metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C20⋊3M4(2), C42.13F5, (C4×C20).5C4, C20⋊C8⋊7C2, C4.9(C4⋊F5), C4⋊2(C4.F5), (C4×D5).78D4, C20.16(C4⋊C4), (C4×D5).20Q8, Dic5.9(C2×Q8), D10.25(C4⋊C4), C5⋊1(C4⋊M4(2)), Dic5.27(C2×D4), (C4×Dic5).35C4, (D5×C42).18C2, C10.6(C2×M4(2)), C22.62(C22×F5), (C4×Dic5).345C22, (C2×Dic5).317C23, C2.6(C2×C4⋊F5), C10.2(C2×C4⋊C4), (C2×C4×D5).36C4, C2.6(C2×C4.F5), (C2×C5⋊C8).1C22, (C2×C4.F5).7C2, (C2×C4).132(C2×F5), (C2×C20).121(C2×C4), (C2×C4×D5).390C22, (C2×C10).19(C22×C4), (C2×Dic5).167(C2×C4), (C22×D5).119(C2×C4), SmallGroup(320,1019)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 426 in 126 conjugacy classes, 60 normal (18 characteristic)
C1, C2, C2 [×2], C2 [×2], C4 [×6], C4 [×4], C22, C22 [×4], C5, C8 [×4], C2×C4, C2×C4 [×2], C2×C4 [×11], C23, D5 [×2], C10, C10 [×2], C42, C42 [×3], C2×C8 [×4], M4(2) [×4], C22×C4 [×3], Dic5 [×2], Dic5 [×2], C20 [×6], D10 [×2], D10 [×2], C2×C10, C4⋊C8 [×4], C2×C42, C2×M4(2) [×2], C5⋊C8 [×4], C4×D5 [×4], C4×D5 [×4], C2×Dic5, C2×Dic5 [×2], C2×C20, C2×C20 [×2], C22×D5, C4⋊M4(2), C4×Dic5, C4×Dic5 [×2], C4×C20, C4.F5 [×4], C2×C5⋊C8 [×4], C2×C4×D5, C2×C4×D5 [×2], C20⋊C8 [×4], D5×C42, C2×C4.F5 [×2], C20⋊3M4(2)
Quotients:
C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×2], Q8 [×2], C23, C4⋊C4 [×4], M4(2) [×4], C22×C4, C2×D4, C2×Q8, F5, C2×C4⋊C4, C2×M4(2) [×2], C2×F5 [×3], C4⋊M4(2), C4.F5 [×4], C4⋊F5 [×2], C22×F5, C2×C4.F5 [×2], C2×C4⋊F5, C20⋊3M4(2)
Generators and relations
G = < a,b,c | a20=b8=c2=1, bab-1=a7, cac=a9, cbc=b5 >
►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 25 61 83 136 143 119 54)(2 28 70 90 137 146 108 41)(3 31 79 97 138 149 117 48)(4 34 68 84 139 152 106 55)(5 37 77 91 140 155 115 42)(6 40 66 98 121 158 104 49)(7 23 75 85 122 141 113 56)(8 26 64 92 123 144 102 43)(9 29 73 99 124 147 111 50)(10 32 62 86 125 150 120 57)(11 35 71 93 126 153 109 44)(12 38 80 100 127 156 118 51)(13 21 69 87 128 159 107 58)(14 24 78 94 129 142 116 45)(15 27 67 81 130 145 105 52)(16 30 76 88 131 148 114 59)(17 33 65 95 132 151 103 46)(18 36 74 82 133 154 112 53)(19 39 63 89 134 157 101 60)(20 22 72 96 135 160 110 47)
(2 10)(3 19)(4 8)(5 17)(7 15)(9 13)(12 20)(14 18)(21 147)(22 156)(23 145)(24 154)(25 143)(26 152)(27 141)(28 150)(29 159)(30 148)(31 157)(32 146)(33 155)(34 144)(35 153)(36 142)(37 151)(38 160)(39 149)(40 158)(41 86)(42 95)(43 84)(44 93)(45 82)(46 91)(47 100)(48 89)(49 98)(50 87)(51 96)(52 85)(53 94)(54 83)(55 92)(56 81)(57 90)(58 99)(59 88)(60 97)(62 70)(63 79)(64 68)(65 77)(67 75)(69 73)(72 80)(74 78)(101 117)(102 106)(103 115)(105 113)(107 111)(108 120)(110 118)(112 116)(122 130)(123 139)(124 128)(125 137)(127 135)(129 133)(132 140)(134 138)
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), (1,25,61,83,136,143,119,54)(2,28,70,90,137,146,108,41)(3,31,79,97,138,149,117,48)(4,34,68,84,139,152,106,55)(5,37,77,91,140,155,115,42)(6,40,66,98,121,158,104,49)(7,23,75,85,122,141,113,56)(8,26,64,92,123,144,102,43)(9,29,73,99,124,147,111,50)(10,32,62,86,125,150,120,57)(11,35,71,93,126,153,109,44)(12,38,80,100,127,156,118,51)(13,21,69,87,128,159,107,58)(14,24,78,94,129,142,116,45)(15,27,67,81,130,145,105,52)(16,30,76,88,131,148,114,59)(17,33,65,95,132,151,103,46)(18,36,74,82,133,154,112,53)(19,39,63,89,134,157,101,60)(20,22,72,96,135,160,110,47), (2,10)(3,19)(4,8)(5,17)(7,15)(9,13)(12,20)(14,18)(21,147)(22,156)(23,145)(24,154)(25,143)(26,152)(27,141)(28,150)(29,159)(30,148)(31,157)(32,146)(33,155)(34,144)(35,153)(36,142)(37,151)(38,160)(39,149)(40,158)(41,86)(42,95)(43,84)(44,93)(45,82)(46,91)(47,100)(48,89)(49,98)(50,87)(51,96)(52,85)(53,94)(54,83)(55,92)(56,81)(57,90)(58,99)(59,88)(60,97)(62,70)(63,79)(64,68)(65,77)(67,75)(69,73)(72,80)(74,78)(101,117)(102,106)(103,115)(105,113)(107,111)(108,120)(110,118)(112,116)(122,130)(123,139)(124,128)(125,137)(127,135)(129,133)(132,140)(134,138)>;
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), (1,25,61,83,136,143,119,54)(2,28,70,90,137,146,108,41)(3,31,79,97,138,149,117,48)(4,34,68,84,139,152,106,55)(5,37,77,91,140,155,115,42)(6,40,66,98,121,158,104,49)(7,23,75,85,122,141,113,56)(8,26,64,92,123,144,102,43)(9,29,73,99,124,147,111,50)(10,32,62,86,125,150,120,57)(11,35,71,93,126,153,109,44)(12,38,80,100,127,156,118,51)(13,21,69,87,128,159,107,58)(14,24,78,94,129,142,116,45)(15,27,67,81,130,145,105,52)(16,30,76,88,131,148,114,59)(17,33,65,95,132,151,103,46)(18,36,74,82,133,154,112,53)(19,39,63,89,134,157,101,60)(20,22,72,96,135,160,110,47), (2,10)(3,19)(4,8)(5,17)(7,15)(9,13)(12,20)(14,18)(21,147)(22,156)(23,145)(24,154)(25,143)(26,152)(27,141)(28,150)(29,159)(30,148)(31,157)(32,146)(33,155)(34,144)(35,153)(36,142)(37,151)(38,160)(39,149)(40,158)(41,86)(42,95)(43,84)(44,93)(45,82)(46,91)(47,100)(48,89)(49,98)(50,87)(51,96)(52,85)(53,94)(54,83)(55,92)(56,81)(57,90)(58,99)(59,88)(60,97)(62,70)(63,79)(64,68)(65,77)(67,75)(69,73)(72,80)(74,78)(101,117)(102,106)(103,115)(105,113)(107,111)(108,120)(110,118)(112,116)(122,130)(123,139)(124,128)(125,137)(127,135)(129,133)(132,140)(134,138) );
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)], [(1,25,61,83,136,143,119,54),(2,28,70,90,137,146,108,41),(3,31,79,97,138,149,117,48),(4,34,68,84,139,152,106,55),(5,37,77,91,140,155,115,42),(6,40,66,98,121,158,104,49),(7,23,75,85,122,141,113,56),(8,26,64,92,123,144,102,43),(9,29,73,99,124,147,111,50),(10,32,62,86,125,150,120,57),(11,35,71,93,126,153,109,44),(12,38,80,100,127,156,118,51),(13,21,69,87,128,159,107,58),(14,24,78,94,129,142,116,45),(15,27,67,81,130,145,105,52),(16,30,76,88,131,148,114,59),(17,33,65,95,132,151,103,46),(18,36,74,82,133,154,112,53),(19,39,63,89,134,157,101,60),(20,22,72,96,135,160,110,47)], [(2,10),(3,19),(4,8),(5,17),(7,15),(9,13),(12,20),(14,18),(21,147),(22,156),(23,145),(24,154),(25,143),(26,152),(27,141),(28,150),(29,159),(30,148),(31,157),(32,146),(33,155),(34,144),(35,153),(36,142),(37,151),(38,160),(39,149),(40,158),(41,86),(42,95),(43,84),(44,93),(45,82),(46,91),(47,100),(48,89),(49,98),(50,87),(51,96),(52,85),(53,94),(54,83),(55,92),(56,81),(57,90),(58,99),(59,88),(60,97),(62,70),(63,79),(64,68),(65,77),(67,75),(69,73),(72,80),(74,78),(101,117),(102,106),(103,115),(105,113),(107,111),(108,120),(110,118),(112,116),(122,130),(123,139),(124,128),(125,137),(127,135),(129,133),(132,140),(134,138)])
Matrix representation ►G ⊆ GL6(𝔽41)
| 1 | 2 | 0 | 0 | 0 | 0 |
| 40 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 40 | 35 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 6 |
| 0 | 0 | 0 | 0 | 35 | 6 |
| 39 | 9 | 0 | 0 | 0 | 0 |
| 27 | 2 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 6 |
| 0 | 0 | 0 | 0 | 0 | 40 |
| 0 | 0 | 32 | 0 | 0 | 0 |
| 0 | 0 | 0 | 32 | 0 | 0 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 6 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 35 |
| 0 | 0 | 0 | 0 | 0 | 1 |
G:=sub<GL(6,GF(41))| [1,40,0,0,0,0,2,40,0,0,0,0,0,0,0,40,0,0,0,0,1,35,0,0,0,0,0,0,1,35,0,0,0,0,6,6],[39,27,0,0,0,0,9,2,0,0,0,0,0,0,0,0,32,0,0,0,0,0,0,32,0,0,1,0,0,0,0,0,6,40,0,0],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,6,40,0,0,0,0,0,0,40,0,0,0,0,0,35,1] >;
44 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | ··· | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 5 | 8A | ··· | 8H | 10A | 10B | 10C | 20A | ··· | 20L |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 8 | ··· | 8 | 10 | 10 | 10 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 10 | 10 | 2 | ··· | 2 | 5 | 5 | 5 | 5 | 10 | 10 | 10 | 10 | 4 | 20 | ··· | 20 | 4 | 4 | 4 | 4 | ··· | 4 |
44 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | - | + | + | ||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | C4 | D4 | Q8 | M4(2) | F5 | C2×F5 | C4.F5 | C4⋊F5 |
| kernel | C20⋊3M4(2) | C20⋊C8 | D5×C42 | C2×C4.F5 | C4×Dic5 | C4×C20 | C2×C4×D5 | C4×D5 | C4×D5 | C20 | C42 | C2×C4 | C4 | C4 |
| # reps | 1 | 4 | 1 | 2 | 2 | 2 | 4 | 2 | 2 | 8 | 1 | 3 | 8 | 4 |
In GAP, Magma, Sage, TeX
C_{20}\rtimes_3M_{4(2)} % in TeX
G:=Group("C20:3M4(2)"); // GroupNames label
G:=SmallGroup(320,1019);
// by ID
G=gap.SmallGroup(320,1019);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,253,120,758,184,136,6278,1595]);
// Polycyclic
G:=Group<a,b,c|a^20=b^8=c^2=1,b*a*b^-1=a^7,c*a*c=a^9,c*b*c=b^5>;
// generators/relations
×
⋊
ℤ
𝔽
○
≀