metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.125D10, C10.92- (1+4), (Q8×D5)⋊5C4, (C4×Q8)⋊6D5, (Q8×C20)⋊7C2, (Q8×Dic5)⋊8C2, Q8.12(C4×D5), C4⋊C4.323D10, (C4×Dic10)⋊38C2, C20.70(C22×C4), C10.46(C23×C4), (C2×Q8).200D10, C42⋊D5.3C2, Dic5⋊3Q8⋊18C2, (C2×C10).116C24, (C2×C20).495C23, (C4×C20).168C22, Dic10.35(C2×C4), D10.41(C22×C4), C22.35(C23×D5), C4⋊Dic5.366C22, (Q8×C10).216C22, Dic5.19(C22×C4), (C4×Dic5).92C22, C2.4(D4.10D10), C2.2(Q8.10D10), C5⋊3(C23.32C23), (C2×Dic5).222C23, (C22×D5).185C23, D10⋊C4.124C22, (C2×Dic10).298C22, C10.D4.137C22, C4.35(C2×C4×D5), (C2×Q8×D5).6C2, (C4×D5).9(C2×C4), C2.27(D5×C22×C4), (C5×Q8).31(C2×C4), (C2×C4×D5).78C22, C4⋊C4⋊7D5.10C2, (C5×C4⋊C4).344C22, (C2×C4).288(C22×D5), SmallGroup(320,1244)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 718 in 266 conjugacy classes, 151 normal (22 characteristic)
C1, C2 [×3], C2 [×2], C4 [×6], C4 [×14], C22, C22 [×4], C5, C2×C4, C2×C4 [×6], C2×C4 [×19], Q8 [×4], Q8 [×12], C23, D5 [×2], C10 [×3], C42 [×3], C42 [×9], C22⋊C4 [×4], C4⋊C4 [×3], C4⋊C4 [×9], C22×C4 [×3], C2×Q8, C2×Q8 [×11], Dic5 [×6], Dic5 [×4], C20 [×6], C20 [×4], D10 [×2], D10 [×2], C2×C10, C42⋊C2 [×6], C4×Q8, C4×Q8 [×7], C22×Q8, Dic10 [×12], C4×D5 [×12], C2×Dic5, C2×Dic5 [×6], C2×C20, C2×C20 [×6], C5×Q8 [×4], C22×D5, C23.32C23, C4×Dic5 [×9], C10.D4 [×6], C4⋊Dic5 [×3], D10⋊C4, D10⋊C4 [×3], C4×C20 [×3], C5×C4⋊C4 [×3], C2×Dic10 [×3], C2×C4×D5 [×3], Q8×D5 [×8], Q8×C10, C4×Dic10 [×3], C42⋊D5 [×3], Dic5⋊3Q8 [×3], C4⋊C4⋊7D5 [×3], Q8×Dic5, Q8×C20, C2×Q8×D5, C42.125D10
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, Q8.10D10, D4.10D10, C42.125D10
Generators and relations
G = < a,b,c,d | a4=b4=1, c10=d2=a2, ab=ba, cac-1=dad-1=a-1, bc=cb, dbd-1=a2b, dcd-1=c9 >
►On 160 points
Generators in S160
(1 96 11 86)(2 87 12 97)(3 98 13 88)(4 89 14 99)(5 100 15 90)(6 91 16 81)(7 82 17 92)(8 93 18 83)(9 84 19 94)(10 95 20 85)(21 79 31 69)(22 70 32 80)(23 61 33 71)(24 72 34 62)(25 63 35 73)(26 74 36 64)(27 65 37 75)(28 76 38 66)(29 67 39 77)(30 78 40 68)(41 115 51 105)(42 106 52 116)(43 117 53 107)(44 108 54 118)(45 119 55 109)(46 110 56 120)(47 101 57 111)(48 112 58 102)(49 103 59 113)(50 114 60 104)(121 145 131 155)(122 156 132 146)(123 147 133 157)(124 158 134 148)(125 149 135 159)(126 160 136 150)(127 151 137 141)(128 142 138 152)(129 153 139 143)(130 144 140 154)
(1 118 28 159)(2 119 29 160)(3 120 30 141)(4 101 31 142)(5 102 32 143)(6 103 33 144)(7 104 34 145)(8 105 35 146)(9 106 36 147)(10 107 37 148)(11 108 38 149)(12 109 39 150)(13 110 40 151)(14 111 21 152)(15 112 22 153)(16 113 23 154)(17 114 24 155)(18 115 25 156)(19 116 26 157)(20 117 27 158)(41 73 122 93)(42 74 123 94)(43 75 124 95)(44 76 125 96)(45 77 126 97)(46 78 127 98)(47 79 128 99)(48 80 129 100)(49 61 130 81)(50 62 131 82)(51 63 132 83)(52 64 133 84)(53 65 134 85)(54 66 135 86)(55 67 136 87)(56 68 137 88)(57 69 138 89)(58 70 139 90)(59 71 140 91)(60 72 121 92)
(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 10 11 20)(2 19 12 9)(3 8 13 18)(4 17 14 7)(5 6 15 16)(21 34 31 24)(22 23 32 33)(25 30 35 40)(26 39 36 29)(27 28 37 38)(41 56 51 46)(42 45 52 55)(43 54 53 44)(47 50 57 60)(48 59 58 49)(61 70 71 80)(62 79 72 69)(63 68 73 78)(64 77 74 67)(65 66 75 76)(81 90 91 100)(82 99 92 89)(83 88 93 98)(84 97 94 87)(85 86 95 96)(101 104 111 114)(102 113 112 103)(105 120 115 110)(106 109 116 119)(107 118 117 108)(121 128 131 138)(122 137 132 127)(123 126 133 136)(124 135 134 125)(129 140 139 130)(141 156 151 146)(142 145 152 155)(143 154 153 144)(147 150 157 160)(148 159 158 149)
G:=sub<Sym(160)| (1,96,11,86)(2,87,12,97)(3,98,13,88)(4,89,14,99)(5,100,15,90)(6,91,16,81)(7,82,17,92)(8,93,18,83)(9,84,19,94)(10,95,20,85)(21,79,31,69)(22,70,32,80)(23,61,33,71)(24,72,34,62)(25,63,35,73)(26,74,36,64)(27,65,37,75)(28,76,38,66)(29,67,39,77)(30,78,40,68)(41,115,51,105)(42,106,52,116)(43,117,53,107)(44,108,54,118)(45,119,55,109)(46,110,56,120)(47,101,57,111)(48,112,58,102)(49,103,59,113)(50,114,60,104)(121,145,131,155)(122,156,132,146)(123,147,133,157)(124,158,134,148)(125,149,135,159)(126,160,136,150)(127,151,137,141)(128,142,138,152)(129,153,139,143)(130,144,140,154), (1,118,28,159)(2,119,29,160)(3,120,30,141)(4,101,31,142)(5,102,32,143)(6,103,33,144)(7,104,34,145)(8,105,35,146)(9,106,36,147)(10,107,37,148)(11,108,38,149)(12,109,39,150)(13,110,40,151)(14,111,21,152)(15,112,22,153)(16,113,23,154)(17,114,24,155)(18,115,25,156)(19,116,26,157)(20,117,27,158)(41,73,122,93)(42,74,123,94)(43,75,124,95)(44,76,125,96)(45,77,126,97)(46,78,127,98)(47,79,128,99)(48,80,129,100)(49,61,130,81)(50,62,131,82)(51,63,132,83)(52,64,133,84)(53,65,134,85)(54,66,135,86)(55,67,136,87)(56,68,137,88)(57,69,138,89)(58,70,139,90)(59,71,140,91)(60,72,121,92), (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), (1,10,11,20)(2,19,12,9)(3,8,13,18)(4,17,14,7)(5,6,15,16)(21,34,31,24)(22,23,32,33)(25,30,35,40)(26,39,36,29)(27,28,37,38)(41,56,51,46)(42,45,52,55)(43,54,53,44)(47,50,57,60)(48,59,58,49)(61,70,71,80)(62,79,72,69)(63,68,73,78)(64,77,74,67)(65,66,75,76)(81,90,91,100)(82,99,92,89)(83,88,93,98)(84,97,94,87)(85,86,95,96)(101,104,111,114)(102,113,112,103)(105,120,115,110)(106,109,116,119)(107,118,117,108)(121,128,131,138)(122,137,132,127)(123,126,133,136)(124,135,134,125)(129,140,139,130)(141,156,151,146)(142,145,152,155)(143,154,153,144)(147,150,157,160)(148,159,158,149)>;
G:=Group( (1,96,11,86)(2,87,12,97)(3,98,13,88)(4,89,14,99)(5,100,15,90)(6,91,16,81)(7,82,17,92)(8,93,18,83)(9,84,19,94)(10,95,20,85)(21,79,31,69)(22,70,32,80)(23,61,33,71)(24,72,34,62)(25,63,35,73)(26,74,36,64)(27,65,37,75)(28,76,38,66)(29,67,39,77)(30,78,40,68)(41,115,51,105)(42,106,52,116)(43,117,53,107)(44,108,54,118)(45,119,55,109)(46,110,56,120)(47,101,57,111)(48,112,58,102)(49,103,59,113)(50,114,60,104)(121,145,131,155)(122,156,132,146)(123,147,133,157)(124,158,134,148)(125,149,135,159)(126,160,136,150)(127,151,137,141)(128,142,138,152)(129,153,139,143)(130,144,140,154), (1,118,28,159)(2,119,29,160)(3,120,30,141)(4,101,31,142)(5,102,32,143)(6,103,33,144)(7,104,34,145)(8,105,35,146)(9,106,36,147)(10,107,37,148)(11,108,38,149)(12,109,39,150)(13,110,40,151)(14,111,21,152)(15,112,22,153)(16,113,23,154)(17,114,24,155)(18,115,25,156)(19,116,26,157)(20,117,27,158)(41,73,122,93)(42,74,123,94)(43,75,124,95)(44,76,125,96)(45,77,126,97)(46,78,127,98)(47,79,128,99)(48,80,129,100)(49,61,130,81)(50,62,131,82)(51,63,132,83)(52,64,133,84)(53,65,134,85)(54,66,135,86)(55,67,136,87)(56,68,137,88)(57,69,138,89)(58,70,139,90)(59,71,140,91)(60,72,121,92), (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), (1,10,11,20)(2,19,12,9)(3,8,13,18)(4,17,14,7)(5,6,15,16)(21,34,31,24)(22,23,32,33)(25,30,35,40)(26,39,36,29)(27,28,37,38)(41,56,51,46)(42,45,52,55)(43,54,53,44)(47,50,57,60)(48,59,58,49)(61,70,71,80)(62,79,72,69)(63,68,73,78)(64,77,74,67)(65,66,75,76)(81,90,91,100)(82,99,92,89)(83,88,93,98)(84,97,94,87)(85,86,95,96)(101,104,111,114)(102,113,112,103)(105,120,115,110)(106,109,116,119)(107,118,117,108)(121,128,131,138)(122,137,132,127)(123,126,133,136)(124,135,134,125)(129,140,139,130)(141,156,151,146)(142,145,152,155)(143,154,153,144)(147,150,157,160)(148,159,158,149) );
G=PermutationGroup([(1,96,11,86),(2,87,12,97),(3,98,13,88),(4,89,14,99),(5,100,15,90),(6,91,16,81),(7,82,17,92),(8,93,18,83),(9,84,19,94),(10,95,20,85),(21,79,31,69),(22,70,32,80),(23,61,33,71),(24,72,34,62),(25,63,35,73),(26,74,36,64),(27,65,37,75),(28,76,38,66),(29,67,39,77),(30,78,40,68),(41,115,51,105),(42,106,52,116),(43,117,53,107),(44,108,54,118),(45,119,55,109),(46,110,56,120),(47,101,57,111),(48,112,58,102),(49,103,59,113),(50,114,60,104),(121,145,131,155),(122,156,132,146),(123,147,133,157),(124,158,134,148),(125,149,135,159),(126,160,136,150),(127,151,137,141),(128,142,138,152),(129,153,139,143),(130,144,140,154)], [(1,118,28,159),(2,119,29,160),(3,120,30,141),(4,101,31,142),(5,102,32,143),(6,103,33,144),(7,104,34,145),(8,105,35,146),(9,106,36,147),(10,107,37,148),(11,108,38,149),(12,109,39,150),(13,110,40,151),(14,111,21,152),(15,112,22,153),(16,113,23,154),(17,114,24,155),(18,115,25,156),(19,116,26,157),(20,117,27,158),(41,73,122,93),(42,74,123,94),(43,75,124,95),(44,76,125,96),(45,77,126,97),(46,78,127,98),(47,79,128,99),(48,80,129,100),(49,61,130,81),(50,62,131,82),(51,63,132,83),(52,64,133,84),(53,65,134,85),(54,66,135,86),(55,67,136,87),(56,68,137,88),(57,69,138,89),(58,70,139,90),(59,71,140,91),(60,72,121,92)], [(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)], [(1,10,11,20),(2,19,12,9),(3,8,13,18),(4,17,14,7),(5,6,15,16),(21,34,31,24),(22,23,32,33),(25,30,35,40),(26,39,36,29),(27,28,37,38),(41,56,51,46),(42,45,52,55),(43,54,53,44),(47,50,57,60),(48,59,58,49),(61,70,71,80),(62,79,72,69),(63,68,73,78),(64,77,74,67),(65,66,75,76),(81,90,91,100),(82,99,92,89),(83,88,93,98),(84,97,94,87),(85,86,95,96),(101,104,111,114),(102,113,112,103),(105,120,115,110),(106,109,116,119),(107,118,117,108),(121,128,131,138),(122,137,132,127),(123,126,133,136),(124,135,134,125),(129,140,139,130),(141,156,151,146),(142,145,152,155),(143,154,153,144),(147,150,157,160),(148,159,158,149)])
Matrix representation ►G ⊆ GL6(𝔽41)
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 11 | 0 | 1 | 0 |
| 0 | 0 | 0 | 11 | 0 | 1 |
| 0 | 0 | 1 | 0 | 30 | 0 |
| 0 | 0 | 0 | 1 | 0 | 30 |
| 9 | 0 | 0 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 | 0 | 0 |
| 0 | 0 | 24 | 40 | 0 | 0 |
| 0 | 0 | 1 | 17 | 0 | 0 |
| 0 | 0 | 0 | 0 | 24 | 40 |
| 0 | 0 | 0 | 0 | 1 | 17 |
| 1 | 6 | 0 | 0 | 0 | 0 |
| 35 | 6 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 7 |
| 0 | 0 | 0 | 0 | 34 | 7 |
| 0 | 0 | 1 | 34 | 0 | 0 |
| 0 | 0 | 7 | 34 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 35 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 7 | 40 |
| 0 | 0 | 0 | 0 | 7 | 34 |
| 0 | 0 | 34 | 1 | 0 | 0 |
| 0 | 0 | 34 | 7 | 0 | 0 |
G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,11,0,1,0,0,0,0,11,0,1,0,0,1,0,30,0,0,0,0,1,0,30],[9,0,0,0,0,0,0,9,0,0,0,0,0,0,24,1,0,0,0,0,40,17,0,0,0,0,0,0,24,1,0,0,0,0,40,17],[1,35,0,0,0,0,6,6,0,0,0,0,0,0,0,0,1,7,0,0,0,0,34,34,0,0,40,34,0,0,0,0,7,7,0,0],[1,35,0,0,0,0,0,40,0,0,0,0,0,0,0,0,34,34,0,0,0,0,1,7,0,0,7,7,0,0,0,0,40,34,0,0] >;
74 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | ··· | 4N | 4O | ··· | 4AB | 5A | 5B | 10A | ··· | 10F | 20A | ··· | 20H | 20I | ··· | 20AF |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 5 | 5 | 10 | ··· | 10 | 20 | ··· | 20 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 10 | 10 | 2 | ··· | 2 | 10 | ··· | 10 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 |
74 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | - | - | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | D5 | D10 | D10 | D10 | C4×D5 | 2- (1+4) | Q8.10D10 | D4.10D10 |
| kernel | C42.125D10 | C4×Dic10 | C42⋊D5 | Dic5⋊3Q8 | C4⋊C4⋊7D5 | Q8×Dic5 | Q8×C20 | C2×Q8×D5 | Q8×D5 | C4×Q8 | C42 | C4⋊C4 | C2×Q8 | Q8 | C10 | C2 | C2 |
| # reps | 1 | 3 | 3 | 3 | 3 | 1 | 1 | 1 | 16 | 2 | 6 | 6 | 2 | 16 | 2 | 4 | 4 |
In GAP, Magma, Sage, TeX
C_4^2._{125}D_{10} % in TeX
G:=Group("C4^2.125D10"); // GroupNames label
G:=SmallGroup(320,1244);
// by ID
G=gap.SmallGroup(320,1244);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,232,387,184,1123,80,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^10=d^2=a^2,a*b=b*a,c*a*c^-1=d*a*d^-1=a^-1,b*c=c*b,d*b*d^-1=a^2*b,d*c*d^-1=c^9>;
// generators/relations
×
⋊
ℤ
𝔽
○
≀