metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C20⋊3M4(2), C42.13F5, (C4×C20).5C4, C20⋊C8⋊7C2, C4⋊2(C4.F5), C4.9(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)), (C4×Dic5).35C4, Dic5.27(C2×D4), (D5×C42).18C2, C10.6(C2×M4(2)), C22.62(C22×F5), (C2×Dic5).317C23, (C4×Dic5).345C22, 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
Generators and relations for C20⋊3M4(2)
G = < a,b,c | a20=b8=c2=1, bab-1=a7, cac=a9, cbc=b5 >
Subgroups: 426 in 126 conjugacy classes, 60 normal (18 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, C2×C4, C23, D5, C10, C10, C42, C42, C2×C8, M4(2), C22×C4, Dic5, Dic5, C20, D10, D10, C2×C10, C4⋊C8, C2×C42, C2×M4(2), C5⋊C8, C4×D5, C4×D5, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C22×D5, C4⋊M4(2), C4×Dic5, C4×Dic5, C4×C20, C4.F5, C2×C5⋊C8, C2×C4×D5, C2×C4×D5, C20⋊C8, D5×C42, C2×C4.F5, C20⋊3M4(2)
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C23, C4⋊C4, M4(2), C22×C4, C2×D4, C2×Q8, F5, C2×C4⋊C4, C2×M4(2), C2×F5, C4⋊M4(2), C4.F5, C4⋊F5, C22×F5, C2×C4.F5, C2×C4⋊F5, C20⋊3M4(2)
►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)
(1 122 144 81 21 79 106 52)(2 125 153 88 22 62 115 59)(3 128 142 95 23 65 104 46)(4 131 151 82 24 68 113 53)(5 134 160 89 25 71 102 60)(6 137 149 96 26 74 111 47)(7 140 158 83 27 77 120 54)(8 123 147 90 28 80 109 41)(9 126 156 97 29 63 118 48)(10 129 145 84 30 66 107 55)(11 132 154 91 31 69 116 42)(12 135 143 98 32 72 105 49)(13 138 152 85 33 75 114 56)(14 121 141 92 34 78 103 43)(15 124 150 99 35 61 112 50)(16 127 159 86 36 64 101 57)(17 130 148 93 37 67 110 44)(18 133 157 100 38 70 119 51)(19 136 146 87 39 73 108 58)(20 139 155 94 40 76 117 45)
(2 10)(3 19)(4 8)(5 17)(7 15)(9 13)(12 20)(14 18)(22 30)(23 39)(24 28)(25 37)(27 35)(29 33)(32 40)(34 38)(41 82)(42 91)(43 100)(44 89)(45 98)(46 87)(47 96)(48 85)(49 94)(50 83)(51 92)(52 81)(53 90)(54 99)(55 88)(56 97)(57 86)(58 95)(59 84)(60 93)(61 140)(62 129)(63 138)(64 127)(65 136)(66 125)(67 134)(68 123)(69 132)(70 121)(71 130)(72 139)(73 128)(74 137)(75 126)(76 135)(77 124)(78 133)(79 122)(80 131)(102 110)(103 119)(104 108)(105 117)(107 115)(109 113)(112 120)(114 118)(141 157)(142 146)(143 155)(145 153)(147 151)(148 160)(150 158)(152 156)
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), (1,122,144,81,21,79,106,52)(2,125,153,88,22,62,115,59)(3,128,142,95,23,65,104,46)(4,131,151,82,24,68,113,53)(5,134,160,89,25,71,102,60)(6,137,149,96,26,74,111,47)(7,140,158,83,27,77,120,54)(8,123,147,90,28,80,109,41)(9,126,156,97,29,63,118,48)(10,129,145,84,30,66,107,55)(11,132,154,91,31,69,116,42)(12,135,143,98,32,72,105,49)(13,138,152,85,33,75,114,56)(14,121,141,92,34,78,103,43)(15,124,150,99,35,61,112,50)(16,127,159,86,36,64,101,57)(17,130,148,93,37,67,110,44)(18,133,157,100,38,70,119,51)(19,136,146,87,39,73,108,58)(20,139,155,94,40,76,117,45), (2,10)(3,19)(4,8)(5,17)(7,15)(9,13)(12,20)(14,18)(22,30)(23,39)(24,28)(25,37)(27,35)(29,33)(32,40)(34,38)(41,82)(42,91)(43,100)(44,89)(45,98)(46,87)(47,96)(48,85)(49,94)(50,83)(51,92)(52,81)(53,90)(54,99)(55,88)(56,97)(57,86)(58,95)(59,84)(60,93)(61,140)(62,129)(63,138)(64,127)(65,136)(66,125)(67,134)(68,123)(69,132)(70,121)(71,130)(72,139)(73,128)(74,137)(75,126)(76,135)(77,124)(78,133)(79,122)(80,131)(102,110)(103,119)(104,108)(105,117)(107,115)(109,113)(112,120)(114,118)(141,157)(142,146)(143,155)(145,153)(147,151)(148,160)(150,158)(152,156)>;
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), (1,122,144,81,21,79,106,52)(2,125,153,88,22,62,115,59)(3,128,142,95,23,65,104,46)(4,131,151,82,24,68,113,53)(5,134,160,89,25,71,102,60)(6,137,149,96,26,74,111,47)(7,140,158,83,27,77,120,54)(8,123,147,90,28,80,109,41)(9,126,156,97,29,63,118,48)(10,129,145,84,30,66,107,55)(11,132,154,91,31,69,116,42)(12,135,143,98,32,72,105,49)(13,138,152,85,33,75,114,56)(14,121,141,92,34,78,103,43)(15,124,150,99,35,61,112,50)(16,127,159,86,36,64,101,57)(17,130,148,93,37,67,110,44)(18,133,157,100,38,70,119,51)(19,136,146,87,39,73,108,58)(20,139,155,94,40,76,117,45), (2,10)(3,19)(4,8)(5,17)(7,15)(9,13)(12,20)(14,18)(22,30)(23,39)(24,28)(25,37)(27,35)(29,33)(32,40)(34,38)(41,82)(42,91)(43,100)(44,89)(45,98)(46,87)(47,96)(48,85)(49,94)(50,83)(51,92)(52,81)(53,90)(54,99)(55,88)(56,97)(57,86)(58,95)(59,84)(60,93)(61,140)(62,129)(63,138)(64,127)(65,136)(66,125)(67,134)(68,123)(69,132)(70,121)(71,130)(72,139)(73,128)(74,137)(75,126)(76,135)(77,124)(78,133)(79,122)(80,131)(102,110)(103,119)(104,108)(105,117)(107,115)(109,113)(112,120)(114,118)(141,157)(142,146)(143,155)(145,153)(147,151)(148,160)(150,158)(152,156) );
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)], [(1,122,144,81,21,79,106,52),(2,125,153,88,22,62,115,59),(3,128,142,95,23,65,104,46),(4,131,151,82,24,68,113,53),(5,134,160,89,25,71,102,60),(6,137,149,96,26,74,111,47),(7,140,158,83,27,77,120,54),(8,123,147,90,28,80,109,41),(9,126,156,97,29,63,118,48),(10,129,145,84,30,66,107,55),(11,132,154,91,31,69,116,42),(12,135,143,98,32,72,105,49),(13,138,152,85,33,75,114,56),(14,121,141,92,34,78,103,43),(15,124,150,99,35,61,112,50),(16,127,159,86,36,64,101,57),(17,130,148,93,37,67,110,44),(18,133,157,100,38,70,119,51),(19,136,146,87,39,73,108,58),(20,139,155,94,40,76,117,45)], [(2,10),(3,19),(4,8),(5,17),(7,15),(9,13),(12,20),(14,18),(22,30),(23,39),(24,28),(25,37),(27,35),(29,33),(32,40),(34,38),(41,82),(42,91),(43,100),(44,89),(45,98),(46,87),(47,96),(48,85),(49,94),(50,83),(51,92),(52,81),(53,90),(54,99),(55,88),(56,97),(57,86),(58,95),(59,84),(60,93),(61,140),(62,129),(63,138),(64,127),(65,136),(66,125),(67,134),(68,123),(69,132),(70,121),(71,130),(72,139),(73,128),(74,137),(75,126),(76,135),(77,124),(78,133),(79,122),(80,131),(102,110),(103,119),(104,108),(105,117),(107,115),(109,113),(112,120),(114,118),(141,157),(142,146),(143,155),(145,153),(147,151),(148,160),(150,158),(152,156)]])
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 |
Matrix representation of C20⋊3M4(2) ►in 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] >;
C20⋊3M4(2) 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