metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.232D10, (C4×D5)⋊6Q8, C20⋊Q8⋊48C2, (C4×Q8)⋊11D5, C4.59(Q8×D5), (Q8×C20)⋊13C2, D10.1(C2×Q8), C4⋊C4.297D10, C20.117(C2×Q8), Dic5.2(C2×Q8), (D5×C42).6C2, (C4×Dic10)⋊39C2, C4.47(C4○D20), D10⋊Q8.5C2, (C2×Q8).177D10, C42⋊D5.4C2, Dic5⋊Q8⋊33C2, C20.117(C4○D4), C10.30(C22×Q8), (C2×C10).122C24, (C2×C20).499C23, (C4×C20).174C22, D10⋊3Q8.15C2, D10⋊2Q8.15C2, Dic5.10(C4○D4), Dic5.Q8⋊46C2, C4⋊Dic5.307C22, (Q8×C10).222C22, C22.143(C23×D5), C5⋊3(C23.37C23), (C4×Dic5).284C22, (C2×Dic5).226C23, (C22×D5).189C23, D10⋊C4.102C22, (C2×Dic10).299C22, C10.D4.155C22, C2.13(C2×Q8×D5), C2.30(D5×C4○D4), C2.61(C2×C4○D20), C10.54(C2×C4○D4), (C2×C4×D5).378C22, (C5×C4⋊C4).350C22, (C2×C4).584(C22×D5), SmallGroup(320,1250)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 670 in 222 conjugacy classes, 109 normal (43 characteristic)
C1, C2 [×3], C2 [×2], C4 [×4], C4 [×14], C22, C22 [×4], C5, C2×C4 [×3], C2×C4 [×4], C2×C4 [×15], Q8 [×8], C23, D5 [×2], C10 [×3], C42, C42 [×2], C42 [×5], C22⋊C4 [×4], C4⋊C4, C4⋊C4 [×2], C4⋊C4 [×13], C22×C4 [×3], C2×Q8, C2×Q8 [×3], Dic5 [×4], Dic5 [×5], C20 [×4], C20 [×5], D10 [×2], D10 [×2], C2×C10, C2×C42, C42⋊C2 [×2], C4×Q8, C4×Q8 [×3], C22⋊Q8 [×4], C42.C2 [×2], C4⋊Q8 [×2], Dic10 [×6], C4×D5 [×4], C4×D5 [×4], C2×Dic5 [×3], C2×Dic5 [×4], C2×C20 [×3], C2×C20 [×4], C5×Q8 [×2], C22×D5, C23.37C23, C4×Dic5 [×3], C4×Dic5 [×2], C10.D4 [×10], C4⋊Dic5, C4⋊Dic5 [×2], D10⋊C4 [×4], C4×C20, C4×C20 [×2], C5×C4⋊C4, C5×C4⋊C4 [×2], C2×Dic10, C2×Dic10 [×2], C2×C4×D5 [×3], Q8×C10, C4×Dic10, C4×Dic10 [×2], D5×C42, C42⋊D5 [×2], C20⋊Q8, Dic5.Q8 [×2], D10⋊Q8 [×2], D10⋊2Q8, Dic5⋊Q8, D10⋊3Q8, Q8×C20, C42.232D10
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], C22×D5 [×7], C23.37C23, C4○D20 [×2], Q8×D5 [×2], C23×D5, C2×C4○D20, C2×Q8×D5, D5×C4○D4, C42.232D10
Generators and relations
G = < a,b,c,d | a4=b4=1, c10=a2, d2=a2b2, ab=ba, cac-1=dad-1=a-1, bc=cb, bd=db, dcd-1=b2c9 >
►On 160 points
Generators in S160
(1 116 11 106)(2 107 12 117)(3 118 13 108)(4 109 14 119)(5 120 15 110)(6 111 16 101)(7 102 17 112)(8 113 18 103)(9 104 19 114)(10 115 20 105)(21 44 31 54)(22 55 32 45)(23 46 33 56)(24 57 34 47)(25 48 35 58)(26 59 36 49)(27 50 37 60)(28 41 38 51)(29 52 39 42)(30 43 40 53)(61 125 71 135)(62 136 72 126)(63 127 73 137)(64 138 74 128)(65 129 75 139)(66 140 76 130)(67 131 77 121)(68 122 78 132)(69 133 79 123)(70 124 80 134)(81 143 91 153)(82 154 92 144)(83 145 93 155)(84 156 94 146)(85 147 95 157)(86 158 96 148)(87 149 97 159)(88 160 98 150)(89 151 99 141)(90 142 100 152)
(1 69 25 83)(2 70 26 84)(3 71 27 85)(4 72 28 86)(5 73 29 87)(6 74 30 88)(7 75 31 89)(8 76 32 90)(9 77 33 91)(10 78 34 92)(11 79 35 93)(12 80 36 94)(13 61 37 95)(14 62 38 96)(15 63 39 97)(16 64 40 98)(17 65 21 99)(18 66 22 100)(19 67 23 81)(20 68 24 82)(41 158 109 126)(42 159 110 127)(43 160 111 128)(44 141 112 129)(45 142 113 130)(46 143 114 131)(47 144 115 132)(48 145 116 133)(49 146 117 134)(50 147 118 135)(51 148 119 136)(52 149 120 137)(53 150 101 138)(54 151 102 139)(55 152 103 140)(56 153 104 121)(57 154 105 122)(58 155 106 123)(59 156 107 124)(60 157 108 125)
(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 35 24)(2 23 36 9)(3 8 37 22)(4 21 38 7)(5 6 39 40)(11 20 25 34)(12 33 26 19)(13 18 27 32)(14 31 28 17)(15 16 29 30)(41 102 119 44)(42 43 120 101)(45 118 103 60)(46 59 104 117)(47 116 105 58)(48 57 106 115)(49 114 107 56)(50 55 108 113)(51 112 109 54)(52 53 110 111)(61 66 85 90)(62 89 86 65)(63 64 87 88)(67 80 91 84)(68 83 92 79)(69 78 93 82)(70 81 94 77)(71 76 95 100)(72 99 96 75)(73 74 97 98)(121 134 143 156)(122 155 144 133)(123 132 145 154)(124 153 146 131)(125 130 147 152)(126 151 148 129)(127 128 149 150)(135 140 157 142)(136 141 158 139)(137 138 159 160)
G:=sub<Sym(160)| (1,116,11,106)(2,107,12,117)(3,118,13,108)(4,109,14,119)(5,120,15,110)(6,111,16,101)(7,102,17,112)(8,113,18,103)(9,104,19,114)(10,115,20,105)(21,44,31,54)(22,55,32,45)(23,46,33,56)(24,57,34,47)(25,48,35,58)(26,59,36,49)(27,50,37,60)(28,41,38,51)(29,52,39,42)(30,43,40,53)(61,125,71,135)(62,136,72,126)(63,127,73,137)(64,138,74,128)(65,129,75,139)(66,140,76,130)(67,131,77,121)(68,122,78,132)(69,133,79,123)(70,124,80,134)(81,143,91,153)(82,154,92,144)(83,145,93,155)(84,156,94,146)(85,147,95,157)(86,158,96,148)(87,149,97,159)(88,160,98,150)(89,151,99,141)(90,142,100,152), (1,69,25,83)(2,70,26,84)(3,71,27,85)(4,72,28,86)(5,73,29,87)(6,74,30,88)(7,75,31,89)(8,76,32,90)(9,77,33,91)(10,78,34,92)(11,79,35,93)(12,80,36,94)(13,61,37,95)(14,62,38,96)(15,63,39,97)(16,64,40,98)(17,65,21,99)(18,66,22,100)(19,67,23,81)(20,68,24,82)(41,158,109,126)(42,159,110,127)(43,160,111,128)(44,141,112,129)(45,142,113,130)(46,143,114,131)(47,144,115,132)(48,145,116,133)(49,146,117,134)(50,147,118,135)(51,148,119,136)(52,149,120,137)(53,150,101,138)(54,151,102,139)(55,152,103,140)(56,153,104,121)(57,154,105,122)(58,155,106,123)(59,156,107,124)(60,157,108,125), (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), (1,10,35,24)(2,23,36,9)(3,8,37,22)(4,21,38,7)(5,6,39,40)(11,20,25,34)(12,33,26,19)(13,18,27,32)(14,31,28,17)(15,16,29,30)(41,102,119,44)(42,43,120,101)(45,118,103,60)(46,59,104,117)(47,116,105,58)(48,57,106,115)(49,114,107,56)(50,55,108,113)(51,112,109,54)(52,53,110,111)(61,66,85,90)(62,89,86,65)(63,64,87,88)(67,80,91,84)(68,83,92,79)(69,78,93,82)(70,81,94,77)(71,76,95,100)(72,99,96,75)(73,74,97,98)(121,134,143,156)(122,155,144,133)(123,132,145,154)(124,153,146,131)(125,130,147,152)(126,151,148,129)(127,128,149,150)(135,140,157,142)(136,141,158,139)(137,138,159,160)>;
G:=Group( (1,116,11,106)(2,107,12,117)(3,118,13,108)(4,109,14,119)(5,120,15,110)(6,111,16,101)(7,102,17,112)(8,113,18,103)(9,104,19,114)(10,115,20,105)(21,44,31,54)(22,55,32,45)(23,46,33,56)(24,57,34,47)(25,48,35,58)(26,59,36,49)(27,50,37,60)(28,41,38,51)(29,52,39,42)(30,43,40,53)(61,125,71,135)(62,136,72,126)(63,127,73,137)(64,138,74,128)(65,129,75,139)(66,140,76,130)(67,131,77,121)(68,122,78,132)(69,133,79,123)(70,124,80,134)(81,143,91,153)(82,154,92,144)(83,145,93,155)(84,156,94,146)(85,147,95,157)(86,158,96,148)(87,149,97,159)(88,160,98,150)(89,151,99,141)(90,142,100,152), (1,69,25,83)(2,70,26,84)(3,71,27,85)(4,72,28,86)(5,73,29,87)(6,74,30,88)(7,75,31,89)(8,76,32,90)(9,77,33,91)(10,78,34,92)(11,79,35,93)(12,80,36,94)(13,61,37,95)(14,62,38,96)(15,63,39,97)(16,64,40,98)(17,65,21,99)(18,66,22,100)(19,67,23,81)(20,68,24,82)(41,158,109,126)(42,159,110,127)(43,160,111,128)(44,141,112,129)(45,142,113,130)(46,143,114,131)(47,144,115,132)(48,145,116,133)(49,146,117,134)(50,147,118,135)(51,148,119,136)(52,149,120,137)(53,150,101,138)(54,151,102,139)(55,152,103,140)(56,153,104,121)(57,154,105,122)(58,155,106,123)(59,156,107,124)(60,157,108,125), (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), (1,10,35,24)(2,23,36,9)(3,8,37,22)(4,21,38,7)(5,6,39,40)(11,20,25,34)(12,33,26,19)(13,18,27,32)(14,31,28,17)(15,16,29,30)(41,102,119,44)(42,43,120,101)(45,118,103,60)(46,59,104,117)(47,116,105,58)(48,57,106,115)(49,114,107,56)(50,55,108,113)(51,112,109,54)(52,53,110,111)(61,66,85,90)(62,89,86,65)(63,64,87,88)(67,80,91,84)(68,83,92,79)(69,78,93,82)(70,81,94,77)(71,76,95,100)(72,99,96,75)(73,74,97,98)(121,134,143,156)(122,155,144,133)(123,132,145,154)(124,153,146,131)(125,130,147,152)(126,151,148,129)(127,128,149,150)(135,140,157,142)(136,141,158,139)(137,138,159,160) );
G=PermutationGroup([(1,116,11,106),(2,107,12,117),(3,118,13,108),(4,109,14,119),(5,120,15,110),(6,111,16,101),(7,102,17,112),(8,113,18,103),(9,104,19,114),(10,115,20,105),(21,44,31,54),(22,55,32,45),(23,46,33,56),(24,57,34,47),(25,48,35,58),(26,59,36,49),(27,50,37,60),(28,41,38,51),(29,52,39,42),(30,43,40,53),(61,125,71,135),(62,136,72,126),(63,127,73,137),(64,138,74,128),(65,129,75,139),(66,140,76,130),(67,131,77,121),(68,122,78,132),(69,133,79,123),(70,124,80,134),(81,143,91,153),(82,154,92,144),(83,145,93,155),(84,156,94,146),(85,147,95,157),(86,158,96,148),(87,149,97,159),(88,160,98,150),(89,151,99,141),(90,142,100,152)], [(1,69,25,83),(2,70,26,84),(3,71,27,85),(4,72,28,86),(5,73,29,87),(6,74,30,88),(7,75,31,89),(8,76,32,90),(9,77,33,91),(10,78,34,92),(11,79,35,93),(12,80,36,94),(13,61,37,95),(14,62,38,96),(15,63,39,97),(16,64,40,98),(17,65,21,99),(18,66,22,100),(19,67,23,81),(20,68,24,82),(41,158,109,126),(42,159,110,127),(43,160,111,128),(44,141,112,129),(45,142,113,130),(46,143,114,131),(47,144,115,132),(48,145,116,133),(49,146,117,134),(50,147,118,135),(51,148,119,136),(52,149,120,137),(53,150,101,138),(54,151,102,139),(55,152,103,140),(56,153,104,121),(57,154,105,122),(58,155,106,123),(59,156,107,124),(60,157,108,125)], [(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)], [(1,10,35,24),(2,23,36,9),(3,8,37,22),(4,21,38,7),(5,6,39,40),(11,20,25,34),(12,33,26,19),(13,18,27,32),(14,31,28,17),(15,16,29,30),(41,102,119,44),(42,43,120,101),(45,118,103,60),(46,59,104,117),(47,116,105,58),(48,57,106,115),(49,114,107,56),(50,55,108,113),(51,112,109,54),(52,53,110,111),(61,66,85,90),(62,89,86,65),(63,64,87,88),(67,80,91,84),(68,83,92,79),(69,78,93,82),(70,81,94,77),(71,76,95,100),(72,99,96,75),(73,74,97,98),(121,134,143,156),(122,155,144,133),(123,132,145,154),(124,153,146,131),(125,130,147,152),(126,151,148,129),(127,128,149,150),(135,140,157,142),(136,141,158,139),(137,138,159,160)])
Matrix representation ►G ⊆ GL4(𝔽41) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 9 | 0 |
| 0 | 0 | 28 | 32 |
| 9 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 |
| 0 | 0 | 9 | 0 |
| 0 | 0 | 0 | 9 |
| 3 | 38 | 0 | 0 |
| 3 | 24 | 0 | 0 |
| 0 | 0 | 29 | 37 |
| 0 | 0 | 26 | 12 |
| 24 | 3 | 0 | 0 |
| 40 | 17 | 0 | 0 |
| 0 | 0 | 12 | 4 |
| 0 | 0 | 36 | 29 |
G:=sub<GL(4,GF(41))| [1,0,0,0,0,1,0,0,0,0,9,28,0,0,0,32],[9,0,0,0,0,9,0,0,0,0,9,0,0,0,0,9],[3,3,0,0,38,24,0,0,0,0,29,26,0,0,37,12],[24,40,0,0,3,17,0,0,0,0,12,36,0,0,4,29] >;
68 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 4M | ··· | 4R | 4S | 4T | 4U | 4V | 5A | 5B | 10A | ··· | 10F | 20A | ··· | 20H | 20I | ··· | 20AF |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 10 | ··· | 10 | 20 | ··· | 20 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 10 | 10 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 10 | ··· | 10 | 20 | 20 | 20 | 20 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 |
68 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | - | + | + | + | + | - | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | Q8 | D5 | C4○D4 | C4○D4 | D10 | D10 | D10 | C4○D20 | Q8×D5 | D5×C4○D4 |
| kernel | C42.232D10 | C4×Dic10 | D5×C42 | C42⋊D5 | C20⋊Q8 | Dic5.Q8 | D10⋊Q8 | D10⋊2Q8 | Dic5⋊Q8 | D10⋊3Q8 | Q8×C20 | C4×D5 | C4×Q8 | Dic5 | C20 | C42 | C4⋊C4 | C2×Q8 | C4 | C4 | C2 |
| # reps | 1 | 3 | 1 | 2 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 4 | 2 | 4 | 4 | 6 | 6 | 2 | 16 | 4 | 4 |
In GAP, Magma, Sage, TeX
C_4^2._{232}D_{10} % in TeX
G:=Group("C4^2.232D10"); // GroupNames label
G:=SmallGroup(320,1250);
// by ID
G=gap.SmallGroup(320,1250);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,232,100,675,570,136,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^10=a^2,d^2=a^2*b^2,a*b=b*a,c*a*c^-1=d*a*d^-1=a^-1,b*c=c*b,b*d=d*b,d*c*d^-1=b^2*c^9>;
// generators/relations
×
⋊
ℤ
𝔽
○
≀