metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.236D10, (C4×D5)⋊3Q8, C20⋊Q8⋊34C2, C4.38(Q8×D5), D10.3(C2×Q8), C20.49(C2×Q8), C4⋊C4.204D10, C42.C2⋊15D5, (D5×C42).8C2, (C2×C20).86C23, D10⋊Q8.1C2, C20.6Q8⋊22C2, Dic5.33(C2×Q8), Dic5⋊3Q8⋊34C2, C10.41(C22×Q8), (C2×C10).232C24, (C4×C20).192C22, Dic5.18(C4○D4), C4⋊Dic5.239C22, C22.253(C23×D5), D10⋊C4.38C22, C5⋊5(C23.37C23), (C2×Dic5).378C23, (C4×Dic5).147C22, C10.D4.50C22, (C22×D5).229C23, (C2×Dic10).184C22, C2.24(C2×Q8×D5), C2.84(D5×C4○D4), (C5×C42.C2)⋊5C2, C4⋊C4⋊7D5.11C2, C10.195(C2×C4○D4), (C2×C4×D5).319C22, (C2×C4).77(C22×D5), (C5×C4⋊C4).187C22, SmallGroup(320,1360)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 686 in 222 conjugacy classes, 107 normal (19 characteristic)
C1, C2, C2 [×2], C2 [×2], C4 [×2], C4 [×16], C22, C22 [×4], C5, C2×C4, C2×C4 [×6], C2×C4 [×15], Q8 [×8], C23, D5 [×2], C10, C10 [×2], C42, C42 [×7], C22⋊C4 [×4], C4⋊C4 [×6], C4⋊C4 [×10], C22×C4 [×3], C2×Q8 [×4], Dic5 [×6], Dic5 [×4], C20 [×2], C20 [×6], D10 [×2], D10 [×2], C2×C10, C2×C42, C42⋊C2 [×2], C4×Q8 [×4], C22⋊Q8 [×4], C42.C2, C42.C2, C4⋊Q8 [×2], Dic10 [×8], C4×D5 [×4], C4×D5 [×4], C2×Dic5, C2×Dic5 [×6], C2×C20, C2×C20 [×6], C22×D5, C23.37C23, C4×Dic5, C4×Dic5 [×6], C10.D4 [×8], C4⋊Dic5 [×2], D10⋊C4 [×4], C4×C20, C5×C4⋊C4 [×6], C2×Dic10 [×4], C2×C4×D5, C2×C4×D5 [×2], C20.6Q8, D5×C42, Dic5⋊3Q8 [×4], C20⋊Q8 [×2], C4⋊C4⋊7D5 [×2], D10⋊Q8 [×4], C5×C42.C2, C42.236D10
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, Q8×D5 [×2], C23×D5, C2×Q8×D5, D5×C4○D4 [×2], C42.236D10
Generators and relations
G = < a,b,c,d | a4=b4=1, c10=b2, d2=a2b2, ab=ba, cac-1=dad-1=a-1, cbc-1=dbd-1=a2b-1, dcd-1=a2c9 >
►On 160 points
Generators in S160
(1 22 65 153)(2 154 66 23)(3 24 67 155)(4 156 68 25)(5 26 69 157)(6 158 70 27)(7 28 71 159)(8 160 72 29)(9 30 73 141)(10 142 74 31)(11 32 75 143)(12 144 76 33)(13 34 77 145)(14 146 78 35)(15 36 79 147)(16 148 80 37)(17 38 61 149)(18 150 62 39)(19 40 63 151)(20 152 64 21)(41 98 135 117)(42 118 136 99)(43 100 137 119)(44 120 138 81)(45 82 139 101)(46 102 140 83)(47 84 121 103)(48 104 122 85)(49 86 123 105)(50 106 124 87)(51 88 125 107)(52 108 126 89)(53 90 127 109)(54 110 128 91)(55 92 129 111)(56 112 130 93)(57 94 131 113)(58 114 132 95)(59 96 133 115)(60 116 134 97)
(1 135 11 125)(2 52 12 42)(3 137 13 127)(4 54 14 44)(5 139 15 129)(6 56 16 46)(7 121 17 131)(8 58 18 48)(9 123 19 133)(10 60 20 50)(21 87 31 97)(22 117 32 107)(23 89 33 99)(24 119 34 109)(25 91 35 81)(26 101 36 111)(27 93 37 83)(28 103 38 113)(29 95 39 85)(30 105 40 115)(41 75 51 65)(43 77 53 67)(45 79 55 69)(47 61 57 71)(49 63 59 73)(62 122 72 132)(64 124 74 134)(66 126 76 136)(68 128 78 138)(70 130 80 140)(82 147 92 157)(84 149 94 159)(86 151 96 141)(88 153 98 143)(90 155 100 145)(102 158 112 148)(104 160 114 150)(106 142 116 152)(108 144 118 154)(110 146 120 156)
(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 75 64)(2 63 76 9)(3 8 77 62)(4 61 78 7)(5 6 79 80)(11 20 65 74)(12 73 66 19)(13 18 67 72)(14 71 68 17)(15 16 69 70)(21 153 142 32)(22 31 143 152)(23 151 144 30)(24 29 145 150)(25 149 146 28)(26 27 147 148)(33 141 154 40)(34 39 155 160)(35 159 156 38)(36 37 157 158)(41 50 125 134)(42 133 126 49)(43 48 127 132)(44 131 128 47)(45 46 129 130)(51 60 135 124)(52 123 136 59)(53 58 137 122)(54 121 138 57)(55 56 139 140)(81 113 110 84)(82 83 111 112)(85 109 114 100)(86 99 115 108)(87 107 116 98)(88 97 117 106)(89 105 118 96)(90 95 119 104)(91 103 120 94)(92 93 101 102)
G:=sub<Sym(160)| (1,22,65,153)(2,154,66,23)(3,24,67,155)(4,156,68,25)(5,26,69,157)(6,158,70,27)(7,28,71,159)(8,160,72,29)(9,30,73,141)(10,142,74,31)(11,32,75,143)(12,144,76,33)(13,34,77,145)(14,146,78,35)(15,36,79,147)(16,148,80,37)(17,38,61,149)(18,150,62,39)(19,40,63,151)(20,152,64,21)(41,98,135,117)(42,118,136,99)(43,100,137,119)(44,120,138,81)(45,82,139,101)(46,102,140,83)(47,84,121,103)(48,104,122,85)(49,86,123,105)(50,106,124,87)(51,88,125,107)(52,108,126,89)(53,90,127,109)(54,110,128,91)(55,92,129,111)(56,112,130,93)(57,94,131,113)(58,114,132,95)(59,96,133,115)(60,116,134,97), (1,135,11,125)(2,52,12,42)(3,137,13,127)(4,54,14,44)(5,139,15,129)(6,56,16,46)(7,121,17,131)(8,58,18,48)(9,123,19,133)(10,60,20,50)(21,87,31,97)(22,117,32,107)(23,89,33,99)(24,119,34,109)(25,91,35,81)(26,101,36,111)(27,93,37,83)(28,103,38,113)(29,95,39,85)(30,105,40,115)(41,75,51,65)(43,77,53,67)(45,79,55,69)(47,61,57,71)(49,63,59,73)(62,122,72,132)(64,124,74,134)(66,126,76,136)(68,128,78,138)(70,130,80,140)(82,147,92,157)(84,149,94,159)(86,151,96,141)(88,153,98,143)(90,155,100,145)(102,158,112,148)(104,160,114,150)(106,142,116,152)(108,144,118,154)(110,146,120,156), (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), (1,10,75,64)(2,63,76,9)(3,8,77,62)(4,61,78,7)(5,6,79,80)(11,20,65,74)(12,73,66,19)(13,18,67,72)(14,71,68,17)(15,16,69,70)(21,153,142,32)(22,31,143,152)(23,151,144,30)(24,29,145,150)(25,149,146,28)(26,27,147,148)(33,141,154,40)(34,39,155,160)(35,159,156,38)(36,37,157,158)(41,50,125,134)(42,133,126,49)(43,48,127,132)(44,131,128,47)(45,46,129,130)(51,60,135,124)(52,123,136,59)(53,58,137,122)(54,121,138,57)(55,56,139,140)(81,113,110,84)(82,83,111,112)(85,109,114,100)(86,99,115,108)(87,107,116,98)(88,97,117,106)(89,105,118,96)(90,95,119,104)(91,103,120,94)(92,93,101,102)>;
G:=Group( (1,22,65,153)(2,154,66,23)(3,24,67,155)(4,156,68,25)(5,26,69,157)(6,158,70,27)(7,28,71,159)(8,160,72,29)(9,30,73,141)(10,142,74,31)(11,32,75,143)(12,144,76,33)(13,34,77,145)(14,146,78,35)(15,36,79,147)(16,148,80,37)(17,38,61,149)(18,150,62,39)(19,40,63,151)(20,152,64,21)(41,98,135,117)(42,118,136,99)(43,100,137,119)(44,120,138,81)(45,82,139,101)(46,102,140,83)(47,84,121,103)(48,104,122,85)(49,86,123,105)(50,106,124,87)(51,88,125,107)(52,108,126,89)(53,90,127,109)(54,110,128,91)(55,92,129,111)(56,112,130,93)(57,94,131,113)(58,114,132,95)(59,96,133,115)(60,116,134,97), (1,135,11,125)(2,52,12,42)(3,137,13,127)(4,54,14,44)(5,139,15,129)(6,56,16,46)(7,121,17,131)(8,58,18,48)(9,123,19,133)(10,60,20,50)(21,87,31,97)(22,117,32,107)(23,89,33,99)(24,119,34,109)(25,91,35,81)(26,101,36,111)(27,93,37,83)(28,103,38,113)(29,95,39,85)(30,105,40,115)(41,75,51,65)(43,77,53,67)(45,79,55,69)(47,61,57,71)(49,63,59,73)(62,122,72,132)(64,124,74,134)(66,126,76,136)(68,128,78,138)(70,130,80,140)(82,147,92,157)(84,149,94,159)(86,151,96,141)(88,153,98,143)(90,155,100,145)(102,158,112,148)(104,160,114,150)(106,142,116,152)(108,144,118,154)(110,146,120,156), (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), (1,10,75,64)(2,63,76,9)(3,8,77,62)(4,61,78,7)(5,6,79,80)(11,20,65,74)(12,73,66,19)(13,18,67,72)(14,71,68,17)(15,16,69,70)(21,153,142,32)(22,31,143,152)(23,151,144,30)(24,29,145,150)(25,149,146,28)(26,27,147,148)(33,141,154,40)(34,39,155,160)(35,159,156,38)(36,37,157,158)(41,50,125,134)(42,133,126,49)(43,48,127,132)(44,131,128,47)(45,46,129,130)(51,60,135,124)(52,123,136,59)(53,58,137,122)(54,121,138,57)(55,56,139,140)(81,113,110,84)(82,83,111,112)(85,109,114,100)(86,99,115,108)(87,107,116,98)(88,97,117,106)(89,105,118,96)(90,95,119,104)(91,103,120,94)(92,93,101,102) );
G=PermutationGroup([(1,22,65,153),(2,154,66,23),(3,24,67,155),(4,156,68,25),(5,26,69,157),(6,158,70,27),(7,28,71,159),(8,160,72,29),(9,30,73,141),(10,142,74,31),(11,32,75,143),(12,144,76,33),(13,34,77,145),(14,146,78,35),(15,36,79,147),(16,148,80,37),(17,38,61,149),(18,150,62,39),(19,40,63,151),(20,152,64,21),(41,98,135,117),(42,118,136,99),(43,100,137,119),(44,120,138,81),(45,82,139,101),(46,102,140,83),(47,84,121,103),(48,104,122,85),(49,86,123,105),(50,106,124,87),(51,88,125,107),(52,108,126,89),(53,90,127,109),(54,110,128,91),(55,92,129,111),(56,112,130,93),(57,94,131,113),(58,114,132,95),(59,96,133,115),(60,116,134,97)], [(1,135,11,125),(2,52,12,42),(3,137,13,127),(4,54,14,44),(5,139,15,129),(6,56,16,46),(7,121,17,131),(8,58,18,48),(9,123,19,133),(10,60,20,50),(21,87,31,97),(22,117,32,107),(23,89,33,99),(24,119,34,109),(25,91,35,81),(26,101,36,111),(27,93,37,83),(28,103,38,113),(29,95,39,85),(30,105,40,115),(41,75,51,65),(43,77,53,67),(45,79,55,69),(47,61,57,71),(49,63,59,73),(62,122,72,132),(64,124,74,134),(66,126,76,136),(68,128,78,138),(70,130,80,140),(82,147,92,157),(84,149,94,159),(86,151,96,141),(88,153,98,143),(90,155,100,145),(102,158,112,148),(104,160,114,150),(106,142,116,152),(108,144,118,154),(110,146,120,156)], [(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)], [(1,10,75,64),(2,63,76,9),(3,8,77,62),(4,61,78,7),(5,6,79,80),(11,20,65,74),(12,73,66,19),(13,18,67,72),(14,71,68,17),(15,16,69,70),(21,153,142,32),(22,31,143,152),(23,151,144,30),(24,29,145,150),(25,149,146,28),(26,27,147,148),(33,141,154,40),(34,39,155,160),(35,159,156,38),(36,37,157,158),(41,50,125,134),(42,133,126,49),(43,48,127,132),(44,131,128,47),(45,46,129,130),(51,60,135,124),(52,123,136,59),(53,58,137,122),(54,121,138,57),(55,56,139,140),(81,113,110,84),(82,83,111,112),(85,109,114,100),(86,99,115,108),(87,107,116,98),(88,97,117,106),(89,105,118,96),(90,95,119,104),(91,103,120,94),(92,93,101,102)])
Matrix representation ►G ⊆ GL6(𝔽41)
| 9 | 0 | 0 | 0 | 0 | 0 |
| 0 | 32 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 9 | 0 |
| 0 | 0 | 0 | 0 | 1 | 32 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 32 | 0 |
| 0 | 0 | 0 | 0 | 0 | 32 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 6 | 0 | 0 |
| 0 | 0 | 34 | 7 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 18 |
| 0 | 0 | 0 | 0 | 9 | 1 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 34 | 6 | 0 | 0 |
| 0 | 0 | 33 | 7 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 23 |
| 0 | 0 | 0 | 0 | 0 | 40 |
G:=sub<GL(6,GF(41))| [9,0,0,0,0,0,0,32,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,9,1,0,0,0,0,0,32],[1,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,32,0,0,0,0,0,0,32],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,34,0,0,0,0,6,7,0,0,0,0,0,0,40,9,0,0,0,0,18,1],[0,1,0,0,0,0,40,0,0,0,0,0,0,0,34,33,0,0,0,0,6,7,0,0,0,0,0,0,1,0,0,0,0,0,23,40] >;
56 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | ··· | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 4O | 4P | 4Q | 4R | 4S | 4T | 4U | 4V | 5A | 5B | 10A | ··· | 10F | 20A | ··· | 20L | 20M | ··· | 20T |
| 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 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 10 | 10 | 10 | 10 | 20 | 20 | 20 | 20 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | ··· | 8 |
56 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | - | + | + | + | - | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | Q8 | D5 | C4○D4 | D10 | D10 | Q8×D5 | D5×C4○D4 |
| kernel | C42.236D10 | C20.6Q8 | D5×C42 | Dic5⋊3Q8 | C20⋊Q8 | C4⋊C4⋊7D5 | D10⋊Q8 | C5×C42.C2 | C4×D5 | C42.C2 | Dic5 | C42 | C4⋊C4 | C4 | C2 |
| # reps | 1 | 1 | 1 | 4 | 2 | 2 | 4 | 1 | 4 | 2 | 8 | 2 | 12 | 4 | 8 |
In GAP, Magma, Sage, TeX
C_4^2._{236}D_{10} % in TeX
G:=Group("C4^2.236D10"); // GroupNames label
G:=SmallGroup(320,1360);
// by ID
G=gap.SmallGroup(320,1360);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,232,100,1123,570,409,80,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^10=b^2,d^2=a^2*b^2,a*b=b*a,c*a*c^-1=d*a*d^-1=a^-1,c*b*c^-1=d*b*d^-1=a^2*b^-1,d*c*d^-1=a^2*c^9>;
// generators/relations
×
⋊
ℤ
𝔽
○
≀