metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.140D10, C10.722+ (1+4), C10.892- (1+4), (C2×Q8).83D10, C4.4D4.9D5, (C2×D4).109D10, (C2×C20).78C23, C22⋊C4.34D10, C20.6Q8⋊28C2, Dic5⋊Q8⋊23C2, (C4×C20).221C22, (C2×C10).216C24, C4⋊Dic5.50C22, C2.74(D4⋊6D10), C23.38(C22×D5), (D4×C10).209C22, C23.D10⋊38C2, (C22×C10).46C23, (Q8×C10).125C22, C22.237(C23×D5), Dic5.14D4⋊39C2, C23.D5.53C22, C5⋊3(C22.57C24), (C4×Dic5).140C22, (C2×Dic5).111C23, C10.D4.83C22, C23.18D10.6C2, C2.50(D4.10D10), (C2×Dic10).182C22, (C22×Dic5).141C22, (C5×C4.4D4).7C2, (C2×C4).192(C22×D5), (C5×C22⋊C4).63C22, SmallGroup(320,1344)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 614 in 196 conjugacy classes, 91 normal (17 characteristic)
C1, C2, C2 [×2], C2 [×2], C4 [×13], C22, C22 [×6], C5, C2×C4, C2×C4 [×4], C2×C4 [×10], D4, Q8 [×3], C23 [×2], C10, C10 [×2], C10 [×2], C42, C42 [×2], C22⋊C4 [×4], C22⋊C4 [×6], C4⋊C4 [×16], C22×C4 [×2], C2×D4, C2×Q8, C2×Q8 [×2], Dic5 [×8], C20 [×5], C2×C10, C2×C10 [×6], C22⋊Q8 [×4], C22.D4 [×2], C4.4D4, C42.C2 [×2], C42⋊2C2 [×4], C4⋊Q8 [×2], Dic10 [×2], C2×Dic5 [×8], C2×Dic5 [×2], C2×C20, C2×C20 [×4], C5×D4, C5×Q8, C22×C10 [×2], C22.57C24, C4×Dic5 [×2], C10.D4 [×12], C4⋊Dic5 [×4], C23.D5 [×6], C4×C20, C5×C22⋊C4 [×4], C2×Dic10 [×2], C22×Dic5 [×2], D4×C10, Q8×C10, C20.6Q8 [×2], Dic5.14D4 [×4], C23.D10 [×4], C23.18D10 [×2], Dic5⋊Q8 [×2], C5×C4.4D4, C42.140D10
Quotients:
C1, C2 [×15], C22 [×35], C23 [×15], D5, C24, D10 [×7], 2+ (1+4), 2- (1+4) [×2], C22×D5 [×7], C22.57C24, C23×D5, D4⋊6D10, D4.10D10 [×2], C42.140D10
Generators and relations
G = < a,b,c,d | a4=b4=c10=1, d2=b2, ab=ba, cac-1=a-1b2, dad-1=a-1, cbc-1=b-1, dbd-1=a2b-1, dcd-1=c-1 >
►On 160 points
Generators in S160
(1 135 40 142)(2 131 36 148)(3 137 37 144)(4 133 38 150)(5 139 39 146)(6 61 31 42)(7 67 32 48)(8 63 33 44)(9 69 34 50)(10 65 35 46)(11 143 16 136)(12 149 17 132)(13 145 18 138)(14 141 19 134)(15 147 20 140)(21 47 29 66)(22 43 30 62)(23 49 26 68)(24 45 27 64)(25 41 28 70)(51 103 56 155)(52 151 57 109)(53 105 58 157)(54 153 59 101)(55 107 60 159)(71 106 76 158)(72 154 77 102)(73 108 78 160)(74 156 79 104)(75 110 80 152)(81 115 86 128)(82 124 87 111)(83 117 88 130)(84 126 89 113)(85 119 90 122)(91 114 96 127)(92 123 97 120)(93 116 98 129)(94 125 99 112)(95 118 100 121)
(1 117 15 112)(2 113 11 118)(3 119 12 114)(4 115 13 120)(5 111 14 116)(6 104 21 109)(7 110 22 105)(8 106 23 101)(9 102 24 107)(10 108 25 103)(16 121 36 126)(17 127 37 122)(18 123 38 128)(19 129 39 124)(20 125 40 130)(26 153 33 158)(27 159 34 154)(28 155 35 160)(29 151 31 156)(30 157 32 152)(41 56 65 78)(42 79 66 57)(43 58 67 80)(44 71 68 59)(45 60 69 72)(46 73 70 51)(47 52 61 74)(48 75 62 53)(49 54 63 76)(50 77 64 55)(81 138 97 150)(82 141 98 139)(83 140 99 142)(84 143 100 131)(85 132 91 144)(86 145 92 133)(87 134 93 146)(88 147 94 135)(89 136 95 148)(90 149 96 137)
(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 27 15 34)(2 26 11 33)(3 30 12 32)(4 29 13 31)(5 28 14 35)(6 38 21 18)(7 37 22 17)(8 36 23 16)(9 40 24 20)(10 39 25 19)(41 141 65 139)(42 150 66 138)(43 149 67 137)(44 148 68 136)(45 147 69 135)(46 146 70 134)(47 145 61 133)(48 144 62 132)(49 143 63 131)(50 142 64 140)(51 82 73 98)(52 81 74 97)(53 90 75 96)(54 89 76 95)(55 88 77 94)(56 87 78 93)(57 86 79 92)(58 85 80 91)(59 84 71 100)(60 83 72 99)(101 113 106 118)(102 112 107 117)(103 111 108 116)(104 120 109 115)(105 119 110 114)(121 153 126 158)(122 152 127 157)(123 151 128 156)(124 160 129 155)(125 159 130 154)
G:=sub<Sym(160)| (1,135,40,142)(2,131,36,148)(3,137,37,144)(4,133,38,150)(5,139,39,146)(6,61,31,42)(7,67,32,48)(8,63,33,44)(9,69,34,50)(10,65,35,46)(11,143,16,136)(12,149,17,132)(13,145,18,138)(14,141,19,134)(15,147,20,140)(21,47,29,66)(22,43,30,62)(23,49,26,68)(24,45,27,64)(25,41,28,70)(51,103,56,155)(52,151,57,109)(53,105,58,157)(54,153,59,101)(55,107,60,159)(71,106,76,158)(72,154,77,102)(73,108,78,160)(74,156,79,104)(75,110,80,152)(81,115,86,128)(82,124,87,111)(83,117,88,130)(84,126,89,113)(85,119,90,122)(91,114,96,127)(92,123,97,120)(93,116,98,129)(94,125,99,112)(95,118,100,121), (1,117,15,112)(2,113,11,118)(3,119,12,114)(4,115,13,120)(5,111,14,116)(6,104,21,109)(7,110,22,105)(8,106,23,101)(9,102,24,107)(10,108,25,103)(16,121,36,126)(17,127,37,122)(18,123,38,128)(19,129,39,124)(20,125,40,130)(26,153,33,158)(27,159,34,154)(28,155,35,160)(29,151,31,156)(30,157,32,152)(41,56,65,78)(42,79,66,57)(43,58,67,80)(44,71,68,59)(45,60,69,72)(46,73,70,51)(47,52,61,74)(48,75,62,53)(49,54,63,76)(50,77,64,55)(81,138,97,150)(82,141,98,139)(83,140,99,142)(84,143,100,131)(85,132,91,144)(86,145,92,133)(87,134,93,146)(88,147,94,135)(89,136,95,148)(90,149,96,137), (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), (1,27,15,34)(2,26,11,33)(3,30,12,32)(4,29,13,31)(5,28,14,35)(6,38,21,18)(7,37,22,17)(8,36,23,16)(9,40,24,20)(10,39,25,19)(41,141,65,139)(42,150,66,138)(43,149,67,137)(44,148,68,136)(45,147,69,135)(46,146,70,134)(47,145,61,133)(48,144,62,132)(49,143,63,131)(50,142,64,140)(51,82,73,98)(52,81,74,97)(53,90,75,96)(54,89,76,95)(55,88,77,94)(56,87,78,93)(57,86,79,92)(58,85,80,91)(59,84,71,100)(60,83,72,99)(101,113,106,118)(102,112,107,117)(103,111,108,116)(104,120,109,115)(105,119,110,114)(121,153,126,158)(122,152,127,157)(123,151,128,156)(124,160,129,155)(125,159,130,154)>;
G:=Group( (1,135,40,142)(2,131,36,148)(3,137,37,144)(4,133,38,150)(5,139,39,146)(6,61,31,42)(7,67,32,48)(8,63,33,44)(9,69,34,50)(10,65,35,46)(11,143,16,136)(12,149,17,132)(13,145,18,138)(14,141,19,134)(15,147,20,140)(21,47,29,66)(22,43,30,62)(23,49,26,68)(24,45,27,64)(25,41,28,70)(51,103,56,155)(52,151,57,109)(53,105,58,157)(54,153,59,101)(55,107,60,159)(71,106,76,158)(72,154,77,102)(73,108,78,160)(74,156,79,104)(75,110,80,152)(81,115,86,128)(82,124,87,111)(83,117,88,130)(84,126,89,113)(85,119,90,122)(91,114,96,127)(92,123,97,120)(93,116,98,129)(94,125,99,112)(95,118,100,121), (1,117,15,112)(2,113,11,118)(3,119,12,114)(4,115,13,120)(5,111,14,116)(6,104,21,109)(7,110,22,105)(8,106,23,101)(9,102,24,107)(10,108,25,103)(16,121,36,126)(17,127,37,122)(18,123,38,128)(19,129,39,124)(20,125,40,130)(26,153,33,158)(27,159,34,154)(28,155,35,160)(29,151,31,156)(30,157,32,152)(41,56,65,78)(42,79,66,57)(43,58,67,80)(44,71,68,59)(45,60,69,72)(46,73,70,51)(47,52,61,74)(48,75,62,53)(49,54,63,76)(50,77,64,55)(81,138,97,150)(82,141,98,139)(83,140,99,142)(84,143,100,131)(85,132,91,144)(86,145,92,133)(87,134,93,146)(88,147,94,135)(89,136,95,148)(90,149,96,137), (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), (1,27,15,34)(2,26,11,33)(3,30,12,32)(4,29,13,31)(5,28,14,35)(6,38,21,18)(7,37,22,17)(8,36,23,16)(9,40,24,20)(10,39,25,19)(41,141,65,139)(42,150,66,138)(43,149,67,137)(44,148,68,136)(45,147,69,135)(46,146,70,134)(47,145,61,133)(48,144,62,132)(49,143,63,131)(50,142,64,140)(51,82,73,98)(52,81,74,97)(53,90,75,96)(54,89,76,95)(55,88,77,94)(56,87,78,93)(57,86,79,92)(58,85,80,91)(59,84,71,100)(60,83,72,99)(101,113,106,118)(102,112,107,117)(103,111,108,116)(104,120,109,115)(105,119,110,114)(121,153,126,158)(122,152,127,157)(123,151,128,156)(124,160,129,155)(125,159,130,154) );
G=PermutationGroup([(1,135,40,142),(2,131,36,148),(3,137,37,144),(4,133,38,150),(5,139,39,146),(6,61,31,42),(7,67,32,48),(8,63,33,44),(9,69,34,50),(10,65,35,46),(11,143,16,136),(12,149,17,132),(13,145,18,138),(14,141,19,134),(15,147,20,140),(21,47,29,66),(22,43,30,62),(23,49,26,68),(24,45,27,64),(25,41,28,70),(51,103,56,155),(52,151,57,109),(53,105,58,157),(54,153,59,101),(55,107,60,159),(71,106,76,158),(72,154,77,102),(73,108,78,160),(74,156,79,104),(75,110,80,152),(81,115,86,128),(82,124,87,111),(83,117,88,130),(84,126,89,113),(85,119,90,122),(91,114,96,127),(92,123,97,120),(93,116,98,129),(94,125,99,112),(95,118,100,121)], [(1,117,15,112),(2,113,11,118),(3,119,12,114),(4,115,13,120),(5,111,14,116),(6,104,21,109),(7,110,22,105),(8,106,23,101),(9,102,24,107),(10,108,25,103),(16,121,36,126),(17,127,37,122),(18,123,38,128),(19,129,39,124),(20,125,40,130),(26,153,33,158),(27,159,34,154),(28,155,35,160),(29,151,31,156),(30,157,32,152),(41,56,65,78),(42,79,66,57),(43,58,67,80),(44,71,68,59),(45,60,69,72),(46,73,70,51),(47,52,61,74),(48,75,62,53),(49,54,63,76),(50,77,64,55),(81,138,97,150),(82,141,98,139),(83,140,99,142),(84,143,100,131),(85,132,91,144),(86,145,92,133),(87,134,93,146),(88,147,94,135),(89,136,95,148),(90,149,96,137)], [(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)], [(1,27,15,34),(2,26,11,33),(3,30,12,32),(4,29,13,31),(5,28,14,35),(6,38,21,18),(7,37,22,17),(8,36,23,16),(9,40,24,20),(10,39,25,19),(41,141,65,139),(42,150,66,138),(43,149,67,137),(44,148,68,136),(45,147,69,135),(46,146,70,134),(47,145,61,133),(48,144,62,132),(49,143,63,131),(50,142,64,140),(51,82,73,98),(52,81,74,97),(53,90,75,96),(54,89,76,95),(55,88,77,94),(56,87,78,93),(57,86,79,92),(58,85,80,91),(59,84,71,100),(60,83,72,99),(101,113,106,118),(102,112,107,117),(103,111,108,116),(104,120,109,115),(105,119,110,114),(121,153,126,158),(122,152,127,157),(123,151,128,156),(124,160,129,155),(125,159,130,154)])
Matrix representation ►G ⊆ GL8(𝔽41)
| 11 | 32 | 0 | 0 | 0 | 0 | 0 | 0 |
| 9 | 30 | 0 | 0 | 0 | 0 | 0 | 0 |
| 32 | 0 | 2 | 32 | 0 | 0 | 0 | 0 |
| 28 | 9 | 37 | 39 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 2 | 13 | 33 | 35 |
| 0 | 0 | 0 | 0 | 28 | 39 | 33 | 39 |
| 0 | 0 | 0 | 0 | 36 | 15 | 30 | 28 |
| 0 | 0 | 0 | 0 | 20 | 21 | 22 | 11 |
| 0 | 0 | 34 | 1 | 0 | 0 | 0 | 0 |
| 40 | 40 | 39 | 40 | 0 | 0 | 0 | 0 |
| 23 | 23 | 1 | 0 | 0 | 0 | 0 | 0 |
| 37 | 38 | 7 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 38 | 38 |
| 0 | 0 | 0 | 0 | 0 | 1 | 3 | 0 |
| 0 | 0 | 0 | 0 | 0 | 13 | 40 | 0 |
| 0 | 0 | 0 | 0 | 28 | 28 | 0 | 40 |
| 40 | 7 | 0 | 0 | 0 | 0 | 0 | 0 |
| 34 | 7 | 0 | 0 | 0 | 0 | 0 | 0 |
| 13 | 6 | 35 | 34 | 0 | 0 | 0 | 0 |
| 16 | 8 | 6 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 35 | 35 | 0 | 0 |
| 0 | 0 | 0 | 0 | 6 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 37 | 28 | 7 | 6 |
| 0 | 0 | 0 | 0 | 0 | 9 | 34 | 0 |
| 24 | 40 | 0 | 0 | 0 | 0 | 0 | 0 |
| 3 | 17 | 0 | 0 | 0 | 0 | 0 | 0 |
| 17 | 0 | 20 | 17 | 0 | 0 | 0 | 0 |
| 38 | 1 | 15 | 21 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 32 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 28 | 9 | 0 | 0 |
| 0 | 0 | 0 | 0 | 36 | 35 | 19 | 28 |
| 0 | 0 | 0 | 0 | 40 | 6 | 31 | 22 |
G:=sub<GL(8,GF(41))| [11,9,32,28,0,0,0,0,32,30,0,9,0,0,0,0,0,0,2,37,0,0,0,0,0,0,32,39,0,0,0,0,0,0,0,0,2,28,36,20,0,0,0,0,13,39,15,21,0,0,0,0,33,33,30,22,0,0,0,0,35,39,28,11],[0,40,23,37,0,0,0,0,0,40,23,38,0,0,0,0,34,39,1,7,0,0,0,0,1,40,0,0,0,0,0,0,0,0,0,0,1,0,0,28,0,0,0,0,0,1,13,28,0,0,0,0,38,3,40,0,0,0,0,0,38,0,0,40],[40,34,13,16,0,0,0,0,7,7,6,8,0,0,0,0,0,0,35,6,0,0,0,0,0,0,34,0,0,0,0,0,0,0,0,0,35,6,37,0,0,0,0,0,35,40,28,9,0,0,0,0,0,0,7,34,0,0,0,0,0,0,6,0],[24,3,17,38,0,0,0,0,40,17,0,1,0,0,0,0,0,0,20,15,0,0,0,0,0,0,17,21,0,0,0,0,0,0,0,0,32,28,36,40,0,0,0,0,0,9,35,6,0,0,0,0,0,0,19,31,0,0,0,0,0,0,28,22] >;
47 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | ··· | 4E | 4F | ··· | 4M | 5A | 5B | 10A | ··· | 10F | 10G | 10H | 10I | 10J | 20A | ··· | 20L | 20M | 20N | 20O | 20P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 5 | 5 | 10 | ··· | 10 | 10 | 10 | 10 | 10 | 20 | ··· | 20 | 20 | 20 | 20 | 20 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 4 | ··· | 4 | 20 | ··· | 20 | 2 | 2 | 2 | ··· | 2 | 8 | 8 | 8 | 8 | 4 | ··· | 4 | 8 | 8 | 8 | 8 |
47 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | - | - | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | D5 | D10 | D10 | D10 | D10 | 2+ (1+4) | 2- (1+4) | D4⋊6D10 | D4.10D10 |
| kernel | C42.140D10 | C20.6Q8 | Dic5.14D4 | C23.D10 | C23.18D10 | Dic5⋊Q8 | C5×C4.4D4 | C4.4D4 | C42 | C22⋊C4 | C2×D4 | C2×Q8 | C10 | C10 | C2 | C2 |
| # reps | 1 | 2 | 4 | 4 | 2 | 2 | 1 | 2 | 2 | 8 | 2 | 2 | 1 | 2 | 4 | 8 |
In GAP, Magma, Sage, TeX
C_4^2._{140}D_{10} % in TeX
G:=Group("C4^2.140D10"); // GroupNames label
G:=SmallGroup(320,1344);
// by ID
G=gap.SmallGroup(320,1344);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,224,758,219,184,1571,570,297,136,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^10=1,d^2=b^2,a*b=b*a,c*a*c^-1=a^-1*b^2,d*a*d^-1=a^-1,c*b*c^-1=b^-1,d*b*d^-1=a^2*b^-1,d*c*d^-1=c^-1>;
// generators/relations
×
⋊
ℤ
𝔽
○
≀