metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C4⋊C4.9F5, C20.5(C4⋊C4), (C4×D5).7Q8, C20⋊C8⋊14C2, (C4×D5).28D4, C4.20(C4⋊F5), D10.9(C4⋊C4), C4⋊Dic5.12C4, C2.7(Q8.F5), D10⋊C4.4C4, C10.12(C8○D4), Dic5.12(C2×Q8), Dic5.30(C2×D4), C2.13(D4.F5), Dic5⋊C8⋊11C2, C22.80(C22×F5), C5⋊2(C42.6C22), (C2×Dic5).335C23, (C4×Dic5).245C22, C10.9(C2×C4⋊C4), C2.12(C2×C4⋊F5), (C5×C4⋊C4).12C4, (C2×D5⋊C8).4C2, (C2×C5⋊C8).6C22, (C2×C4).28(C2×F5), (C2×C4.F5).4C2, (C2×C20).86(C2×C4), C4⋊C4⋊7D5.19C2, (C2×C4×D5).191C22, (C2×C10).46(C22×C4), (C2×Dic5).61(C2×C4), (C22×D5).51(C2×C4), SmallGroup(320,1046)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 378 in 114 conjugacy classes, 52 normal (26 characteristic)
C1, C2 [×3], C2 [×2], C4 [×2], C4 [×6], C22, C22 [×4], C5, C8 [×4], C2×C4, C2×C4 [×2], C2×C4 [×7], C23, D5 [×2], C10 [×3], C42 [×2], C22⋊C4 [×2], C4⋊C4, C4⋊C4, C2×C8 [×6], M4(2) [×2], C22×C4, Dic5 [×2], Dic5 [×2], C20 [×2], C20 [×2], D10 [×2], D10 [×2], C2×C10, C4⋊C8 [×4], C42⋊C2, C22×C8, C2×M4(2), C5⋊C8 [×4], C4×D5 [×4], C2×Dic5, C2×Dic5 [×2], C2×C20, C2×C20 [×2], C22×D5, C42.6C22, C4×Dic5 [×2], C4⋊Dic5, D10⋊C4 [×2], C5×C4⋊C4, D5⋊C8 [×2], C4.F5 [×2], C2×C5⋊C8 [×2], C2×C5⋊C8 [×2], C2×C4×D5, C20⋊C8 [×2], Dic5⋊C8 [×2], C4⋊C4⋊7D5, C2×D5⋊C8, C2×C4.F5, C4⋊C4.9F5
Quotients:
C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×2], Q8 [×2], C23, C4⋊C4 [×4], C22×C4, C2×D4, C2×Q8, F5, C2×C4⋊C4, C8○D4 [×2], C2×F5 [×3], C42.6C22, C4⋊F5 [×2], C22×F5, C2×C4⋊F5, D4.F5, Q8.F5, C4⋊C4.9F5
Generators and relations
G = < a,b,c,d | a4=b4=c5=1, d4=a2, bab-1=a-1, ac=ca, ad=da, bc=cb, dbd-1=b-1, dcd-1=c3 >
►On 160 points
Generators in S160
(1 143 5 139)(2 144 6 140)(3 137 7 141)(4 138 8 142)(9 63 13 59)(10 64 14 60)(11 57 15 61)(12 58 16 62)(17 37 21 33)(18 38 22 34)(19 39 23 35)(20 40 24 36)(25 42 29 46)(26 43 30 47)(27 44 31 48)(28 45 32 41)(49 84 53 88)(50 85 54 81)(51 86 55 82)(52 87 56 83)(65 109 69 105)(66 110 70 106)(67 111 71 107)(68 112 72 108)(73 117 77 113)(74 118 78 114)(75 119 79 115)(76 120 80 116)(89 160 93 156)(90 153 94 157)(91 154 95 158)(92 155 96 159)(97 148 101 152)(98 149 102 145)(99 150 103 146)(100 151 104 147)(121 136 125 132)(122 129 126 133)(123 130 127 134)(124 131 128 135)
(1 18 137 36)(2 37 138 19)(3 20 139 38)(4 39 140 21)(5 22 141 40)(6 33 142 23)(7 24 143 34)(8 35 144 17)(9 100 134 46)(10 47 135 101)(11 102 136 48)(12 41 129 103)(13 104 130 42)(14 43 131 97)(15 98 132 44)(16 45 133 99)(25 59 151 127)(26 128 152 60)(27 61 145 121)(28 122 146 62)(29 63 147 123)(30 124 148 64)(31 57 149 125)(32 126 150 58)(49 160 106 114)(50 115 107 153)(51 154 108 116)(52 117 109 155)(53 156 110 118)(54 119 111 157)(55 158 112 120)(56 113 105 159)(65 96 83 77)(66 78 84 89)(67 90 85 79)(68 80 86 91)(69 92 87 73)(70 74 88 93)(71 94 81 75)(72 76 82 95)
(1 94 132 63 113)(2 64 95 114 133)(3 115 57 134 96)(4 135 116 89 58)(5 90 136 59 117)(6 60 91 118 129)(7 119 61 130 92)(8 131 120 93 62)(9 77 139 153 125)(10 154 78 126 140)(11 127 155 141 79)(12 142 128 80 156)(13 73 143 157 121)(14 158 74 122 144)(15 123 159 137 75)(16 138 124 76 160)(17 43 112 88 146)(18 81 44 147 105)(19 148 82 106 45)(20 107 149 46 83)(21 47 108 84 150)(22 85 48 151 109)(23 152 86 110 41)(24 111 145 42 87)(25 52 40 67 102)(26 68 53 103 33)(27 104 69 34 54)(28 35 97 55 70)(29 56 36 71 98)(30 72 49 99 37)(31 100 65 38 50)(32 39 101 51 66)
(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)(2,144,6,140)(3,137,7,141)(4,138,8,142)(9,63,13,59)(10,64,14,60)(11,57,15,61)(12,58,16,62)(17,37,21,33)(18,38,22,34)(19,39,23,35)(20,40,24,36)(25,42,29,46)(26,43,30,47)(27,44,31,48)(28,45,32,41)(49,84,53,88)(50,85,54,81)(51,86,55,82)(52,87,56,83)(65,109,69,105)(66,110,70,106)(67,111,71,107)(68,112,72,108)(73,117,77,113)(74,118,78,114)(75,119,79,115)(76,120,80,116)(89,160,93,156)(90,153,94,157)(91,154,95,158)(92,155,96,159)(97,148,101,152)(98,149,102,145)(99,150,103,146)(100,151,104,147)(121,136,125,132)(122,129,126,133)(123,130,127,134)(124,131,128,135), (1,18,137,36)(2,37,138,19)(3,20,139,38)(4,39,140,21)(5,22,141,40)(6,33,142,23)(7,24,143,34)(8,35,144,17)(9,100,134,46)(10,47,135,101)(11,102,136,48)(12,41,129,103)(13,104,130,42)(14,43,131,97)(15,98,132,44)(16,45,133,99)(25,59,151,127)(26,128,152,60)(27,61,145,121)(28,122,146,62)(29,63,147,123)(30,124,148,64)(31,57,149,125)(32,126,150,58)(49,160,106,114)(50,115,107,153)(51,154,108,116)(52,117,109,155)(53,156,110,118)(54,119,111,157)(55,158,112,120)(56,113,105,159)(65,96,83,77)(66,78,84,89)(67,90,85,79)(68,80,86,91)(69,92,87,73)(70,74,88,93)(71,94,81,75)(72,76,82,95), (1,94,132,63,113)(2,64,95,114,133)(3,115,57,134,96)(4,135,116,89,58)(5,90,136,59,117)(6,60,91,118,129)(7,119,61,130,92)(8,131,120,93,62)(9,77,139,153,125)(10,154,78,126,140)(11,127,155,141,79)(12,142,128,80,156)(13,73,143,157,121)(14,158,74,122,144)(15,123,159,137,75)(16,138,124,76,160)(17,43,112,88,146)(18,81,44,147,105)(19,148,82,106,45)(20,107,149,46,83)(21,47,108,84,150)(22,85,48,151,109)(23,152,86,110,41)(24,111,145,42,87)(25,52,40,67,102)(26,68,53,103,33)(27,104,69,34,54)(28,35,97,55,70)(29,56,36,71,98)(30,72,49,99,37)(31,100,65,38,50)(32,39,101,51,66), (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)(2,144,6,140)(3,137,7,141)(4,138,8,142)(9,63,13,59)(10,64,14,60)(11,57,15,61)(12,58,16,62)(17,37,21,33)(18,38,22,34)(19,39,23,35)(20,40,24,36)(25,42,29,46)(26,43,30,47)(27,44,31,48)(28,45,32,41)(49,84,53,88)(50,85,54,81)(51,86,55,82)(52,87,56,83)(65,109,69,105)(66,110,70,106)(67,111,71,107)(68,112,72,108)(73,117,77,113)(74,118,78,114)(75,119,79,115)(76,120,80,116)(89,160,93,156)(90,153,94,157)(91,154,95,158)(92,155,96,159)(97,148,101,152)(98,149,102,145)(99,150,103,146)(100,151,104,147)(121,136,125,132)(122,129,126,133)(123,130,127,134)(124,131,128,135), (1,18,137,36)(2,37,138,19)(3,20,139,38)(4,39,140,21)(5,22,141,40)(6,33,142,23)(7,24,143,34)(8,35,144,17)(9,100,134,46)(10,47,135,101)(11,102,136,48)(12,41,129,103)(13,104,130,42)(14,43,131,97)(15,98,132,44)(16,45,133,99)(25,59,151,127)(26,128,152,60)(27,61,145,121)(28,122,146,62)(29,63,147,123)(30,124,148,64)(31,57,149,125)(32,126,150,58)(49,160,106,114)(50,115,107,153)(51,154,108,116)(52,117,109,155)(53,156,110,118)(54,119,111,157)(55,158,112,120)(56,113,105,159)(65,96,83,77)(66,78,84,89)(67,90,85,79)(68,80,86,91)(69,92,87,73)(70,74,88,93)(71,94,81,75)(72,76,82,95), (1,94,132,63,113)(2,64,95,114,133)(3,115,57,134,96)(4,135,116,89,58)(5,90,136,59,117)(6,60,91,118,129)(7,119,61,130,92)(8,131,120,93,62)(9,77,139,153,125)(10,154,78,126,140)(11,127,155,141,79)(12,142,128,80,156)(13,73,143,157,121)(14,158,74,122,144)(15,123,159,137,75)(16,138,124,76,160)(17,43,112,88,146)(18,81,44,147,105)(19,148,82,106,45)(20,107,149,46,83)(21,47,108,84,150)(22,85,48,151,109)(23,152,86,110,41)(24,111,145,42,87)(25,52,40,67,102)(26,68,53,103,33)(27,104,69,34,54)(28,35,97,55,70)(29,56,36,71,98)(30,72,49,99,37)(31,100,65,38,50)(32,39,101,51,66), (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),(2,144,6,140),(3,137,7,141),(4,138,8,142),(9,63,13,59),(10,64,14,60),(11,57,15,61),(12,58,16,62),(17,37,21,33),(18,38,22,34),(19,39,23,35),(20,40,24,36),(25,42,29,46),(26,43,30,47),(27,44,31,48),(28,45,32,41),(49,84,53,88),(50,85,54,81),(51,86,55,82),(52,87,56,83),(65,109,69,105),(66,110,70,106),(67,111,71,107),(68,112,72,108),(73,117,77,113),(74,118,78,114),(75,119,79,115),(76,120,80,116),(89,160,93,156),(90,153,94,157),(91,154,95,158),(92,155,96,159),(97,148,101,152),(98,149,102,145),(99,150,103,146),(100,151,104,147),(121,136,125,132),(122,129,126,133),(123,130,127,134),(124,131,128,135)], [(1,18,137,36),(2,37,138,19),(3,20,139,38),(4,39,140,21),(5,22,141,40),(6,33,142,23),(7,24,143,34),(8,35,144,17),(9,100,134,46),(10,47,135,101),(11,102,136,48),(12,41,129,103),(13,104,130,42),(14,43,131,97),(15,98,132,44),(16,45,133,99),(25,59,151,127),(26,128,152,60),(27,61,145,121),(28,122,146,62),(29,63,147,123),(30,124,148,64),(31,57,149,125),(32,126,150,58),(49,160,106,114),(50,115,107,153),(51,154,108,116),(52,117,109,155),(53,156,110,118),(54,119,111,157),(55,158,112,120),(56,113,105,159),(65,96,83,77),(66,78,84,89),(67,90,85,79),(68,80,86,91),(69,92,87,73),(70,74,88,93),(71,94,81,75),(72,76,82,95)], [(1,94,132,63,113),(2,64,95,114,133),(3,115,57,134,96),(4,135,116,89,58),(5,90,136,59,117),(6,60,91,118,129),(7,119,61,130,92),(8,131,120,93,62),(9,77,139,153,125),(10,154,78,126,140),(11,127,155,141,79),(12,142,128,80,156),(13,73,143,157,121),(14,158,74,122,144),(15,123,159,137,75),(16,138,124,76,160),(17,43,112,88,146),(18,81,44,147,105),(19,148,82,106,45),(20,107,149,46,83),(21,47,108,84,150),(22,85,48,151,109),(23,152,86,110,41),(24,111,145,42,87),(25,52,40,67,102),(26,68,53,103,33),(27,104,69,34,54),(28,35,97,55,70),(29,56,36,71,98),(30,72,49,99,37),(31,100,65,38,50),(32,39,101,51,66)], [(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)
| 32 | 0 | 0 | 0 | 0 | 0 |
| 38 | 9 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 6 | 5 | 0 | 0 | 0 | 0 |
| 9 | 35 | 0 | 0 | 0 | 0 |
| 0 | 0 | 34 | 14 | 0 | 27 |
| 0 | 0 | 0 | 7 | 14 | 27 |
| 0 | 0 | 27 | 14 | 7 | 0 |
| 0 | 0 | 27 | 0 | 14 | 34 |
| 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 |
| 14 | 0 | 0 | 0 | 0 | 0 |
| 32 | 27 | 0 | 0 | 0 | 0 |
| 0 | 0 | 34 | 7 | 27 | 0 |
| 0 | 0 | 20 | 7 | 0 | 34 |
| 0 | 0 | 34 | 0 | 7 | 20 |
| 0 | 0 | 0 | 27 | 7 | 34 |
G:=sub<GL(6,GF(41))| [32,38,0,0,0,0,0,9,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[6,9,0,0,0,0,5,35,0,0,0,0,0,0,34,0,27,27,0,0,14,7,14,0,0,0,0,14,7,14,0,0,27,27,0,34],[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],[14,32,0,0,0,0,0,27,0,0,0,0,0,0,34,20,34,0,0,0,7,7,0,27,0,0,27,0,7,7,0,0,0,34,20,34] >;
38 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 5 | 8A | ··· | 8H | 8I | 8J | 8K | 8L | 10A | 10B | 10C | 20A | ··· | 20F |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 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 | 10 | 10 | 2 | 2 | 4 | 4 | 5 | 5 | 5 | 5 | 20 | 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 | 2 | 2 | 2 | 4 | 4 | 4 | 8 | 8 |
| type | + | + | + | + | + | + | + | - | + | + | - | + | |||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | D4 | Q8 | C8○D4 | F5 | C2×F5 | C4⋊F5 | D4.F5 | Q8.F5 |
| kernel | C4⋊C4.9F5 | C20⋊C8 | Dic5⋊C8 | C4⋊C4⋊7D5 | C2×D5⋊C8 | C2×C4.F5 | C4⋊Dic5 | D10⋊C4 | C5×C4⋊C4 | C4×D5 | C4×D5 | C10 | C4⋊C4 | C2×C4 | C4 | C2 | C2 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 2 | 4 | 2 | 2 | 2 | 8 | 1 | 3 | 4 | 1 | 1 |
In GAP, Magma, Sage, TeX
C_4\rtimes C_4._9F_5
% in TeX
G:=Group("C4:C4.9F5"); // GroupNames label
G:=SmallGroup(320,1046);
// by ID
G=gap.SmallGroup(320,1046);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,232,422,387,100,136,6278,1595]);
// 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,a*d=d*a,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=c^3>;
// generators/relations
×
⋊
ℤ
𝔽
○
≀