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), (C4×Dic5).244C22, (C2×Dic5).333C23, 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
Generators and relations for C4⋊C4.7F5
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 >
Subgroups: 330 in 96 conjugacy classes, 42 normal (all characteristic)
C1, C2, C2, C4, C22, C22, C5, C8, C2×C4, C2×C4, C23, D5, C10, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C22×C4, Dic5, Dic5, C20, D10, C2×C10, C4×C8, C8⋊C4, C22⋊C8, C4⋊C8, C42⋊C2, C5⋊C8, C4×D5, C2×Dic5, C2×C20, C22×D5, C42.7C22, C4×Dic5, C4⋊Dic5, D10⋊C4, C5×C4⋊C4, C2×C5⋊C8, C2×C4×D5, C4×C5⋊C8, C20⋊C8, C10.C42, D10⋊C8, Dic5⋊C8, C4⋊C4⋊7D5, C4⋊C4.7F5
Quotients: C1, C2, C4, C22, C2×C4, C23, C22×C4, C4○D4, F5, C42⋊C2, C8○D4, C2×F5, C42.7C22, C22×F5, D10.C23, D4.F5, Q8.F5, C4⋊C4.7F5
►On 160 points
Generators in S160
(1 143 5 139)(2 8 6 4)(3 137 7 141)(9 63 13 59)(10 124 14 128)(11 57 15 61)(12 126 16 122)(17 19 21 23)(18 38 22 34)(20 40 24 36)(25 42 29 46)(26 97 30 101)(27 44 31 48)(28 99 32 103)(33 35 37 39)(41 146 45 150)(43 148 47 152)(49 66 53 70)(50 85 54 81)(51 68 55 72)(52 87 56 83)(58 133 62 129)(60 135 64 131)(65 109 69 105)(67 111 71 107)(73 117 77 113)(74 156 78 160)(75 119 79 115)(76 158 80 154)(82 108 86 112)(84 110 88 106)(89 114 93 118)(90 153 94 157)(91 116 95 120)(92 155 96 159)(98 149 102 145)(100 151 104 147)(121 136 125 132)(123 130 127 134)(138 144 142 140)
(1 18 137 36)(2 19 138 37)(3 20 139 38)(4 21 140 39)(5 22 141 40)(6 23 142 33)(7 24 143 34)(8 17 144 35)(9 100 134 46)(10 101 135 47)(11 102 136 48)(12 103 129 41)(13 104 130 42)(14 97 131 43)(15 98 132 44)(16 99 133 45)(25 59 151 127)(26 60 152 128)(27 61 145 121)(28 62 146 122)(29 63 147 123)(30 64 148 124)(31 57 149 125)(32 58 150 126)(49 114 106 160)(50 115 107 153)(51 116 108 154)(52 117 109 155)(53 118 110 156)(54 119 111 157)(55 120 112 158)(56 113 105 159)(65 96 83 77)(66 89 84 78)(67 90 85 79)(68 91 86 80)(69 92 87 73)(70 93 88 74)(71 94 81 75)(72 95 82 76)
(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)(2,8,6,4)(3,137,7,141)(9,63,13,59)(10,124,14,128)(11,57,15,61)(12,126,16,122)(17,19,21,23)(18,38,22,34)(20,40,24,36)(25,42,29,46)(26,97,30,101)(27,44,31,48)(28,99,32,103)(33,35,37,39)(41,146,45,150)(43,148,47,152)(49,66,53,70)(50,85,54,81)(51,68,55,72)(52,87,56,83)(58,133,62,129)(60,135,64,131)(65,109,69,105)(67,111,71,107)(73,117,77,113)(74,156,78,160)(75,119,79,115)(76,158,80,154)(82,108,86,112)(84,110,88,106)(89,114,93,118)(90,153,94,157)(91,116,95,120)(92,155,96,159)(98,149,102,145)(100,151,104,147)(121,136,125,132)(123,130,127,134)(138,144,142,140), (1,18,137,36)(2,19,138,37)(3,20,139,38)(4,21,140,39)(5,22,141,40)(6,23,142,33)(7,24,143,34)(8,17,144,35)(9,100,134,46)(10,101,135,47)(11,102,136,48)(12,103,129,41)(13,104,130,42)(14,97,131,43)(15,98,132,44)(16,99,133,45)(25,59,151,127)(26,60,152,128)(27,61,145,121)(28,62,146,122)(29,63,147,123)(30,64,148,124)(31,57,149,125)(32,58,150,126)(49,114,106,160)(50,115,107,153)(51,116,108,154)(52,117,109,155)(53,118,110,156)(54,119,111,157)(55,120,112,158)(56,113,105,159)(65,96,83,77)(66,89,84,78)(67,90,85,79)(68,91,86,80)(69,92,87,73)(70,93,88,74)(71,94,81,75)(72,95,82,76), (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)(2,8,6,4)(3,137,7,141)(9,63,13,59)(10,124,14,128)(11,57,15,61)(12,126,16,122)(17,19,21,23)(18,38,22,34)(20,40,24,36)(25,42,29,46)(26,97,30,101)(27,44,31,48)(28,99,32,103)(33,35,37,39)(41,146,45,150)(43,148,47,152)(49,66,53,70)(50,85,54,81)(51,68,55,72)(52,87,56,83)(58,133,62,129)(60,135,64,131)(65,109,69,105)(67,111,71,107)(73,117,77,113)(74,156,78,160)(75,119,79,115)(76,158,80,154)(82,108,86,112)(84,110,88,106)(89,114,93,118)(90,153,94,157)(91,116,95,120)(92,155,96,159)(98,149,102,145)(100,151,104,147)(121,136,125,132)(123,130,127,134)(138,144,142,140), (1,18,137,36)(2,19,138,37)(3,20,139,38)(4,21,140,39)(5,22,141,40)(6,23,142,33)(7,24,143,34)(8,17,144,35)(9,100,134,46)(10,101,135,47)(11,102,136,48)(12,103,129,41)(13,104,130,42)(14,97,131,43)(15,98,132,44)(16,99,133,45)(25,59,151,127)(26,60,152,128)(27,61,145,121)(28,62,146,122)(29,63,147,123)(30,64,148,124)(31,57,149,125)(32,58,150,126)(49,114,106,160)(50,115,107,153)(51,116,108,154)(52,117,109,155)(53,118,110,156)(54,119,111,157)(55,120,112,158)(56,113,105,159)(65,96,83,77)(66,89,84,78)(67,90,85,79)(68,91,86,80)(69,92,87,73)(70,93,88,74)(71,94,81,75)(72,95,82,76), (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),(2,8,6,4),(3,137,7,141),(9,63,13,59),(10,124,14,128),(11,57,15,61),(12,126,16,122),(17,19,21,23),(18,38,22,34),(20,40,24,36),(25,42,29,46),(26,97,30,101),(27,44,31,48),(28,99,32,103),(33,35,37,39),(41,146,45,150),(43,148,47,152),(49,66,53,70),(50,85,54,81),(51,68,55,72),(52,87,56,83),(58,133,62,129),(60,135,64,131),(65,109,69,105),(67,111,71,107),(73,117,77,113),(74,156,78,160),(75,119,79,115),(76,158,80,154),(82,108,86,112),(84,110,88,106),(89,114,93,118),(90,153,94,157),(91,116,95,120),(92,155,96,159),(98,149,102,145),(100,151,104,147),(121,136,125,132),(123,130,127,134),(138,144,142,140)], [(1,18,137,36),(2,19,138,37),(3,20,139,38),(4,21,140,39),(5,22,141,40),(6,23,142,33),(7,24,143,34),(8,17,144,35),(9,100,134,46),(10,101,135,47),(11,102,136,48),(12,103,129,41),(13,104,130,42),(14,97,131,43),(15,98,132,44),(16,99,133,45),(25,59,151,127),(26,60,152,128),(27,61,145,121),(28,62,146,122),(29,63,147,123),(30,64,148,124),(31,57,149,125),(32,58,150,126),(49,114,106,160),(50,115,107,153),(51,116,108,154),(52,117,109,155),(53,118,110,156),(54,119,111,157),(55,120,112,158),(56,113,105,159),(65,96,83,77),(66,89,84,78),(67,90,85,79),(68,91,86,80),(69,92,87,73),(70,93,88,74),(71,94,81,75),(72,95,82,76)], [(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)]])
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 |
Matrix representation of C4⋊C4.7F5 ►in 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] >;
C4⋊C4.7F5 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