], C2×Dic10 [×4], C22×C20, C22×C20 [×2], C4×Dic10 [×4], C20⋊2Q8 [×2], C20.6Q8 [×2], C20.48D4 [×4], C23.21D10 [×2], C2×C4×C20, C42.274D10
Quotients:
C1, C2 [×15], C22 [×35], Q8 [×4], C23 [×15], D5, C2×Q8 [×6], C4○D4 [×4], C24, D10 [×7], C22×Q8, C2×C4○D4 [×2], Dic10 [×4], C22×D5 [×7], C23.37C23, C2×Dic10 [×6], C4○D20 [×4], C23×D5, C22×Dic10, C2×C4○D20 [×2], C42.274D10
Generators and relations
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 >
►On 160 points
Generators in S160
(1 48 9 43)(2 49 10 44)(3 50 6 45)(4 46 7 41)(5 47 8 42)(11 56 16 51)(12 57 17 52)(13 58 18 53)(14 59 19 54)(15 60 20 55)(21 69 26 61)(22 70 27 62)(23 66 28 63)(24 67 29 64)(25 68 30 65)(31 79 36 74)(32 80 37 75)(33 76 38 71)(34 77 39 72)(35 78 40 73)(81 132 105 91)(82 133 106 92)(83 134 107 93)(84 135 108 94)(85 136 109 95)(86 137 110 96)(87 138 101 97)(88 139 102 98)(89 140 103 99)(90 131 104 100)(111 148 124 157)(112 149 125 158)(113 150 126 159)(114 141 127 160)(115 142 128 151)(116 143 129 152)(117 144 130 153)(118 145 121 154)(119 146 122 155)(120 147 123 156)
(1 33 13 23)(2 34 14 24)(3 35 15 25)(4 31 11 21)(5 32 12 22)(6 40 20 30)(7 36 16 26)(8 37 17 27)(9 38 18 28)(10 39 19 29)(41 74 51 61)(42 75 52 62)(43 71 53 63)(44 72 54 64)(45 73 55 65)(46 79 56 69)(47 80 57 70)(48 76 58 66)(49 77 59 67)(50 78 60 68)(81 117 86 112)(82 118 87 113)(83 119 88 114)(84 120 89 115)(85 111 90 116)(91 153 96 158)(92 154 97 159)(93 155 98 160)(94 156 99 151)(95 157 100 152)(101 126 106 121)(102 127 107 122)(103 128 108 123)(104 129 109 124)(105 130 110 125)(131 143 136 148)(132 144 137 149)(133 145 138 150)(134 146 139 141)(135 147 140 142)
(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)
(1 104 18 85)(2 108 19 89)(3 102 20 83)(4 106 16 87)(5 110 17 81)(6 88 15 107)(7 82 11 101)(8 86 12 105)(9 90 13 109)(10 84 14 103)(21 126 36 118)(22 130 37 112)(23 124 38 116)(24 128 39 120)(25 122 40 114)(26 113 31 121)(27 117 32 125)(28 111 33 129)(29 115 34 123)(30 119 35 127)(41 92 56 138)(42 96 57 132)(43 100 58 136)(44 94 59 140)(45 98 60 134)(46 133 51 97)(47 137 52 91)(48 131 53 95)(49 135 54 99)(50 139 55 93)(61 159 79 145)(62 153 80 149)(63 157 76 143)(64 151 77 147)(65 155 78 141)(66 148 71 152)(67 142 72 156)(68 146 73 160)(69 150 74 154)(70 144 75 158)
G:=sub<Sym(160)| (1,48,9,43)(2,49,10,44)(3,50,6,45)(4,46,7,41)(5,47,8,42)(11,56,16,51)(12,57,17,52)(13,58,18,53)(14,59,19,54)(15,60,20,55)(21,69,26,61)(22,70,27,62)(23,66,28,63)(24,67,29,64)(25,68,30,65)(31,79,36,74)(32,80,37,75)(33,76,38,71)(34,77,39,72)(35,78,40,73)(81,132,105,91)(82,133,106,92)(83,134,107,93)(84,135,108,94)(85,136,109,95)(86,137,110,96)(87,138,101,97)(88,139,102,98)(89,140,103,99)(90,131,104,100)(111,148,124,157)(112,149,125,158)(113,150,126,159)(114,141,127,160)(115,142,128,151)(116,143,129,152)(117,144,130,153)(118,145,121,154)(119,146,122,155)(120,147,123,156), (1,33,13,23)(2,34,14,24)(3,35,15,25)(4,31,11,21)(5,32,12,22)(6,40,20,30)(7,36,16,26)(8,37,17,27)(9,38,18,28)(10,39,19,29)(41,74,51,61)(42,75,52,62)(43,71,53,63)(44,72,54,64)(45,73,55,65)(46,79,56,69)(47,80,57,70)(48,76,58,66)(49,77,59,67)(50,78,60,68)(81,117,86,112)(82,118,87,113)(83,119,88,114)(84,120,89,115)(85,111,90,116)(91,153,96,158)(92,154,97,159)(93,155,98,160)(94,156,99,151)(95,157,100,152)(101,126,106,121)(102,127,107,122)(103,128,108,123)(104,129,109,124)(105,130,110,125)(131,143,136,148)(132,144,137,149)(133,145,138,150)(134,146,139,141)(135,147,140,142), (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), (1,104,18,85)(2,108,19,89)(3,102,20,83)(4,106,16,87)(5,110,17,81)(6,88,15,107)(7,82,11,101)(8,86,12,105)(9,90,13,109)(10,84,14,103)(21,126,36,118)(22,130,37,112)(23,124,38,116)(24,128,39,120)(25,122,40,114)(26,113,31,121)(27,117,32,125)(28,111,33,129)(29,115,34,123)(30,119,35,127)(41,92,56,138)(42,96,57,132)(43,100,58,136)(44,94,59,140)(45,98,60,134)(46,133,51,97)(47,137,52,91)(48,131,53,95)(49,135,54,99)(50,139,55,93)(61,159,79,145)(62,153,80,149)(63,157,76,143)(64,151,77,147)(65,155,78,141)(66,148,71,152)(67,142,72,156)(68,146,73,160)(69,150,74,154)(70,144,75,158)>;
G:=Group( (1,48,9,43)(2,49,10,44)(3,50,6,45)(4,46,7,41)(5,47,8,42)(11,56,16,51)(12,57,17,52)(13,58,18,53)(14,59,19,54)(15,60,20,55)(21,69,26,61)(22,70,27,62)(23,66,28,63)(24,67,29,64)(25,68,30,65)(31,79,36,74)(32,80,37,75)(33,76,38,71)(34,77,39,72)(35,78,40,73)(81,132,105,91)(82,133,106,92)(83,134,107,93)(84,135,108,94)(85,136,109,95)(86,137,110,96)(87,138,101,97)(88,139,102,98)(89,140,103,99)(90,131,104,100)(111,148,124,157)(112,149,125,158)(113,150,126,159)(114,141,127,160)(115,142,128,151)(116,143,129,152)(117,144,130,153)(118,145,121,154)(119,146,122,155)(120,147,123,156), (1,33,13,23)(2,34,14,24)(3,35,15,25)(4,31,11,21)(5,32,12,22)(6,40,20,30)(7,36,16,26)(8,37,17,27)(9,38,18,28)(10,39,19,29)(41,74,51,61)(42,75,52,62)(43,71,53,63)(44,72,54,64)(45,73,55,65)(46,79,56,69)(47,80,57,70)(48,76,58,66)(49,77,59,67)(50,78,60,68)(81,117,86,112)(82,118,87,113)(83,119,88,114)(84,120,89,115)(85,111,90,116)(91,153,96,158)(92,154,97,159)(93,155,98,160)(94,156,99,151)(95,157,100,152)(101,126,106,121)(102,127,107,122)(103,128,108,123)(104,129,109,124)(105,130,110,125)(131,143,136,148)(132,144,137,149)(133,145,138,150)(134,146,139,141)(135,147,140,142), (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), (1,104,18,85)(2,108,19,89)(3,102,20,83)(4,106,16,87)(5,110,17,81)(6,88,15,107)(7,82,11,101)(8,86,12,105)(9,90,13,109)(10,84,14,103)(21,126,36,118)(22,130,37,112)(23,124,38,116)(24,128,39,120)(25,122,40,114)(26,113,31,121)(27,117,32,125)(28,111,33,129)(29,115,34,123)(30,119,35,127)(41,92,56,138)(42,96,57,132)(43,100,58,136)(44,94,59,140)(45,98,60,134)(46,133,51,97)(47,137,52,91)(48,131,53,95)(49,135,54,99)(50,139,55,93)(61,159,79,145)(62,153,80,149)(63,157,76,143)(64,151,77,147)(65,155,78,141)(66,148,71,152)(67,142,72,156)(68,146,73,160)(69,150,74,154)(70,144,75,158) );
G=PermutationGroup([(1,48,9,43),(2,49,10,44),(3,50,6,45),(4,46,7,41),(5,47,8,42),(11,56,16,51),(12,57,17,52),(13,58,18,53),(14,59,19,54),(15,60,20,55),(21,69,26,61),(22,70,27,62),(23,66,28,63),(24,67,29,64),(25,68,30,65),(31,79,36,74),(32,80,37,75),(33,76,38,71),(34,77,39,72),(35,78,40,73),(81,132,105,91),(82,133,106,92),(83,134,107,93),(84,135,108,94),(85,136,109,95),(86,137,110,96),(87,138,101,97),(88,139,102,98),(89,140,103,99),(90,131,104,100),(111,148,124,157),(112,149,125,158),(113,150,126,159),(114,141,127,160),(115,142,128,151),(116,143,129,152),(117,144,130,153),(118,145,121,154),(119,146,122,155),(120,147,123,156)], [(1,33,13,23),(2,34,14,24),(3,35,15,25),(4,31,11,21),(5,32,12,22),(6,40,20,30),(7,36,16,26),(8,37,17,27),(9,38,18,28),(10,39,19,29),(41,74,51,61),(42,75,52,62),(43,71,53,63),(44,72,54,64),(45,73,55,65),(46,79,56,69),(47,80,57,70),(48,76,58,66),(49,77,59,67),(50,78,60,68),(81,117,86,112),(82,118,87,113),(83,119,88,114),(84,120,89,115),(85,111,90,116),(91,153,96,158),(92,154,97,159),(93,155,98,160),(94,156,99,151),(95,157,100,152),(101,126,106,121),(102,127,107,122),(103,128,108,123),(104,129,109,124),(105,130,110,125),(131,143,136,148),(132,144,137,149),(133,145,138,150),(134,146,139,141),(135,147,140,142)], [(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)], [(1,104,18,85),(2,108,19,89),(3,102,20,83),(4,106,16,87),(5,110,17,81),(6,88,15,107),(7,82,11,101),(8,86,12,105),(9,90,13,109),(10,84,14,103),(21,126,36,118),(22,130,37,112),(23,124,38,116),(24,128,39,120),(25,122,40,114),(26,113,31,121),(27,117,32,125),(28,111,33,129),(29,115,34,123),(30,119,35,127),(41,92,56,138),(42,96,57,132),(43,100,58,136),(44,94,59,140),(45,98,60,134),(46,133,51,97),(47,137,52,91),(48,131,53,95),(49,135,54,99),(50,139,55,93),(61,159,79,145),(62,153,80,149),(63,157,76,143),(64,151,77,147),(65,155,78,141),(66,148,71,152),(67,142,72,156),(68,146,73,160),(69,150,74,154),(70,144,75,158)])
Matrix representation ►G ⊆ 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] >;
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 |
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
×
⋊
ℤ
𝔽
○
≀