metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.117D10, C10.1062+ (1+4), (C4×D4)⋊25D5, (D4×C20)⋊27C2, (C4×D20)⋊34C2, C20⋊2D4⋊11C2, C20⋊7D4⋊20C2, C4⋊C4.320D10, C20⋊2Q8⋊26C2, (C2×D4).224D10, C4.66(C4○D20), C20.114(C4○D4), (C4×C20).161C22, (C2×C20).165C23, (C2×C10).107C24, C22⋊C4.119D10, (C22×C4).215D10, C4.118(D4⋊2D5), C2.19(D4⋊8D10), Dic5.5D4⋊11C2, (C2×D20).222C22, (D4×C10).266C22, C23.21D10⋊9C2, C4⋊Dic5.302C22, (C22×D5).41C23, C23.104(C22×D5), C22.132(C23×D5), D10⋊C4.55C22, (C22×C10).177C23, (C22×C20).111C22, C5⋊2(C22.49C24), (C2×Dic10).30C22, (C2×Dic5).219C23, (C4×Dic5).226C22, C23.D5.108C22, C4⋊C4⋊7D5⋊16C2, C2.56(C2×C4○D20), C10.49(C2×C4○D4), (C2×C4×D5).77C22, C2.24(C2×D4⋊2D5), (C5×C4⋊C4).335C22, (C2×C4).163(C22×D5), (C2×C5⋊D4).20C22, (C5×C22⋊C4).130C22, SmallGroup(320,1235)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 838 in 236 conjugacy classes, 99 normal (29 characteristic)
C1, C2 [×3], C2 [×4], C4 [×4], C4 [×9], C22, C22 [×12], C5, C2×C4 [×3], C2×C4 [×2], C2×C4 [×14], D4 [×8], Q8 [×2], C23 [×2], C23 [×2], D5 [×2], C10 [×3], C10 [×2], C42, C42 [×4], C22⋊C4 [×2], C22⋊C4 [×10], C4⋊C4, C4⋊C4 [×5], C22×C4 [×2], C22×C4 [×2], C2×D4, C2×D4 [×5], C2×Q8 [×2], Dic5 [×6], C20 [×4], C20 [×3], D10 [×6], C2×C10, C2×C10 [×6], C42⋊C2 [×4], C4×D4, C4×D4, C4⋊D4 [×4], C4.4D4 [×4], C4⋊Q8, Dic10 [×2], C4×D5 [×4], D20 [×2], C2×Dic5 [×6], C5⋊D4 [×4], C2×C20 [×3], C2×C20 [×2], C2×C20 [×4], C5×D4 [×2], C22×D5 [×2], C22×C10 [×2], C22.49C24, C4×Dic5 [×4], C4⋊Dic5, C4⋊Dic5 [×4], D10⋊C4 [×6], C23.D5 [×4], C4×C20, C5×C22⋊C4 [×2], C5×C4⋊C4, C2×Dic10 [×2], C2×C4×D5 [×2], C2×D20, C2×C5⋊D4 [×4], C22×C20 [×2], D4×C10, C20⋊2Q8, C4×D20, Dic5.5D4 [×4], C4⋊C4⋊7D5 [×2], C23.21D10 [×2], C20⋊7D4 [×2], C20⋊2D4 [×2], D4×C20, C42.117D10
Quotients:
C1, C2 [×15], C22 [×35], C23 [×15], D5, C4○D4 [×4], C24, D10 [×7], C2×C4○D4 [×2], 2+ (1+4), C22×D5 [×7], C22.49C24, C4○D20 [×2], D4⋊2D5 [×2], C23×D5, C2×C4○D20, C2×D4⋊2D5, D4⋊8D10, C42.117D10
Generators and relations
G = < a,b,c,d | a4=b4=c10=1, d2=a2b2, ab=ba, cac-1=dad-1=a-1, bc=cb, dbd-1=b-1, dcd-1=a2b2c-1 >
►On 160 points
Generators in S160
(1 45 30 118)(2 119 21 46)(3 47 22 120)(4 111 23 48)(5 49 24 112)(6 113 25 50)(7 41 26 114)(8 115 27 42)(9 43 28 116)(10 117 29 44)(11 33 93 106)(12 107 94 34)(13 35 95 108)(14 109 96 36)(15 37 97 110)(16 101 98 38)(17 39 99 102)(18 103 100 40)(19 31 91 104)(20 105 92 32)(51 76 124 149)(52 150 125 77)(53 78 126 141)(54 142 127 79)(55 80 128 143)(56 144 129 71)(57 72 130 145)(58 146 121 73)(59 74 122 147)(60 148 123 75)(61 82 134 159)(62 160 135 83)(63 84 136 151)(64 152 137 85)(65 86 138 153)(66 154 139 87)(67 88 140 155)(68 156 131 89)(69 90 132 157)(70 158 133 81)
(1 70 60 103)(2 61 51 104)(3 62 52 105)(4 63 53 106)(5 64 54 107)(6 65 55 108)(7 66 56 109)(8 67 57 110)(9 68 58 101)(10 69 59 102)(11 111 84 78)(12 112 85 79)(13 113 86 80)(14 114 87 71)(15 115 88 72)(16 116 89 73)(17 117 90 74)(18 118 81 75)(19 119 82 76)(20 120 83 77)(21 134 124 31)(22 135 125 32)(23 136 126 33)(24 137 127 34)(25 138 128 35)(26 139 129 36)(27 140 130 37)(28 131 121 38)(29 132 122 39)(30 133 123 40)(41 154 144 96)(42 155 145 97)(43 156 146 98)(44 157 147 99)(45 158 148 100)(46 159 149 91)(47 160 150 92)(48 151 141 93)(49 152 142 94)(50 153 143 95)
(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 10 123 122)(2 121 124 9)(3 8 125 130)(4 129 126 7)(5 6 127 128)(11 96 151 87)(12 86 152 95)(13 94 153 85)(14 84 154 93)(15 92 155 83)(16 82 156 91)(17 100 157 81)(18 90 158 99)(19 98 159 89)(20 88 160 97)(21 58 51 28)(22 27 52 57)(23 56 53 26)(24 25 54 55)(29 60 59 30)(31 101 61 131)(32 140 62 110)(33 109 63 139)(34 138 64 108)(35 107 65 137)(36 136 66 106)(37 105 67 135)(38 134 68 104)(39 103 69 133)(40 132 70 102)(41 48 71 78)(42 77 72 47)(43 46 73 76)(44 75 74 45)(49 50 79 80)(111 144 141 114)(112 113 142 143)(115 150 145 120)(116 119 146 149)(117 148 147 118)
G:=sub<Sym(160)| (1,45,30,118)(2,119,21,46)(3,47,22,120)(4,111,23,48)(5,49,24,112)(6,113,25,50)(7,41,26,114)(8,115,27,42)(9,43,28,116)(10,117,29,44)(11,33,93,106)(12,107,94,34)(13,35,95,108)(14,109,96,36)(15,37,97,110)(16,101,98,38)(17,39,99,102)(18,103,100,40)(19,31,91,104)(20,105,92,32)(51,76,124,149)(52,150,125,77)(53,78,126,141)(54,142,127,79)(55,80,128,143)(56,144,129,71)(57,72,130,145)(58,146,121,73)(59,74,122,147)(60,148,123,75)(61,82,134,159)(62,160,135,83)(63,84,136,151)(64,152,137,85)(65,86,138,153)(66,154,139,87)(67,88,140,155)(68,156,131,89)(69,90,132,157)(70,158,133,81), (1,70,60,103)(2,61,51,104)(3,62,52,105)(4,63,53,106)(5,64,54,107)(6,65,55,108)(7,66,56,109)(8,67,57,110)(9,68,58,101)(10,69,59,102)(11,111,84,78)(12,112,85,79)(13,113,86,80)(14,114,87,71)(15,115,88,72)(16,116,89,73)(17,117,90,74)(18,118,81,75)(19,119,82,76)(20,120,83,77)(21,134,124,31)(22,135,125,32)(23,136,126,33)(24,137,127,34)(25,138,128,35)(26,139,129,36)(27,140,130,37)(28,131,121,38)(29,132,122,39)(30,133,123,40)(41,154,144,96)(42,155,145,97)(43,156,146,98)(44,157,147,99)(45,158,148,100)(46,159,149,91)(47,160,150,92)(48,151,141,93)(49,152,142,94)(50,153,143,95), (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), (1,10,123,122)(2,121,124,9)(3,8,125,130)(4,129,126,7)(5,6,127,128)(11,96,151,87)(12,86,152,95)(13,94,153,85)(14,84,154,93)(15,92,155,83)(16,82,156,91)(17,100,157,81)(18,90,158,99)(19,98,159,89)(20,88,160,97)(21,58,51,28)(22,27,52,57)(23,56,53,26)(24,25,54,55)(29,60,59,30)(31,101,61,131)(32,140,62,110)(33,109,63,139)(34,138,64,108)(35,107,65,137)(36,136,66,106)(37,105,67,135)(38,134,68,104)(39,103,69,133)(40,132,70,102)(41,48,71,78)(42,77,72,47)(43,46,73,76)(44,75,74,45)(49,50,79,80)(111,144,141,114)(112,113,142,143)(115,150,145,120)(116,119,146,149)(117,148,147,118)>;
G:=Group( (1,45,30,118)(2,119,21,46)(3,47,22,120)(4,111,23,48)(5,49,24,112)(6,113,25,50)(7,41,26,114)(8,115,27,42)(9,43,28,116)(10,117,29,44)(11,33,93,106)(12,107,94,34)(13,35,95,108)(14,109,96,36)(15,37,97,110)(16,101,98,38)(17,39,99,102)(18,103,100,40)(19,31,91,104)(20,105,92,32)(51,76,124,149)(52,150,125,77)(53,78,126,141)(54,142,127,79)(55,80,128,143)(56,144,129,71)(57,72,130,145)(58,146,121,73)(59,74,122,147)(60,148,123,75)(61,82,134,159)(62,160,135,83)(63,84,136,151)(64,152,137,85)(65,86,138,153)(66,154,139,87)(67,88,140,155)(68,156,131,89)(69,90,132,157)(70,158,133,81), (1,70,60,103)(2,61,51,104)(3,62,52,105)(4,63,53,106)(5,64,54,107)(6,65,55,108)(7,66,56,109)(8,67,57,110)(9,68,58,101)(10,69,59,102)(11,111,84,78)(12,112,85,79)(13,113,86,80)(14,114,87,71)(15,115,88,72)(16,116,89,73)(17,117,90,74)(18,118,81,75)(19,119,82,76)(20,120,83,77)(21,134,124,31)(22,135,125,32)(23,136,126,33)(24,137,127,34)(25,138,128,35)(26,139,129,36)(27,140,130,37)(28,131,121,38)(29,132,122,39)(30,133,123,40)(41,154,144,96)(42,155,145,97)(43,156,146,98)(44,157,147,99)(45,158,148,100)(46,159,149,91)(47,160,150,92)(48,151,141,93)(49,152,142,94)(50,153,143,95), (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), (1,10,123,122)(2,121,124,9)(3,8,125,130)(4,129,126,7)(5,6,127,128)(11,96,151,87)(12,86,152,95)(13,94,153,85)(14,84,154,93)(15,92,155,83)(16,82,156,91)(17,100,157,81)(18,90,158,99)(19,98,159,89)(20,88,160,97)(21,58,51,28)(22,27,52,57)(23,56,53,26)(24,25,54,55)(29,60,59,30)(31,101,61,131)(32,140,62,110)(33,109,63,139)(34,138,64,108)(35,107,65,137)(36,136,66,106)(37,105,67,135)(38,134,68,104)(39,103,69,133)(40,132,70,102)(41,48,71,78)(42,77,72,47)(43,46,73,76)(44,75,74,45)(49,50,79,80)(111,144,141,114)(112,113,142,143)(115,150,145,120)(116,119,146,149)(117,148,147,118) );
G=PermutationGroup([(1,45,30,118),(2,119,21,46),(3,47,22,120),(4,111,23,48),(5,49,24,112),(6,113,25,50),(7,41,26,114),(8,115,27,42),(9,43,28,116),(10,117,29,44),(11,33,93,106),(12,107,94,34),(13,35,95,108),(14,109,96,36),(15,37,97,110),(16,101,98,38),(17,39,99,102),(18,103,100,40),(19,31,91,104),(20,105,92,32),(51,76,124,149),(52,150,125,77),(53,78,126,141),(54,142,127,79),(55,80,128,143),(56,144,129,71),(57,72,130,145),(58,146,121,73),(59,74,122,147),(60,148,123,75),(61,82,134,159),(62,160,135,83),(63,84,136,151),(64,152,137,85),(65,86,138,153),(66,154,139,87),(67,88,140,155),(68,156,131,89),(69,90,132,157),(70,158,133,81)], [(1,70,60,103),(2,61,51,104),(3,62,52,105),(4,63,53,106),(5,64,54,107),(6,65,55,108),(7,66,56,109),(8,67,57,110),(9,68,58,101),(10,69,59,102),(11,111,84,78),(12,112,85,79),(13,113,86,80),(14,114,87,71),(15,115,88,72),(16,116,89,73),(17,117,90,74),(18,118,81,75),(19,119,82,76),(20,120,83,77),(21,134,124,31),(22,135,125,32),(23,136,126,33),(24,137,127,34),(25,138,128,35),(26,139,129,36),(27,140,130,37),(28,131,121,38),(29,132,122,39),(30,133,123,40),(41,154,144,96),(42,155,145,97),(43,156,146,98),(44,157,147,99),(45,158,148,100),(46,159,149,91),(47,160,150,92),(48,151,141,93),(49,152,142,94),(50,153,143,95)], [(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)], [(1,10,123,122),(2,121,124,9),(3,8,125,130),(4,129,126,7),(5,6,127,128),(11,96,151,87),(12,86,152,95),(13,94,153,85),(14,84,154,93),(15,92,155,83),(16,82,156,91),(17,100,157,81),(18,90,158,99),(19,98,159,89),(20,88,160,97),(21,58,51,28),(22,27,52,57),(23,56,53,26),(24,25,54,55),(29,60,59,30),(31,101,61,131),(32,140,62,110),(33,109,63,139),(34,138,64,108),(35,107,65,137),(36,136,66,106),(37,105,67,135),(38,134,68,104),(39,103,69,133),(40,132,70,102),(41,48,71,78),(42,77,72,47),(43,46,73,76),(44,75,74,45),(49,50,79,80),(111,144,141,114),(112,113,142,143),(115,150,145,120),(116,119,146,149),(117,148,147,118)])
Matrix representation ►G ⊆ GL4(𝔽41) generated by
| 40 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 0 | 0 | 32 | 0 |
| 0 | 0 | 29 | 9 |
| 39 | 28 | 0 | 0 |
| 13 | 2 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 20 | 21 | 0 | 0 |
| 20 | 23 | 0 | 0 |
| 0 | 0 | 15 | 39 |
| 0 | 0 | 30 | 26 |
| 20 | 21 | 0 | 0 |
| 18 | 21 | 0 | 0 |
| 0 | 0 | 26 | 2 |
| 0 | 0 | 10 | 15 |
G:=sub<GL(4,GF(41))| [40,0,0,0,0,40,0,0,0,0,32,29,0,0,0,9],[39,13,0,0,28,2,0,0,0,0,1,0,0,0,0,1],[20,20,0,0,21,23,0,0,0,0,15,30,0,0,39,26],[20,18,0,0,21,21,0,0,0,0,26,10,0,0,2,15] >;
65 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | ··· | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 4O | 4P | 4Q | 5A | 5B | 10A | ··· | 10F | 10G | ··· | 10N | 20A | ··· | 20H | 20I | ··· | 20X |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 10 | ··· | 10 | 10 | ··· | 10 | 20 | ··· | 20 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 20 | 20 | 2 | ··· | 2 | 4 | 10 | 10 | 10 | 10 | 20 | 20 | 20 | 20 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
65 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D5 | C4○D4 | D10 | D10 | D10 | D10 | D10 | C4○D20 | 2+ (1+4) | D4⋊2D5 | D4⋊8D10 |
| kernel | C42.117D10 | C20⋊2Q8 | C4×D20 | Dic5.5D4 | C4⋊C4⋊7D5 | C23.21D10 | C20⋊7D4 | C20⋊2D4 | D4×C20 | C4×D4 | C20 | C42 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×D4 | C4 | C10 | C4 | C2 |
| # reps | 1 | 1 | 1 | 4 | 2 | 2 | 2 | 2 | 1 | 2 | 8 | 2 | 4 | 2 | 4 | 2 | 16 | 1 | 4 | 4 |
In GAP, Magma, Sage, TeX
C_4^2._{117}D_{10} % in TeX
G:=Group("C4^2.117D10"); // GroupNames label
G:=SmallGroup(320,1235);
// by ID
G=gap.SmallGroup(320,1235);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,477,758,219,1571,192,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^10=1,d^2=a^2*b^2,a*b=b*a,c*a*c^-1=d*a*d^-1=a^-1,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=a^2*b^2*c^-1>;
// generators/relations
×
⋊
ℤ
𝔽
○
≀