metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C20⋊2M4(2), C5⋊C8⋊3D4, C5⋊3(C8⋊6D4), C20⋊C8⋊6C2, (C2×D4).6F5, C2.28(D4×F5), (D4×C10).9C4, C10.28(C4×D4), C4⋊Dic5.14C4, C4⋊1(C22.F5), C23.12(C2×F5), C23.D5.7C4, C10.16(C8○D4), (D4×Dic5).18C2, Dic5.80(C2×D4), C2.16(D4.F5), C10.31(C2×M4(2)), C23.2F5⋊10C2, Dic5.59(C4○D4), C22.92(C22×F5), (C4×Dic5).195C22, (C2×Dic5).353C23, (C22×Dic5).186C22, (C4×C5⋊C8)⋊6C2, (C2×C4).81(C2×F5), (C2×C20).55(C2×C4), (C2×C5⋊C8).10C22, (C2×C22.F5)⋊5C2, C2.10(C2×C22.F5), (C22×C10).25(C2×C4), (C2×C10).77(C22×C4), (C2×Dic5).72(C2×C4), SmallGroup(320,1112)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C20⋊2M4(2)
G = < a,b,c | a20=b8=c2=1, bab-1=a3, cac=a11, cbc=b5 >
Subgroups: 394 in 122 conjugacy classes, 48 normal (26 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, D4, C23, C10, C10, C42, C22⋊C4, C4⋊C4, C2×C8, M4(2), C22×C4, C2×D4, Dic5, Dic5, C20, C2×C10, C2×C10, C4×C8, C22⋊C8, C4⋊C8, C4×D4, C2×M4(2), C5⋊C8, C5⋊C8, C2×Dic5, C2×Dic5, C2×Dic5, C2×C20, C5×D4, C22×C10, C8⋊6D4, C4×Dic5, C4⋊Dic5, C23.D5, C2×C5⋊C8, C2×C5⋊C8, C22.F5, C22×Dic5, D4×C10, C4×C5⋊C8, C20⋊C8, C23.2F5, D4×Dic5, C2×C22.F5, C20⋊2M4(2)
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, M4(2), C22×C4, C2×D4, C4○D4, F5, C4×D4, C2×M4(2), C8○D4, C2×F5, C8⋊6D4, C22.F5, C22×F5, D4.F5, D4×F5, C2×C22.F5, C20⋊2M4(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)
(1 155 28 107 52 72 98 133)(2 142 37 110 53 79 87 136)(3 149 26 113 54 66 96 139)(4 156 35 116 55 73 85 122)(5 143 24 119 56 80 94 125)(6 150 33 102 57 67 83 128)(7 157 22 105 58 74 92 131)(8 144 31 108 59 61 81 134)(9 151 40 111 60 68 90 137)(10 158 29 114 41 75 99 140)(11 145 38 117 42 62 88 123)(12 152 27 120 43 69 97 126)(13 159 36 103 44 76 86 129)(14 146 25 106 45 63 95 132)(15 153 34 109 46 70 84 135)(16 160 23 112 47 77 93 138)(17 147 32 115 48 64 82 121)(18 154 21 118 49 71 91 124)(19 141 30 101 50 78 100 127)(20 148 39 104 51 65 89 130)
(1 6)(2 17)(3 8)(4 19)(5 10)(7 12)(9 14)(11 16)(13 18)(15 20)(21 36)(22 27)(23 38)(24 29)(25 40)(26 31)(28 33)(30 35)(32 37)(34 39)(41 56)(42 47)(43 58)(44 49)(45 60)(46 51)(48 53)(50 55)(52 57)(54 59)(61 149)(62 160)(63 151)(64 142)(65 153)(66 144)(67 155)(68 146)(69 157)(70 148)(71 159)(72 150)(73 141)(74 152)(75 143)(76 154)(77 145)(78 156)(79 147)(80 158)(81 96)(82 87)(83 98)(84 89)(85 100)(86 91)(88 93)(90 95)(92 97)(94 99)(101 122)(102 133)(103 124)(104 135)(105 126)(106 137)(107 128)(108 139)(109 130)(110 121)(111 132)(112 123)(113 134)(114 125)(115 136)(116 127)(117 138)(118 129)(119 140)(120 131)
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), (1,155,28,107,52,72,98,133)(2,142,37,110,53,79,87,136)(3,149,26,113,54,66,96,139)(4,156,35,116,55,73,85,122)(5,143,24,119,56,80,94,125)(6,150,33,102,57,67,83,128)(7,157,22,105,58,74,92,131)(8,144,31,108,59,61,81,134)(9,151,40,111,60,68,90,137)(10,158,29,114,41,75,99,140)(11,145,38,117,42,62,88,123)(12,152,27,120,43,69,97,126)(13,159,36,103,44,76,86,129)(14,146,25,106,45,63,95,132)(15,153,34,109,46,70,84,135)(16,160,23,112,47,77,93,138)(17,147,32,115,48,64,82,121)(18,154,21,118,49,71,91,124)(19,141,30,101,50,78,100,127)(20,148,39,104,51,65,89,130), (1,6)(2,17)(3,8)(4,19)(5,10)(7,12)(9,14)(11,16)(13,18)(15,20)(21,36)(22,27)(23,38)(24,29)(25,40)(26,31)(28,33)(30,35)(32,37)(34,39)(41,56)(42,47)(43,58)(44,49)(45,60)(46,51)(48,53)(50,55)(52,57)(54,59)(61,149)(62,160)(63,151)(64,142)(65,153)(66,144)(67,155)(68,146)(69,157)(70,148)(71,159)(72,150)(73,141)(74,152)(75,143)(76,154)(77,145)(78,156)(79,147)(80,158)(81,96)(82,87)(83,98)(84,89)(85,100)(86,91)(88,93)(90,95)(92,97)(94,99)(101,122)(102,133)(103,124)(104,135)(105,126)(106,137)(107,128)(108,139)(109,130)(110,121)(111,132)(112,123)(113,134)(114,125)(115,136)(116,127)(117,138)(118,129)(119,140)(120,131)>;
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), (1,155,28,107,52,72,98,133)(2,142,37,110,53,79,87,136)(3,149,26,113,54,66,96,139)(4,156,35,116,55,73,85,122)(5,143,24,119,56,80,94,125)(6,150,33,102,57,67,83,128)(7,157,22,105,58,74,92,131)(8,144,31,108,59,61,81,134)(9,151,40,111,60,68,90,137)(10,158,29,114,41,75,99,140)(11,145,38,117,42,62,88,123)(12,152,27,120,43,69,97,126)(13,159,36,103,44,76,86,129)(14,146,25,106,45,63,95,132)(15,153,34,109,46,70,84,135)(16,160,23,112,47,77,93,138)(17,147,32,115,48,64,82,121)(18,154,21,118,49,71,91,124)(19,141,30,101,50,78,100,127)(20,148,39,104,51,65,89,130), (1,6)(2,17)(3,8)(4,19)(5,10)(7,12)(9,14)(11,16)(13,18)(15,20)(21,36)(22,27)(23,38)(24,29)(25,40)(26,31)(28,33)(30,35)(32,37)(34,39)(41,56)(42,47)(43,58)(44,49)(45,60)(46,51)(48,53)(50,55)(52,57)(54,59)(61,149)(62,160)(63,151)(64,142)(65,153)(66,144)(67,155)(68,146)(69,157)(70,148)(71,159)(72,150)(73,141)(74,152)(75,143)(76,154)(77,145)(78,156)(79,147)(80,158)(81,96)(82,87)(83,98)(84,89)(85,100)(86,91)(88,93)(90,95)(92,97)(94,99)(101,122)(102,133)(103,124)(104,135)(105,126)(106,137)(107,128)(108,139)(109,130)(110,121)(111,132)(112,123)(113,134)(114,125)(115,136)(116,127)(117,138)(118,129)(119,140)(120,131) );
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)], [(1,155,28,107,52,72,98,133),(2,142,37,110,53,79,87,136),(3,149,26,113,54,66,96,139),(4,156,35,116,55,73,85,122),(5,143,24,119,56,80,94,125),(6,150,33,102,57,67,83,128),(7,157,22,105,58,74,92,131),(8,144,31,108,59,61,81,134),(9,151,40,111,60,68,90,137),(10,158,29,114,41,75,99,140),(11,145,38,117,42,62,88,123),(12,152,27,120,43,69,97,126),(13,159,36,103,44,76,86,129),(14,146,25,106,45,63,95,132),(15,153,34,109,46,70,84,135),(16,160,23,112,47,77,93,138),(17,147,32,115,48,64,82,121),(18,154,21,118,49,71,91,124),(19,141,30,101,50,78,100,127),(20,148,39,104,51,65,89,130)], [(1,6),(2,17),(3,8),(4,19),(5,10),(7,12),(9,14),(11,16),(13,18),(15,20),(21,36),(22,27),(23,38),(24,29),(25,40),(26,31),(28,33),(30,35),(32,37),(34,39),(41,56),(42,47),(43,58),(44,49),(45,60),(46,51),(48,53),(50,55),(52,57),(54,59),(61,149),(62,160),(63,151),(64,142),(65,153),(66,144),(67,155),(68,146),(69,157),(70,148),(71,159),(72,150),(73,141),(74,152),(75,143),(76,154),(77,145),(78,156),(79,147),(80,158),(81,96),(82,87),(83,98),(84,89),(85,100),(86,91),(88,93),(90,95),(92,97),(94,99),(101,122),(102,133),(103,124),(104,135),(105,126),(106,137),(107,128),(108,139),(109,130),(110,121),(111,132),(112,123),(113,134),(114,125),(115,136),(116,127),(117,138),(118,129),(119,140),(120,131)]])
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 | 10D | 10E | 10F | 10G | 20A | 20B |
| 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 | 10 | 10 | 10 | 10 | 20 | 20 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 2 | 2 | 5 | 5 | 5 | 5 | 10 | 10 | 20 | 20 | 4 | 10 | ··· | 10 | 20 | 20 | 20 | 20 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 8 | 8 |
38 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 8 | 8 |
| type | + | + | + | + | + | + | + | + | + | + | - | - | + | ||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | D4 | C4○D4 | M4(2) | C8○D4 | F5 | C2×F5 | C2×F5 | C22.F5 | D4.F5 | D4×F5 |
| kernel | C20⋊2M4(2) | C4×C5⋊C8 | C20⋊C8 | C23.2F5 | D4×Dic5 | C2×C22.F5 | C4⋊Dic5 | C23.D5 | D4×C10 | C5⋊C8 | Dic5 | C20 | C10 | C2×D4 | C2×C4 | C23 | C4 | C2 | C2 |
| # reps | 1 | 1 | 1 | 2 | 1 | 2 | 2 | 4 | 2 | 2 | 2 | 4 | 4 | 1 | 1 | 2 | 4 | 1 | 1 |
Matrix representation of C20⋊2M4(2) ►in GL8(𝔽41)
| 1 | 39 | 0 | 0 | 0 | 0 | 0 | 0 |
| 1 | 40 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 40 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 0 | 40 | 0 | 1 |
| 0 | 0 | 0 | 0 | 0 | 0 | 40 | 1 |
| 40 | 2 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 34 | 19 | 0 | 0 | 0 | 0 |
| 0 | 0 | 25 | 7 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 37 | 14 | 4 | 2 |
| 0 | 0 | 0 | 0 | 0 | 16 | 29 | 39 |
| 0 | 0 | 0 | 0 | 25 | 12 | 2 | 2 |
| 0 | 0 | 0 | 0 | 39 | 16 | 4 | 27 |
| 40 | 2 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 18 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 40 |
G:=sub<GL(8,GF(41))| [1,1,0,0,0,0,0,0,39,40,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,1,1,1,1],[40,0,0,0,0,0,0,0,2,1,0,0,0,0,0,0,0,0,34,25,0,0,0,0,0,0,19,7,0,0,0,0,0,0,0,0,37,0,25,39,0,0,0,0,14,16,12,16,0,0,0,0,4,29,2,4,0,0,0,0,2,39,2,27],[40,0,0,0,0,0,0,0,2,1,0,0,0,0,0,0,0,0,1,18,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40] >;
C20⋊2M4(2) in GAP, Magma, Sage, TeX
C_{20}\rtimes_2M_4(2) % in TeX
G:=Group("C20:2M4(2)"); // GroupNames label
G:=SmallGroup(320,1112);
// by ID
G=gap.SmallGroup(320,1112);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,477,758,219,136,6278,1595]);
// Polycyclic
G:=Group<a,b,c|a^20=b^8=c^2=1,b*a*b^-1=a^3,c*a*c=a^11,c*b*c=b^5>;
// generators/relations