metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.105D10, C10.562- (1+4), (C4×D4).12D5, C4⋊C4.280D10, (D4×C20).13C2, (C4×Dic10)⋊27C2, (C2×D4).209D10, C20.48D4⋊8C2, (C2×C10).85C24, C20.6Q8⋊15C2, C20.291(C4○D4), (C4×C20).147C22, (C2×C20).585C23, C22⋊C4.130D10, Dic5.Q8⋊7C2, C23.D10⋊6C2, (C22×C4).204D10, C4.115(D4⋊2D5), C23.D5.9C22, (D4×C10).303C22, C22.11(C4○D20), C23.21D10⋊6C2, C4⋊Dic5.296C22, (C22×C20).79C22, (C2×Dic5).35C23, C22.113(C23×D5), C23.165(C22×D5), C23.11D10⋊27C2, (C22×C10).155C23, C5⋊4(C22.46C24), (C4×Dic5).222C22, C23.18D10.5C2, C2.14(D4.10D10), (C2×Dic10).244C22, C10.D4.153C22, (C22×Dic5).93C22, (C2×C4⋊Dic5)⋊23C2, C2.41(C2×C4○D20), C10.37(C2×C4○D4), C2.19(C2×D4⋊2D5), (C2×C10).15(C4○D4), (C5×C4⋊C4).321C22, (C2×C4).155(C22×D5), (C5×C22⋊C4).142C22, SmallGroup(320,1213)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 598 in 214 conjugacy classes, 99 normal (51 characteristic)
C1, C2 [×3], C2 [×3], C4 [×2], C4 [×12], C22, C22 [×2], C22 [×5], C5, C2×C4 [×5], C2×C4 [×16], D4 [×2], Q8 [×2], C23 [×2], C10 [×3], C10 [×3], C42, C42 [×4], C22⋊C4 [×2], C22⋊C4 [×6], C4⋊C4, C4⋊C4 [×15], C22×C4 [×2], C22×C4 [×2], C2×D4, C2×Q8, Dic5 [×8], C20 [×2], C20 [×4], C2×C10, C2×C10 [×2], C2×C10 [×5], C2×C4⋊C4, C42⋊C2 [×3], C4×D4, C4×Q8, C22⋊Q8 [×2], C22.D4 [×2], C42.C2 [×3], C42⋊2C2 [×2], Dic10 [×2], C2×Dic5 [×8], C2×Dic5 [×4], C2×C20 [×5], C2×C20 [×4], C5×D4 [×2], C22×C10 [×2], C22.46C24, C4×Dic5 [×4], C10.D4 [×10], C4⋊Dic5 [×3], C4⋊Dic5 [×2], C23.D5 [×6], C4×C20, C5×C22⋊C4 [×2], C5×C4⋊C4, C2×Dic10, C22×Dic5 [×2], C22×C20 [×2], D4×C10, C4×Dic10, C20.6Q8, C23.11D10 [×2], C23.D10 [×2], Dic5.Q8 [×2], C20.48D4 [×2], C2×C4⋊Dic5, C23.21D10, C23.18D10 [×2], D4×C20, C42.105D10
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], C22.46C24, C4○D20 [×2], D4⋊2D5 [×2], C23×D5, C2×C4○D20, C2×D4⋊2D5, D4.10D10, C42.105D10
Generators and relations
G = < a,b,c,d | a4=b4=c10=1, d2=a2, ab=ba, cac-1=a-1, dad-1=a-1b2, bc=cb, dbd-1=b-1, dcd-1=c-1 >
►On 160 points
Generators in S160
(1 68 63 6)(2 7 64 69)(3 70 65 8)(4 9 66 61)(5 62 67 10)(11 16 84 89)(12 90 85 17)(13 18 86 81)(14 82 87 19)(15 20 88 83)(21 26 80 75)(22 76 71 27)(23 28 72 77)(24 78 73 29)(25 30 74 79)(31 36 58 53)(32 54 59 37)(33 38 60 55)(34 56 51 39)(35 40 52 57)(41 153 129 107)(42 108 130 154)(43 155 121 109)(44 110 122 156)(45 157 123 101)(46 102 124 158)(47 159 125 103)(48 104 126 160)(49 151 127 105)(50 106 128 152)(91 120 147 138)(92 139 148 111)(93 112 149 140)(94 131 150 113)(95 114 141 132)(96 133 142 115)(97 116 143 134)(98 135 144 117)(99 118 145 136)(100 137 146 119)
(1 18 57 24)(2 19 58 25)(3 20 59 26)(4 11 60 27)(5 12 51 28)(6 13 52 29)(7 14 53 30)(8 15 54 21)(9 16 55 22)(10 17 56 23)(31 74 64 82)(32 75 65 83)(33 76 66 84)(34 77 67 85)(35 78 68 86)(36 79 69 87)(37 80 70 88)(38 71 61 89)(39 72 62 90)(40 73 63 81)(41 148 158 116)(42 149 159 117)(43 150 160 118)(44 141 151 119)(45 142 152 120)(46 143 153 111)(47 144 154 112)(48 145 155 113)(49 146 156 114)(50 147 157 115)(91 101 133 128)(92 102 134 129)(93 103 135 130)(94 104 136 121)(95 105 137 122)(96 106 138 123)(97 107 139 124)(98 108 140 125)(99 109 131 126)(100 110 132 127)
(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)
(1 146 63 100)(2 145 64 99)(3 144 65 98)(4 143 66 97)(5 142 67 96)(6 141 68 95)(7 150 69 94)(8 149 70 93)(9 148 61 92)(10 147 62 91)(11 46 84 124)(12 45 85 123)(13 44 86 122)(14 43 87 121)(15 42 88 130)(16 41 89 129)(17 50 90 128)(18 49 81 127)(19 48 82 126)(20 47 83 125)(21 159 80 103)(22 158 71 102)(23 157 72 101)(24 156 73 110)(25 155 74 109)(26 154 75 108)(27 153 76 107)(28 152 77 106)(29 151 78 105)(30 160 79 104)(31 131 58 113)(32 140 59 112)(33 139 60 111)(34 138 51 120)(35 137 52 119)(36 136 53 118)(37 135 54 117)(38 134 55 116)(39 133 56 115)(40 132 57 114)
G:=sub<Sym(160)| (1,68,63,6)(2,7,64,69)(3,70,65,8)(4,9,66,61)(5,62,67,10)(11,16,84,89)(12,90,85,17)(13,18,86,81)(14,82,87,19)(15,20,88,83)(21,26,80,75)(22,76,71,27)(23,28,72,77)(24,78,73,29)(25,30,74,79)(31,36,58,53)(32,54,59,37)(33,38,60,55)(34,56,51,39)(35,40,52,57)(41,153,129,107)(42,108,130,154)(43,155,121,109)(44,110,122,156)(45,157,123,101)(46,102,124,158)(47,159,125,103)(48,104,126,160)(49,151,127,105)(50,106,128,152)(91,120,147,138)(92,139,148,111)(93,112,149,140)(94,131,150,113)(95,114,141,132)(96,133,142,115)(97,116,143,134)(98,135,144,117)(99,118,145,136)(100,137,146,119), (1,18,57,24)(2,19,58,25)(3,20,59,26)(4,11,60,27)(5,12,51,28)(6,13,52,29)(7,14,53,30)(8,15,54,21)(9,16,55,22)(10,17,56,23)(31,74,64,82)(32,75,65,83)(33,76,66,84)(34,77,67,85)(35,78,68,86)(36,79,69,87)(37,80,70,88)(38,71,61,89)(39,72,62,90)(40,73,63,81)(41,148,158,116)(42,149,159,117)(43,150,160,118)(44,141,151,119)(45,142,152,120)(46,143,153,111)(47,144,154,112)(48,145,155,113)(49,146,156,114)(50,147,157,115)(91,101,133,128)(92,102,134,129)(93,103,135,130)(94,104,136,121)(95,105,137,122)(96,106,138,123)(97,107,139,124)(98,108,140,125)(99,109,131,126)(100,110,132,127), (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), (1,146,63,100)(2,145,64,99)(3,144,65,98)(4,143,66,97)(5,142,67,96)(6,141,68,95)(7,150,69,94)(8,149,70,93)(9,148,61,92)(10,147,62,91)(11,46,84,124)(12,45,85,123)(13,44,86,122)(14,43,87,121)(15,42,88,130)(16,41,89,129)(17,50,90,128)(18,49,81,127)(19,48,82,126)(20,47,83,125)(21,159,80,103)(22,158,71,102)(23,157,72,101)(24,156,73,110)(25,155,74,109)(26,154,75,108)(27,153,76,107)(28,152,77,106)(29,151,78,105)(30,160,79,104)(31,131,58,113)(32,140,59,112)(33,139,60,111)(34,138,51,120)(35,137,52,119)(36,136,53,118)(37,135,54,117)(38,134,55,116)(39,133,56,115)(40,132,57,114)>;
G:=Group( (1,68,63,6)(2,7,64,69)(3,70,65,8)(4,9,66,61)(5,62,67,10)(11,16,84,89)(12,90,85,17)(13,18,86,81)(14,82,87,19)(15,20,88,83)(21,26,80,75)(22,76,71,27)(23,28,72,77)(24,78,73,29)(25,30,74,79)(31,36,58,53)(32,54,59,37)(33,38,60,55)(34,56,51,39)(35,40,52,57)(41,153,129,107)(42,108,130,154)(43,155,121,109)(44,110,122,156)(45,157,123,101)(46,102,124,158)(47,159,125,103)(48,104,126,160)(49,151,127,105)(50,106,128,152)(91,120,147,138)(92,139,148,111)(93,112,149,140)(94,131,150,113)(95,114,141,132)(96,133,142,115)(97,116,143,134)(98,135,144,117)(99,118,145,136)(100,137,146,119), (1,18,57,24)(2,19,58,25)(3,20,59,26)(4,11,60,27)(5,12,51,28)(6,13,52,29)(7,14,53,30)(8,15,54,21)(9,16,55,22)(10,17,56,23)(31,74,64,82)(32,75,65,83)(33,76,66,84)(34,77,67,85)(35,78,68,86)(36,79,69,87)(37,80,70,88)(38,71,61,89)(39,72,62,90)(40,73,63,81)(41,148,158,116)(42,149,159,117)(43,150,160,118)(44,141,151,119)(45,142,152,120)(46,143,153,111)(47,144,154,112)(48,145,155,113)(49,146,156,114)(50,147,157,115)(91,101,133,128)(92,102,134,129)(93,103,135,130)(94,104,136,121)(95,105,137,122)(96,106,138,123)(97,107,139,124)(98,108,140,125)(99,109,131,126)(100,110,132,127), (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), (1,146,63,100)(2,145,64,99)(3,144,65,98)(4,143,66,97)(5,142,67,96)(6,141,68,95)(7,150,69,94)(8,149,70,93)(9,148,61,92)(10,147,62,91)(11,46,84,124)(12,45,85,123)(13,44,86,122)(14,43,87,121)(15,42,88,130)(16,41,89,129)(17,50,90,128)(18,49,81,127)(19,48,82,126)(20,47,83,125)(21,159,80,103)(22,158,71,102)(23,157,72,101)(24,156,73,110)(25,155,74,109)(26,154,75,108)(27,153,76,107)(28,152,77,106)(29,151,78,105)(30,160,79,104)(31,131,58,113)(32,140,59,112)(33,139,60,111)(34,138,51,120)(35,137,52,119)(36,136,53,118)(37,135,54,117)(38,134,55,116)(39,133,56,115)(40,132,57,114) );
G=PermutationGroup([(1,68,63,6),(2,7,64,69),(3,70,65,8),(4,9,66,61),(5,62,67,10),(11,16,84,89),(12,90,85,17),(13,18,86,81),(14,82,87,19),(15,20,88,83),(21,26,80,75),(22,76,71,27),(23,28,72,77),(24,78,73,29),(25,30,74,79),(31,36,58,53),(32,54,59,37),(33,38,60,55),(34,56,51,39),(35,40,52,57),(41,153,129,107),(42,108,130,154),(43,155,121,109),(44,110,122,156),(45,157,123,101),(46,102,124,158),(47,159,125,103),(48,104,126,160),(49,151,127,105),(50,106,128,152),(91,120,147,138),(92,139,148,111),(93,112,149,140),(94,131,150,113),(95,114,141,132),(96,133,142,115),(97,116,143,134),(98,135,144,117),(99,118,145,136),(100,137,146,119)], [(1,18,57,24),(2,19,58,25),(3,20,59,26),(4,11,60,27),(5,12,51,28),(6,13,52,29),(7,14,53,30),(8,15,54,21),(9,16,55,22),(10,17,56,23),(31,74,64,82),(32,75,65,83),(33,76,66,84),(34,77,67,85),(35,78,68,86),(36,79,69,87),(37,80,70,88),(38,71,61,89),(39,72,62,90),(40,73,63,81),(41,148,158,116),(42,149,159,117),(43,150,160,118),(44,141,151,119),(45,142,152,120),(46,143,153,111),(47,144,154,112),(48,145,155,113),(49,146,156,114),(50,147,157,115),(91,101,133,128),(92,102,134,129),(93,103,135,130),(94,104,136,121),(95,105,137,122),(96,106,138,123),(97,107,139,124),(98,108,140,125),(99,109,131,126),(100,110,132,127)], [(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)], [(1,146,63,100),(2,145,64,99),(3,144,65,98),(4,143,66,97),(5,142,67,96),(6,141,68,95),(7,150,69,94),(8,149,70,93),(9,148,61,92),(10,147,62,91),(11,46,84,124),(12,45,85,123),(13,44,86,122),(14,43,87,121),(15,42,88,130),(16,41,89,129),(17,50,90,128),(18,49,81,127),(19,48,82,126),(20,47,83,125),(21,159,80,103),(22,158,71,102),(23,157,72,101),(24,156,73,110),(25,155,74,109),(26,154,75,108),(27,153,76,107),(28,152,77,106),(29,151,78,105),(30,160,79,104),(31,131,58,113),(32,140,59,112),(33,139,60,111),(34,138,51,120),(35,137,52,119),(36,136,53,118),(37,135,54,117),(38,134,55,116),(39,133,56,115),(40,132,57,114)])
Matrix representation ►G ⊆ GL4(𝔽41) generated by
| 40 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 4 | 12 |
| 0 | 0 | 2 | 37 |
| 32 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 |
| 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 40 |
| 25 | 0 | 0 | 0 |
| 0 | 23 | 0 | 0 |
| 0 | 0 | 37 | 28 |
| 0 | 0 | 39 | 4 |
| 0 | 23 | 0 | 0 |
| 25 | 0 | 0 | 0 |
| 0 | 0 | 5 | 6 |
| 0 | 0 | 23 | 36 |
G:=sub<GL(4,GF(41))| [40,0,0,0,0,1,0,0,0,0,4,2,0,0,12,37],[32,0,0,0,0,9,0,0,0,0,40,0,0,0,0,40],[25,0,0,0,0,23,0,0,0,0,37,39,0,0,28,4],[0,25,0,0,23,0,0,0,0,0,5,23,0,0,6,36] >;
65 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 4A | ··· | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 4M | ··· | 4R | 5A | 5B | 10A | ··· | 10F | 10G | ··· | 10N | 20A | ··· | 20H | 20I | ··· | 20X |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 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 | 2 | 2 | 4 | 2 | ··· | 2 | 4 | 4 | 10 | 10 | 10 | 10 | 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 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | - | - | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D5 | C4○D4 | C4○D4 | D10 | D10 | D10 | D10 | D10 | C4○D20 | 2- (1+4) | D4⋊2D5 | D4.10D10 |
| kernel | C42.105D10 | C4×Dic10 | C20.6Q8 | C23.11D10 | C23.D10 | Dic5.Q8 | C20.48D4 | C2×C4⋊Dic5 | C23.21D10 | C23.18D10 | D4×C20 | C4×D4 | C20 | C2×C10 | C42 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×D4 | C22 | C10 | C4 | C2 |
| # reps | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 1 | 1 | 2 | 1 | 2 | 4 | 4 | 2 | 4 | 2 | 4 | 2 | 16 | 1 | 4 | 4 |
In GAP, Magma, Sage, TeX
C_4^2._{105}D_{10} % in TeX
G:=Group("C4^2.105D10"); // GroupNames label
G:=SmallGroup(320,1213);
// by ID
G=gap.SmallGroup(320,1213);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,224,387,100,675,570,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^10=1,d^2=a^2,a*b=b*a,c*a*c^-1=a^-1,d*a*d^-1=a^-1*b^2,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=c^-1>;
// generators/relations
×
⋊
ℤ
𝔽
○
≀