metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C4⋊C4.7F5, C20⋊C8⋊13C2, C4⋊Dic5.6C4, C2.6(Q8.F5), D10⋊C8.3C2, C10.10(C8○D4), C2.11(D4.F5), Dic5⋊C8⋊10C2, D10⋊C4.10C4, C10.C42⋊14C2, Dic5.28(C4○D4), C22.78(C22×F5), C5⋊2(C42.7C22), C10.11(C42⋊C2), (C2×Dic5).333C23, (C4×Dic5).244C22, C2.14(D10.C23), (C4×C5⋊C8)⋊15C2, (C5×C4⋊C4).10C4, (C2×C4).62(C2×F5), (C2×C20).84(C2×C4), (C2×C5⋊C8).30C22, (C2×C4×D5).59C22, C4⋊C4⋊7D5.18C2, (C2×C10).44(C22×C4), (C2×Dic5).59(C2×C4), (C22×D5).50(C2×C4), SmallGroup(320,1044)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 330 in 96 conjugacy classes, 42 normal (all characteristic)
C1, C2 [×3], C2, C4 [×7], C22, C22 [×3], C5, C8 [×4], C2×C4 [×3], C2×C4 [×5], C23, D5, C10 [×3], C42 [×2], C22⋊C4 [×2], C4⋊C4, C4⋊C4, C2×C8 [×4], C22×C4, Dic5 [×2], Dic5 [×2], C20 [×3], D10 [×3], C2×C10, C4×C8, C8⋊C4, C22⋊C8 [×2], C4⋊C8 [×2], C42⋊C2, C5⋊C8 [×4], C4×D5 [×2], C2×Dic5 [×3], C2×C20 [×3], C22×D5, C42.7C22, C4×Dic5 [×2], C4⋊Dic5, D10⋊C4 [×2], C5×C4⋊C4, C2×C5⋊C8 [×4], C2×C4×D5, C4×C5⋊C8, C20⋊C8, C10.C42, D10⋊C8 [×2], Dic5⋊C8, C4⋊C4⋊7D5, C4⋊C4.7F5
Quotients:
C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], C23, C22×C4, C4○D4 [×2], F5, C42⋊C2, C8○D4 [×2], C2×F5 [×3], C42.7C22, C22×F5, D10.C23, D4.F5, Q8.F5, C4⋊C4.7F5
Generators and relations
G = < a,b,c,d | a4=b4=c5=1, d4=a2, bab-1=a-1, ac=ca, dad-1=ab2, bc=cb, bd=db, dcd-1=c3 >
►On 160 points
Generators in S160
(1 143 5 139)(2 8 6 4)(3 137 7 141)(9 11 13 15)(10 18 14 22)(12 20 16 24)(17 19 21 23)(25 35 29 39)(26 97 30 101)(27 37 31 33)(28 99 32 103)(34 146 38 150)(36 148 40 152)(41 136 45 132)(42 70 46 66)(43 130 47 134)(44 72 48 68)(49 129 53 133)(50 71 54 67)(51 131 55 135)(52 65 56 69)(57 82 61 86)(58 128 62 124)(59 84 63 88)(60 122 64 126)(73 117 77 113)(74 156 78 160)(75 119 79 115)(76 158 80 154)(81 112 85 108)(83 106 87 110)(89 116 93 120)(90 155 94 159)(91 118 95 114)(92 157 96 153)(98 149 102 145)(100 151 104 147)(105 123 109 127)(107 125 111 121)(138 144 142 140)
(1 18 137 12)(2 19 138 13)(3 20 139 14)(4 21 140 15)(5 22 141 16)(6 23 142 9)(7 24 143 10)(8 17 144 11)(25 54 151 47)(26 55 152 48)(27 56 145 41)(28 49 146 42)(29 50 147 43)(30 51 148 44)(31 52 149 45)(32 53 150 46)(33 69 102 136)(34 70 103 129)(35 71 104 130)(36 72 97 131)(37 65 98 132)(38 66 99 133)(39 67 100 134)(40 68 101 135)(57 117 109 155)(58 118 110 156)(59 119 111 157)(60 120 112 158)(61 113 105 159)(62 114 106 160)(63 115 107 153)(64 116 108 154)(73 127 90 82)(74 128 91 83)(75 121 92 84)(76 122 93 85)(77 123 94 86)(78 124 95 87)(79 125 96 88)(80 126 89 81)
(1 92 132 50 113)(2 51 93 114 133)(3 115 52 134 94)(4 135 116 95 53)(5 96 136 54 117)(6 55 89 118 129)(7 119 56 130 90)(8 131 120 91 49)(9 26 126 58 103)(10 59 27 104 127)(11 97 60 128 28)(12 121 98 29 61)(13 30 122 62 99)(14 63 31 100 123)(15 101 64 124 32)(16 125 102 25 57)(17 36 112 83 146)(18 84 37 147 105)(19 148 85 106 38)(20 107 149 39 86)(21 40 108 87 150)(22 88 33 151 109)(23 152 81 110 34)(24 111 145 35 82)(41 71 73 143 157)(42 144 72 158 74)(43 159 137 75 65)(44 76 160 66 138)(45 67 77 139 153)(46 140 68 154 78)(47 155 141 79 69)(48 80 156 70 142)
(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)
G:=sub<Sym(160)| (1,143,5,139)(2,8,6,4)(3,137,7,141)(9,11,13,15)(10,18,14,22)(12,20,16,24)(17,19,21,23)(25,35,29,39)(26,97,30,101)(27,37,31,33)(28,99,32,103)(34,146,38,150)(36,148,40,152)(41,136,45,132)(42,70,46,66)(43,130,47,134)(44,72,48,68)(49,129,53,133)(50,71,54,67)(51,131,55,135)(52,65,56,69)(57,82,61,86)(58,128,62,124)(59,84,63,88)(60,122,64,126)(73,117,77,113)(74,156,78,160)(75,119,79,115)(76,158,80,154)(81,112,85,108)(83,106,87,110)(89,116,93,120)(90,155,94,159)(91,118,95,114)(92,157,96,153)(98,149,102,145)(100,151,104,147)(105,123,109,127)(107,125,111,121)(138,144,142,140), (1,18,137,12)(2,19,138,13)(3,20,139,14)(4,21,140,15)(5,22,141,16)(6,23,142,9)(7,24,143,10)(8,17,144,11)(25,54,151,47)(26,55,152,48)(27,56,145,41)(28,49,146,42)(29,50,147,43)(30,51,148,44)(31,52,149,45)(32,53,150,46)(33,69,102,136)(34,70,103,129)(35,71,104,130)(36,72,97,131)(37,65,98,132)(38,66,99,133)(39,67,100,134)(40,68,101,135)(57,117,109,155)(58,118,110,156)(59,119,111,157)(60,120,112,158)(61,113,105,159)(62,114,106,160)(63,115,107,153)(64,116,108,154)(73,127,90,82)(74,128,91,83)(75,121,92,84)(76,122,93,85)(77,123,94,86)(78,124,95,87)(79,125,96,88)(80,126,89,81), (1,92,132,50,113)(2,51,93,114,133)(3,115,52,134,94)(4,135,116,95,53)(5,96,136,54,117)(6,55,89,118,129)(7,119,56,130,90)(8,131,120,91,49)(9,26,126,58,103)(10,59,27,104,127)(11,97,60,128,28)(12,121,98,29,61)(13,30,122,62,99)(14,63,31,100,123)(15,101,64,124,32)(16,125,102,25,57)(17,36,112,83,146)(18,84,37,147,105)(19,148,85,106,38)(20,107,149,39,86)(21,40,108,87,150)(22,88,33,151,109)(23,152,81,110,34)(24,111,145,35,82)(41,71,73,143,157)(42,144,72,158,74)(43,159,137,75,65)(44,76,160,66,138)(45,67,77,139,153)(46,140,68,154,78)(47,155,141,79,69)(48,80,156,70,142), (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)>;
G:=Group( (1,143,5,139)(2,8,6,4)(3,137,7,141)(9,11,13,15)(10,18,14,22)(12,20,16,24)(17,19,21,23)(25,35,29,39)(26,97,30,101)(27,37,31,33)(28,99,32,103)(34,146,38,150)(36,148,40,152)(41,136,45,132)(42,70,46,66)(43,130,47,134)(44,72,48,68)(49,129,53,133)(50,71,54,67)(51,131,55,135)(52,65,56,69)(57,82,61,86)(58,128,62,124)(59,84,63,88)(60,122,64,126)(73,117,77,113)(74,156,78,160)(75,119,79,115)(76,158,80,154)(81,112,85,108)(83,106,87,110)(89,116,93,120)(90,155,94,159)(91,118,95,114)(92,157,96,153)(98,149,102,145)(100,151,104,147)(105,123,109,127)(107,125,111,121)(138,144,142,140), (1,18,137,12)(2,19,138,13)(3,20,139,14)(4,21,140,15)(5,22,141,16)(6,23,142,9)(7,24,143,10)(8,17,144,11)(25,54,151,47)(26,55,152,48)(27,56,145,41)(28,49,146,42)(29,50,147,43)(30,51,148,44)(31,52,149,45)(32,53,150,46)(33,69,102,136)(34,70,103,129)(35,71,104,130)(36,72,97,131)(37,65,98,132)(38,66,99,133)(39,67,100,134)(40,68,101,135)(57,117,109,155)(58,118,110,156)(59,119,111,157)(60,120,112,158)(61,113,105,159)(62,114,106,160)(63,115,107,153)(64,116,108,154)(73,127,90,82)(74,128,91,83)(75,121,92,84)(76,122,93,85)(77,123,94,86)(78,124,95,87)(79,125,96,88)(80,126,89,81), (1,92,132,50,113)(2,51,93,114,133)(3,115,52,134,94)(4,135,116,95,53)(5,96,136,54,117)(6,55,89,118,129)(7,119,56,130,90)(8,131,120,91,49)(9,26,126,58,103)(10,59,27,104,127)(11,97,60,128,28)(12,121,98,29,61)(13,30,122,62,99)(14,63,31,100,123)(15,101,64,124,32)(16,125,102,25,57)(17,36,112,83,146)(18,84,37,147,105)(19,148,85,106,38)(20,107,149,39,86)(21,40,108,87,150)(22,88,33,151,109)(23,152,81,110,34)(24,111,145,35,82)(41,71,73,143,157)(42,144,72,158,74)(43,159,137,75,65)(44,76,160,66,138)(45,67,77,139,153)(46,140,68,154,78)(47,155,141,79,69)(48,80,156,70,142), (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) );
G=PermutationGroup([(1,143,5,139),(2,8,6,4),(3,137,7,141),(9,11,13,15),(10,18,14,22),(12,20,16,24),(17,19,21,23),(25,35,29,39),(26,97,30,101),(27,37,31,33),(28,99,32,103),(34,146,38,150),(36,148,40,152),(41,136,45,132),(42,70,46,66),(43,130,47,134),(44,72,48,68),(49,129,53,133),(50,71,54,67),(51,131,55,135),(52,65,56,69),(57,82,61,86),(58,128,62,124),(59,84,63,88),(60,122,64,126),(73,117,77,113),(74,156,78,160),(75,119,79,115),(76,158,80,154),(81,112,85,108),(83,106,87,110),(89,116,93,120),(90,155,94,159),(91,118,95,114),(92,157,96,153),(98,149,102,145),(100,151,104,147),(105,123,109,127),(107,125,111,121),(138,144,142,140)], [(1,18,137,12),(2,19,138,13),(3,20,139,14),(4,21,140,15),(5,22,141,16),(6,23,142,9),(7,24,143,10),(8,17,144,11),(25,54,151,47),(26,55,152,48),(27,56,145,41),(28,49,146,42),(29,50,147,43),(30,51,148,44),(31,52,149,45),(32,53,150,46),(33,69,102,136),(34,70,103,129),(35,71,104,130),(36,72,97,131),(37,65,98,132),(38,66,99,133),(39,67,100,134),(40,68,101,135),(57,117,109,155),(58,118,110,156),(59,119,111,157),(60,120,112,158),(61,113,105,159),(62,114,106,160),(63,115,107,153),(64,116,108,154),(73,127,90,82),(74,128,91,83),(75,121,92,84),(76,122,93,85),(77,123,94,86),(78,124,95,87),(79,125,96,88),(80,126,89,81)], [(1,92,132,50,113),(2,51,93,114,133),(3,115,52,134,94),(4,135,116,95,53),(5,96,136,54,117),(6,55,89,118,129),(7,119,56,130,90),(8,131,120,91,49),(9,26,126,58,103),(10,59,27,104,127),(11,97,60,128,28),(12,121,98,29,61),(13,30,122,62,99),(14,63,31,100,123),(15,101,64,124,32),(16,125,102,25,57),(17,36,112,83,146),(18,84,37,147,105),(19,148,85,106,38),(20,107,149,39,86),(21,40,108,87,150),(22,88,33,151,109),(23,152,81,110,34),(24,111,145,35,82),(41,71,73,143,157),(42,144,72,158,74),(43,159,137,75,65),(44,76,160,66,138),(45,67,77,139,153),(46,140,68,154,78),(47,155,141,79,69),(48,80,156,70,142)], [(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)])
Matrix representation ►G ⊆ GL6(𝔽41)
| 9 | 0 | 0 | 0 | 0 | 0 |
| 0 | 32 | 0 | 0 | 0 | 0 |
| 0 | 0 | 19 | 3 | 0 | 38 |
| 0 | 0 | 0 | 22 | 3 | 38 |
| 0 | 0 | 38 | 3 | 22 | 0 |
| 0 | 0 | 38 | 0 | 3 | 19 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 32 | 0 | 0 | 0 |
| 0 | 0 | 0 | 32 | 0 | 0 |
| 0 | 0 | 0 | 0 | 32 | 0 |
| 0 | 0 | 0 | 0 | 0 | 32 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 |
| 0 | 0 | 1 | 0 | 0 | 40 |
| 0 | 0 | 0 | 1 | 0 | 40 |
| 0 | 0 | 0 | 0 | 1 | 40 |
| 27 | 0 | 0 | 0 | 0 | 0 |
| 0 | 27 | 0 | 0 | 0 | 0 |
| 0 | 0 | 6 | 6 | 24 | 22 |
| 0 | 0 | 30 | 28 | 7 | 28 |
| 0 | 0 | 13 | 34 | 13 | 11 |
| 0 | 0 | 19 | 17 | 35 | 35 |
G:=sub<GL(6,GF(41))| [9,0,0,0,0,0,0,32,0,0,0,0,0,0,19,0,38,38,0,0,3,22,3,0,0,0,0,3,22,3,0,0,38,38,0,19],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,32,0,0,0,0,0,0,32,0,0,0,0,0,0,32,0,0,0,0,0,0,32],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,40,40,40,40],[27,0,0,0,0,0,0,27,0,0,0,0,0,0,6,30,13,19,0,0,6,28,34,17,0,0,24,7,13,35,0,0,22,28,11,35] >;
38 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 5 | 8A | ··· | 8H | 8I | 8J | 8K | 8L | 10A | 10B | 10C | 20A | ··· | 20F |
| order | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 8 | ··· | 8 | 8 | 8 | 8 | 8 | 10 | 10 | 10 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 20 | 2 | 2 | 4 | 4 | 5 | 5 | 5 | 5 | 10 | 10 | 20 | 4 | 10 | ··· | 10 | 20 | 20 | 20 | 20 | 4 | 4 | 4 | 8 | ··· | 8 |
38 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 4 | 8 | 8 |
| type | + | + | + | + | + | + | + | + | + | - | + | ||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | C4○D4 | C8○D4 | F5 | C2×F5 | D10.C23 | D4.F5 | Q8.F5 |
| kernel | C4⋊C4.7F5 | C4×C5⋊C8 | C20⋊C8 | C10.C42 | D10⋊C8 | Dic5⋊C8 | C4⋊C4⋊7D5 | C4⋊Dic5 | D10⋊C4 | C5×C4⋊C4 | Dic5 | C10 | C4⋊C4 | C2×C4 | C2 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 2 | 4 | 2 | 4 | 8 | 1 | 3 | 4 | 1 | 1 |
In GAP, Magma, Sage, TeX
C_4\rtimes C_4._7F_5
% in TeX
G:=Group("C4:C4.7F5"); // GroupNames label
G:=SmallGroup(320,1044);
// by ID
G=gap.SmallGroup(320,1044);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,120,422,387,184,136,6278,1595]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^5=1,d^4=a^2,b*a*b^-1=a^-1,a*c=c*a,d*a*d^-1=a*b^2,b*c=c*b,b*d=d*b,d*c*d^-1=c^3>;
// generators/relations
×
⋊
ℤ
𝔽
○
≀