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), (C4×Dic5).245C22, (C2×Dic5).335C23, 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
Generators and relations for C4⋊C4.9F5
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 >
Subgroups: 378 in 114 conjugacy classes, 52 normal (26 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, C2×C4, C23, D5, C10, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, M4(2), C22×C4, Dic5, Dic5, C20, C20, D10, D10, C2×C10, C4⋊C8, C42⋊C2, C22×C8, C2×M4(2), C5⋊C8, C4×D5, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C22×D5, C42.6C22, C4×Dic5, C4⋊Dic5, D10⋊C4, C5×C4⋊C4, D5⋊C8, C4.F5, C2×C5⋊C8, C2×C5⋊C8, C2×C4×D5, C20⋊C8, Dic5⋊C8, C4⋊C4⋊7D5, C2×D5⋊C8, C2×C4.F5, C4⋊C4.9F5
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C23, C4⋊C4, C22×C4, C2×D4, C2×Q8, F5, C2×C4⋊C4, C8○D4, C2×F5, C42.6C22, C4⋊F5, C22×F5, C2×C4⋊F5, D4.F5, Q8.F5, C4⋊C4.9F5
►On 160 points
Generators in S160
(1 130 5 134)(2 131 6 135)(3 132 7 136)(4 133 8 129)(9 154 13 158)(10 155 14 159)(11 156 15 160)(12 157 16 153)(17 96 21 92)(18 89 22 93)(19 90 23 94)(20 91 24 95)(25 149 29 145)(26 150 30 146)(27 151 31 147)(28 152 32 148)(33 81 37 85)(34 82 38 86)(35 83 39 87)(36 84 40 88)(41 61 45 57)(42 62 46 58)(43 63 47 59)(44 64 48 60)(49 99 53 103)(50 100 54 104)(51 101 55 97)(52 102 56 98)(65 141 69 137)(66 142 70 138)(67 143 71 139)(68 144 72 140)(73 117 77 113)(74 118 78 114)(75 119 79 115)(76 120 80 116)(105 123 109 127)(106 124 110 128)(107 125 111 121)(108 126 112 122)
(1 18 132 95)(2 96 133 19)(3 20 134 89)(4 90 135 21)(5 22 136 91)(6 92 129 23)(7 24 130 93)(8 94 131 17)(9 126 80 101)(10 102 73 127)(11 128 74 103)(12 104 75 121)(13 122 76 97)(14 98 77 123)(15 124 78 99)(16 100 79 125)(25 139 33 47)(26 48 34 140)(27 141 35 41)(28 42 36 142)(29 143 37 43)(30 44 38 144)(31 137 39 45)(32 46 40 138)(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)(57 151 65 83)(58 84 66 152)(59 145 67 85)(60 86 68 146)(61 147 69 87)(62 88 70 148)(63 149 71 81)(64 82 72 150)
(1 12 57 143 113)(2 144 13 114 58)(3 115 137 59 14)(4 60 116 15 138)(5 16 61 139 117)(6 140 9 118 62)(7 119 141 63 10)(8 64 120 11 142)(17 150 112 103 36)(18 104 151 37 105)(19 38 97 106 152)(20 107 39 145 98)(21 146 108 99 40)(22 100 147 33 109)(23 34 101 110 148)(24 111 35 149 102)(25 52 91 125 87)(26 126 53 88 92)(27 81 127 93 54)(28 94 82 55 128)(29 56 95 121 83)(30 122 49 84 96)(31 85 123 89 50)(32 90 86 51 124)(41 71 73 130 157)(42 131 72 158 74)(43 159 132 75 65)(44 76 160 66 133)(45 67 77 134 153)(46 135 68 154 78)(47 155 136 79 69)(48 80 156 70 129)
(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)| (1,130,5,134)(2,131,6,135)(3,132,7,136)(4,133,8,129)(9,154,13,158)(10,155,14,159)(11,156,15,160)(12,157,16,153)(17,96,21,92)(18,89,22,93)(19,90,23,94)(20,91,24,95)(25,149,29,145)(26,150,30,146)(27,151,31,147)(28,152,32,148)(33,81,37,85)(34,82,38,86)(35,83,39,87)(36,84,40,88)(41,61,45,57)(42,62,46,58)(43,63,47,59)(44,64,48,60)(49,99,53,103)(50,100,54,104)(51,101,55,97)(52,102,56,98)(65,141,69,137)(66,142,70,138)(67,143,71,139)(68,144,72,140)(73,117,77,113)(74,118,78,114)(75,119,79,115)(76,120,80,116)(105,123,109,127)(106,124,110,128)(107,125,111,121)(108,126,112,122), (1,18,132,95)(2,96,133,19)(3,20,134,89)(4,90,135,21)(5,22,136,91)(6,92,129,23)(7,24,130,93)(8,94,131,17)(9,126,80,101)(10,102,73,127)(11,128,74,103)(12,104,75,121)(13,122,76,97)(14,98,77,123)(15,124,78,99)(16,100,79,125)(25,139,33,47)(26,48,34,140)(27,141,35,41)(28,42,36,142)(29,143,37,43)(30,44,38,144)(31,137,39,45)(32,46,40,138)(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)(57,151,65,83)(58,84,66,152)(59,145,67,85)(60,86,68,146)(61,147,69,87)(62,88,70,148)(63,149,71,81)(64,82,72,150), (1,12,57,143,113)(2,144,13,114,58)(3,115,137,59,14)(4,60,116,15,138)(5,16,61,139,117)(6,140,9,118,62)(7,119,141,63,10)(8,64,120,11,142)(17,150,112,103,36)(18,104,151,37,105)(19,38,97,106,152)(20,107,39,145,98)(21,146,108,99,40)(22,100,147,33,109)(23,34,101,110,148)(24,111,35,149,102)(25,52,91,125,87)(26,126,53,88,92)(27,81,127,93,54)(28,94,82,55,128)(29,56,95,121,83)(30,122,49,84,96)(31,85,123,89,50)(32,90,86,51,124)(41,71,73,130,157)(42,131,72,158,74)(43,159,132,75,65)(44,76,160,66,133)(45,67,77,134,153)(46,135,68,154,78)(47,155,136,79,69)(48,80,156,70,129), (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( (1,130,5,134)(2,131,6,135)(3,132,7,136)(4,133,8,129)(9,154,13,158)(10,155,14,159)(11,156,15,160)(12,157,16,153)(17,96,21,92)(18,89,22,93)(19,90,23,94)(20,91,24,95)(25,149,29,145)(26,150,30,146)(27,151,31,147)(28,152,32,148)(33,81,37,85)(34,82,38,86)(35,83,39,87)(36,84,40,88)(41,61,45,57)(42,62,46,58)(43,63,47,59)(44,64,48,60)(49,99,53,103)(50,100,54,104)(51,101,55,97)(52,102,56,98)(65,141,69,137)(66,142,70,138)(67,143,71,139)(68,144,72,140)(73,117,77,113)(74,118,78,114)(75,119,79,115)(76,120,80,116)(105,123,109,127)(106,124,110,128)(107,125,111,121)(108,126,112,122), (1,18,132,95)(2,96,133,19)(3,20,134,89)(4,90,135,21)(5,22,136,91)(6,92,129,23)(7,24,130,93)(8,94,131,17)(9,126,80,101)(10,102,73,127)(11,128,74,103)(12,104,75,121)(13,122,76,97)(14,98,77,123)(15,124,78,99)(16,100,79,125)(25,139,33,47)(26,48,34,140)(27,141,35,41)(28,42,36,142)(29,143,37,43)(30,44,38,144)(31,137,39,45)(32,46,40,138)(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)(57,151,65,83)(58,84,66,152)(59,145,67,85)(60,86,68,146)(61,147,69,87)(62,88,70,148)(63,149,71,81)(64,82,72,150), (1,12,57,143,113)(2,144,13,114,58)(3,115,137,59,14)(4,60,116,15,138)(5,16,61,139,117)(6,140,9,118,62)(7,119,141,63,10)(8,64,120,11,142)(17,150,112,103,36)(18,104,151,37,105)(19,38,97,106,152)(20,107,39,145,98)(21,146,108,99,40)(22,100,147,33,109)(23,34,101,110,148)(24,111,35,149,102)(25,52,91,125,87)(26,126,53,88,92)(27,81,127,93,54)(28,94,82,55,128)(29,56,95,121,83)(30,122,49,84,96)(31,85,123,89,50)(32,90,86,51,124)(41,71,73,130,157)(42,131,72,158,74)(43,159,132,75,65)(44,76,160,66,133)(45,67,77,134,153)(46,135,68,154,78)(47,155,136,79,69)(48,80,156,70,129), (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([[(1,130,5,134),(2,131,6,135),(3,132,7,136),(4,133,8,129),(9,154,13,158),(10,155,14,159),(11,156,15,160),(12,157,16,153),(17,96,21,92),(18,89,22,93),(19,90,23,94),(20,91,24,95),(25,149,29,145),(26,150,30,146),(27,151,31,147),(28,152,32,148),(33,81,37,85),(34,82,38,86),(35,83,39,87),(36,84,40,88),(41,61,45,57),(42,62,46,58),(43,63,47,59),(44,64,48,60),(49,99,53,103),(50,100,54,104),(51,101,55,97),(52,102,56,98),(65,141,69,137),(66,142,70,138),(67,143,71,139),(68,144,72,140),(73,117,77,113),(74,118,78,114),(75,119,79,115),(76,120,80,116),(105,123,109,127),(106,124,110,128),(107,125,111,121),(108,126,112,122)], [(1,18,132,95),(2,96,133,19),(3,20,134,89),(4,90,135,21),(5,22,136,91),(6,92,129,23),(7,24,130,93),(8,94,131,17),(9,126,80,101),(10,102,73,127),(11,128,74,103),(12,104,75,121),(13,122,76,97),(14,98,77,123),(15,124,78,99),(16,100,79,125),(25,139,33,47),(26,48,34,140),(27,141,35,41),(28,42,36,142),(29,143,37,43),(30,44,38,144),(31,137,39,45),(32,46,40,138),(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),(57,151,65,83),(58,84,66,152),(59,145,67,85),(60,86,68,146),(61,147,69,87),(62,88,70,148),(63,149,71,81),(64,82,72,150)], [(1,12,57,143,113),(2,144,13,114,58),(3,115,137,59,14),(4,60,116,15,138),(5,16,61,139,117),(6,140,9,118,62),(7,119,141,63,10),(8,64,120,11,142),(17,150,112,103,36),(18,104,151,37,105),(19,38,97,106,152),(20,107,39,145,98),(21,146,108,99,40),(22,100,147,33,109),(23,34,101,110,148),(24,111,35,149,102),(25,52,91,125,87),(26,126,53,88,92),(27,81,127,93,54),(28,94,82,55,128),(29,56,95,121,83),(30,122,49,84,96),(31,85,123,89,50),(32,90,86,51,124),(41,71,73,130,157),(42,131,72,158,74),(43,159,132,75,65),(44,76,160,66,133),(45,67,77,134,153),(46,135,68,154,78),(47,155,136,79,69),(48,80,156,70,129)], [(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)]])
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 |
Matrix representation of C4⋊C4.9F5 ►in 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] >;
C4⋊C4.9F5 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