metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C40⋊32D4, C8⋊15(C5⋊D4), C5⋊10(C8⋊9D4), C40⋊8C4⋊23C2, (C22×C8)⋊11D5, D10⋊1C8⋊3C2, (C22×C40)⋊16C2, (C2×C8).294D10, C10.108(C4×D4), C20.437(C2×D4), C20.8Q8⋊3C2, C23.35(C4×D5), C10.43(C8○D4), C22⋊2(C8⋊D5), (C2×C10)⋊11M4(2), C23.D5.16C4, D10⋊C4.13C4, C20.252(C4○D4), C4.136(C4○D20), C20.55D4⋊26C2, (C2×C20).861C23, (C2×C40).355C22, C10.D4.13C4, (C22×C4).402D10, C10.47(C2×M4(2)), C2.19(D20.3C4), (C22×C20).561C22, (C4×Dic5).207C22, (C2×C4).94(C4×D5), C2.23(C4×C5⋊D4), (C2×C8⋊D5)⋊23C2, C2.15(C2×C8⋊D5), (C2×C5⋊D4).18C4, (C4×C5⋊D4).15C2, C4.127(C2×C5⋊D4), C22.142(C2×C4×D5), (C2×C20).383(C2×C4), (C2×C4×D5).235C22, (C2×Dic5).34(C2×C4), (C22×D5).29(C2×C4), (C2×C4).803(C22×D5), (C2×C10).232(C22×C4), (C22×C10).164(C2×C4), (C2×C5⋊2C8).209C22, SmallGroup(320,738)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C40⋊32D4
G = < a,b,c | a40=b4=c2=1, bab-1=cac=a29, cbc=b-1 >
Subgroups: 382 in 124 conjugacy classes, 55 normal (47 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C5, C8, C8, C2×C4, C2×C4, D4, C23, C23, D5, C10, C10, C42, C22⋊C4, C4⋊C4, C2×C8, C2×C8, M4(2), C22×C4, C22×C4, C2×D4, Dic5, C20, C20, D10, C2×C10, C2×C10, C2×C10, C8⋊C4, C22⋊C8, C4⋊C8, C4×D4, C22×C8, C2×M4(2), C5⋊2C8, C40, C40, C4×D5, C2×Dic5, C5⋊D4, C2×C20, C2×C20, C22×D5, C22×C10, C8⋊9D4, C8⋊D5, C2×C5⋊2C8, C4×Dic5, C10.D4, D10⋊C4, C23.D5, C2×C40, C2×C40, C2×C4×D5, C2×C5⋊D4, C22×C20, C20.8Q8, C40⋊8C4, D10⋊1C8, C20.55D4, C2×C8⋊D5, C4×C5⋊D4, C22×C40, C40⋊32D4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, D5, M4(2), C22×C4, C2×D4, C4○D4, D10, C4×D4, C2×M4(2), C8○D4, C4×D5, C5⋊D4, C22×D5, C8⋊9D4, C8⋊D5, C2×C4×D5, C4○D20, C2×C5⋊D4, C2×C8⋊D5, D20.3C4, C4×C5⋊D4, C40⋊32D4
►On 160 points
Generators in S160
(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 67 152 106)(2 56 153 95)(3 45 154 84)(4 74 155 113)(5 63 156 102)(6 52 157 91)(7 41 158 120)(8 70 159 109)(9 59 160 98)(10 48 121 87)(11 77 122 116)(12 66 123 105)(13 55 124 94)(14 44 125 83)(15 73 126 112)(16 62 127 101)(17 51 128 90)(18 80 129 119)(19 69 130 108)(20 58 131 97)(21 47 132 86)(22 76 133 115)(23 65 134 104)(24 54 135 93)(25 43 136 82)(26 72 137 111)(27 61 138 100)(28 50 139 89)(29 79 140 118)(30 68 141 107)(31 57 142 96)(32 46 143 85)(33 75 144 114)(34 64 145 103)(35 53 146 92)(36 42 147 81)(37 71 148 110)(38 60 149 99)(39 49 150 88)(40 78 151 117)
(2 30)(3 19)(4 8)(5 37)(6 26)(7 15)(9 33)(10 22)(12 40)(13 29)(14 18)(16 36)(17 25)(20 32)(23 39)(24 28)(27 35)(34 38)(41 112)(42 101)(43 90)(44 119)(45 108)(46 97)(47 86)(48 115)(49 104)(50 93)(51 82)(52 111)(53 100)(54 89)(55 118)(56 107)(57 96)(58 85)(59 114)(60 103)(61 92)(62 81)(63 110)(64 99)(65 88)(66 117)(67 106)(68 95)(69 84)(70 113)(71 102)(72 91)(73 120)(74 109)(75 98)(76 87)(77 116)(78 105)(79 94)(80 83)(121 133)(123 151)(124 140)(125 129)(126 158)(127 147)(128 136)(130 154)(131 143)(134 150)(135 139)(137 157)(138 146)(141 153)(144 160)(145 149)(148 156)(155 159)
G:=sub<Sym(160)| (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), (1,67,152,106)(2,56,153,95)(3,45,154,84)(4,74,155,113)(5,63,156,102)(6,52,157,91)(7,41,158,120)(8,70,159,109)(9,59,160,98)(10,48,121,87)(11,77,122,116)(12,66,123,105)(13,55,124,94)(14,44,125,83)(15,73,126,112)(16,62,127,101)(17,51,128,90)(18,80,129,119)(19,69,130,108)(20,58,131,97)(21,47,132,86)(22,76,133,115)(23,65,134,104)(24,54,135,93)(25,43,136,82)(26,72,137,111)(27,61,138,100)(28,50,139,89)(29,79,140,118)(30,68,141,107)(31,57,142,96)(32,46,143,85)(33,75,144,114)(34,64,145,103)(35,53,146,92)(36,42,147,81)(37,71,148,110)(38,60,149,99)(39,49,150,88)(40,78,151,117), (2,30)(3,19)(4,8)(5,37)(6,26)(7,15)(9,33)(10,22)(12,40)(13,29)(14,18)(16,36)(17,25)(20,32)(23,39)(24,28)(27,35)(34,38)(41,112)(42,101)(43,90)(44,119)(45,108)(46,97)(47,86)(48,115)(49,104)(50,93)(51,82)(52,111)(53,100)(54,89)(55,118)(56,107)(57,96)(58,85)(59,114)(60,103)(61,92)(62,81)(63,110)(64,99)(65,88)(66,117)(67,106)(68,95)(69,84)(70,113)(71,102)(72,91)(73,120)(74,109)(75,98)(76,87)(77,116)(78,105)(79,94)(80,83)(121,133)(123,151)(124,140)(125,129)(126,158)(127,147)(128,136)(130,154)(131,143)(134,150)(135,139)(137,157)(138,146)(141,153)(144,160)(145,149)(148,156)(155,159)>;
G:=Group( (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), (1,67,152,106)(2,56,153,95)(3,45,154,84)(4,74,155,113)(5,63,156,102)(6,52,157,91)(7,41,158,120)(8,70,159,109)(9,59,160,98)(10,48,121,87)(11,77,122,116)(12,66,123,105)(13,55,124,94)(14,44,125,83)(15,73,126,112)(16,62,127,101)(17,51,128,90)(18,80,129,119)(19,69,130,108)(20,58,131,97)(21,47,132,86)(22,76,133,115)(23,65,134,104)(24,54,135,93)(25,43,136,82)(26,72,137,111)(27,61,138,100)(28,50,139,89)(29,79,140,118)(30,68,141,107)(31,57,142,96)(32,46,143,85)(33,75,144,114)(34,64,145,103)(35,53,146,92)(36,42,147,81)(37,71,148,110)(38,60,149,99)(39,49,150,88)(40,78,151,117), (2,30)(3,19)(4,8)(5,37)(6,26)(7,15)(9,33)(10,22)(12,40)(13,29)(14,18)(16,36)(17,25)(20,32)(23,39)(24,28)(27,35)(34,38)(41,112)(42,101)(43,90)(44,119)(45,108)(46,97)(47,86)(48,115)(49,104)(50,93)(51,82)(52,111)(53,100)(54,89)(55,118)(56,107)(57,96)(58,85)(59,114)(60,103)(61,92)(62,81)(63,110)(64,99)(65,88)(66,117)(67,106)(68,95)(69,84)(70,113)(71,102)(72,91)(73,120)(74,109)(75,98)(76,87)(77,116)(78,105)(79,94)(80,83)(121,133)(123,151)(124,140)(125,129)(126,158)(127,147)(128,136)(130,154)(131,143)(134,150)(135,139)(137,157)(138,146)(141,153)(144,160)(145,149)(148,156)(155,159) );
G=PermutationGroup([[(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)], [(1,67,152,106),(2,56,153,95),(3,45,154,84),(4,74,155,113),(5,63,156,102),(6,52,157,91),(7,41,158,120),(8,70,159,109),(9,59,160,98),(10,48,121,87),(11,77,122,116),(12,66,123,105),(13,55,124,94),(14,44,125,83),(15,73,126,112),(16,62,127,101),(17,51,128,90),(18,80,129,119),(19,69,130,108),(20,58,131,97),(21,47,132,86),(22,76,133,115),(23,65,134,104),(24,54,135,93),(25,43,136,82),(26,72,137,111),(27,61,138,100),(28,50,139,89),(29,79,140,118),(30,68,141,107),(31,57,142,96),(32,46,143,85),(33,75,144,114),(34,64,145,103),(35,53,146,92),(36,42,147,81),(37,71,148,110),(38,60,149,99),(39,49,150,88),(40,78,151,117)], [(2,30),(3,19),(4,8),(5,37),(6,26),(7,15),(9,33),(10,22),(12,40),(13,29),(14,18),(16,36),(17,25),(20,32),(23,39),(24,28),(27,35),(34,38),(41,112),(42,101),(43,90),(44,119),(45,108),(46,97),(47,86),(48,115),(49,104),(50,93),(51,82),(52,111),(53,100),(54,89),(55,118),(56,107),(57,96),(58,85),(59,114),(60,103),(61,92),(62,81),(63,110),(64,99),(65,88),(66,117),(67,106),(68,95),(69,84),(70,113),(71,102),(72,91),(73,120),(74,109),(75,98),(76,87),(77,116),(78,105),(79,94),(80,83),(121,133),(123,151),(124,140),(125,129),(126,158),(127,147),(128,136),(130,154),(131,143),(134,150),(135,139),(137,157),(138,146),(141,153),(144,160),(145,149),(148,156),(155,159)]])
92 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 5A | 5B | 8A | ··· | 8H | 8I | 8J | 8K | 8L | 10A | ··· | 10N | 20A | ··· | 20P | 40A | ··· | 40AF |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | ··· | 8 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 20 | ··· | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 20 | 1 | 1 | 1 | 1 | 2 | 2 | 20 | 20 | 20 | 2 | 2 | 2 | ··· | 2 | 20 | 20 | 20 | 20 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
92 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | |||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | C4 | D4 | D5 | C4○D4 | M4(2) | D10 | D10 | C8○D4 | C5⋊D4 | C4×D5 | C4×D5 | C4○D20 | C8⋊D5 | D20.3C4 |
| kernel | C40⋊32D4 | C20.8Q8 | C40⋊8C4 | D10⋊1C8 | C20.55D4 | C2×C8⋊D5 | C4×C5⋊D4 | C22×C40 | C10.D4 | D10⋊C4 | C23.D5 | C2×C5⋊D4 | C40 | C22×C8 | C20 | C2×C10 | C2×C8 | C22×C4 | C10 | C8 | C2×C4 | C23 | C4 | C22 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 2 | 4 | 8 | 4 | 4 | 8 | 16 | 16 |
Matrix representation of C40⋊32D4 ►in GL4(𝔽41) generated by
| 31 | 9 | 0 | 0 |
| 32 | 9 | 0 | 0 |
| 0 | 0 | 19 | 32 |
| 0 | 0 | 37 | 23 |
| 17 | 40 | 0 | 0 |
| 3 | 24 | 0 | 0 |
| 0 | 0 | 35 | 40 |
| 0 | 0 | 35 | 6 |
| 1 | 0 | 0 | 0 |
| 34 | 40 | 0 | 0 |
| 0 | 0 | 6 | 1 |
| 0 | 0 | 6 | 35 |
G:=sub<GL(4,GF(41))| [31,32,0,0,9,9,0,0,0,0,19,37,0,0,32,23],[17,3,0,0,40,24,0,0,0,0,35,35,0,0,40,6],[1,34,0,0,0,40,0,0,0,0,6,6,0,0,1,35] >;
C40⋊32D4 in GAP, Magma, Sage, TeX
C_{40}\rtimes_{32}D_4 % in TeX
G:=Group("C40:32D4"); // GroupNames label
G:=SmallGroup(320,738);
// by ID
G=gap.SmallGroup(320,738);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,253,758,58,136,12550]);
// Polycyclic
G:=Group<a,b,c|a^40=b^4=c^2=1,b*a*b^-1=c*a*c=a^29,c*b*c=b^-1>;
// generators/relations