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), Dic5.80(C2×D4), (D4×Dic5).18C2, C2.16(D4.F5), C10.31(C2×M4(2)), C23.2F5⋊10C2, Dic5.59(C4○D4), C22.92(C22×F5), (C2×Dic5).353C23, (C4×Dic5).195C22, (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), (C2×C10).77(C22×C4), (C22×C10).25(C2×C4), (C2×Dic5).72(C2×C4), SmallGroup(320,1112)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 394 in 122 conjugacy classes, 48 normal (26 characteristic)
C1, C2 [×3], C2 [×2], C4 [×2], C4 [×5], C22, C22 [×6], C5, C8 [×5], C2×C4, C2×C4 [×8], D4 [×2], C23 [×2], C10 [×3], C10 [×2], C42, C22⋊C4 [×2], C4⋊C4, C2×C8 [×4], M4(2) [×4], C22×C4 [×2], C2×D4, Dic5 [×2], Dic5 [×3], C20 [×2], C2×C10, C2×C10 [×6], C4×C8, C22⋊C8 [×2], C4⋊C8, C4×D4, C2×M4(2) [×2], C5⋊C8 [×2], C5⋊C8 [×3], C2×Dic5 [×2], C2×Dic5 [×2], C2×Dic5 [×4], C2×C20, C5×D4 [×2], C22×C10 [×2], C8⋊6D4, C4×Dic5, C4⋊Dic5, C23.D5 [×2], C2×C5⋊C8 [×2], C2×C5⋊C8 [×2], C22.F5 [×4], C22×Dic5 [×2], D4×C10, C4×C5⋊C8, C20⋊C8, C23.2F5 [×2], D4×Dic5, C2×C22.F5 [×2], C20⋊2M4(2)
Quotients:
C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×2], C23, M4(2) [×2], C22×C4, C2×D4, C4○D4, F5, C4×D4, C2×M4(2), C8○D4, C2×F5 [×3], C8⋊6D4, C22.F5 [×2], C22×F5, D4.F5, D4×F5, C2×C22.F5, C20⋊2M4(2)
Generators and relations
G = < a,b,c | a20=b8=c2=1, bab-1=a3, cac=a11, 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 70 157 100 130 110 41 39)(2 77 146 83 131 117 50 22)(3 64 155 86 132 104 59 25)(4 71 144 89 133 111 48 28)(5 78 153 92 134 118 57 31)(6 65 142 95 135 105 46 34)(7 72 151 98 136 112 55 37)(8 79 160 81 137 119 44 40)(9 66 149 84 138 106 53 23)(10 73 158 87 139 113 42 26)(11 80 147 90 140 120 51 29)(12 67 156 93 121 107 60 32)(13 74 145 96 122 114 49 35)(14 61 154 99 123 101 58 38)(15 68 143 82 124 108 47 21)(16 75 152 85 125 115 56 24)(17 62 141 88 126 102 45 27)(18 69 150 91 127 109 54 30)(19 76 159 94 128 116 43 33)(20 63 148 97 129 103 52 36)
(1 6)(2 17)(3 8)(4 19)(5 10)(7 12)(9 14)(11 16)(13 18)(15 20)(21 97)(22 88)(23 99)(24 90)(25 81)(26 92)(27 83)(28 94)(29 85)(30 96)(31 87)(32 98)(33 89)(34 100)(35 91)(36 82)(37 93)(38 84)(39 95)(40 86)(41 46)(42 57)(43 48)(44 59)(45 50)(47 52)(49 54)(51 56)(53 58)(55 60)(61 106)(62 117)(63 108)(64 119)(65 110)(66 101)(67 112)(68 103)(69 114)(70 105)(71 116)(72 107)(73 118)(74 109)(75 120)(76 111)(77 102)(78 113)(79 104)(80 115)(121 136)(122 127)(123 138)(124 129)(125 140)(126 131)(128 133)(130 135)(132 137)(134 139)(141 146)(142 157)(143 148)(144 159)(145 150)(147 152)(149 154)(151 156)(153 158)(155 160)
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), (1,70,157,100,130,110,41,39)(2,77,146,83,131,117,50,22)(3,64,155,86,132,104,59,25)(4,71,144,89,133,111,48,28)(5,78,153,92,134,118,57,31)(6,65,142,95,135,105,46,34)(7,72,151,98,136,112,55,37)(8,79,160,81,137,119,44,40)(9,66,149,84,138,106,53,23)(10,73,158,87,139,113,42,26)(11,80,147,90,140,120,51,29)(12,67,156,93,121,107,60,32)(13,74,145,96,122,114,49,35)(14,61,154,99,123,101,58,38)(15,68,143,82,124,108,47,21)(16,75,152,85,125,115,56,24)(17,62,141,88,126,102,45,27)(18,69,150,91,127,109,54,30)(19,76,159,94,128,116,43,33)(20,63,148,97,129,103,52,36), (1,6)(2,17)(3,8)(4,19)(5,10)(7,12)(9,14)(11,16)(13,18)(15,20)(21,97)(22,88)(23,99)(24,90)(25,81)(26,92)(27,83)(28,94)(29,85)(30,96)(31,87)(32,98)(33,89)(34,100)(35,91)(36,82)(37,93)(38,84)(39,95)(40,86)(41,46)(42,57)(43,48)(44,59)(45,50)(47,52)(49,54)(51,56)(53,58)(55,60)(61,106)(62,117)(63,108)(64,119)(65,110)(66,101)(67,112)(68,103)(69,114)(70,105)(71,116)(72,107)(73,118)(74,109)(75,120)(76,111)(77,102)(78,113)(79,104)(80,115)(121,136)(122,127)(123,138)(124,129)(125,140)(126,131)(128,133)(130,135)(132,137)(134,139)(141,146)(142,157)(143,148)(144,159)(145,150)(147,152)(149,154)(151,156)(153,158)(155,160)>;
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), (1,70,157,100,130,110,41,39)(2,77,146,83,131,117,50,22)(3,64,155,86,132,104,59,25)(4,71,144,89,133,111,48,28)(5,78,153,92,134,118,57,31)(6,65,142,95,135,105,46,34)(7,72,151,98,136,112,55,37)(8,79,160,81,137,119,44,40)(9,66,149,84,138,106,53,23)(10,73,158,87,139,113,42,26)(11,80,147,90,140,120,51,29)(12,67,156,93,121,107,60,32)(13,74,145,96,122,114,49,35)(14,61,154,99,123,101,58,38)(15,68,143,82,124,108,47,21)(16,75,152,85,125,115,56,24)(17,62,141,88,126,102,45,27)(18,69,150,91,127,109,54,30)(19,76,159,94,128,116,43,33)(20,63,148,97,129,103,52,36), (1,6)(2,17)(3,8)(4,19)(5,10)(7,12)(9,14)(11,16)(13,18)(15,20)(21,97)(22,88)(23,99)(24,90)(25,81)(26,92)(27,83)(28,94)(29,85)(30,96)(31,87)(32,98)(33,89)(34,100)(35,91)(36,82)(37,93)(38,84)(39,95)(40,86)(41,46)(42,57)(43,48)(44,59)(45,50)(47,52)(49,54)(51,56)(53,58)(55,60)(61,106)(62,117)(63,108)(64,119)(65,110)(66,101)(67,112)(68,103)(69,114)(70,105)(71,116)(72,107)(73,118)(74,109)(75,120)(76,111)(77,102)(78,113)(79,104)(80,115)(121,136)(122,127)(123,138)(124,129)(125,140)(126,131)(128,133)(130,135)(132,137)(134,139)(141,146)(142,157)(143,148)(144,159)(145,150)(147,152)(149,154)(151,156)(153,158)(155,160) );
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)], [(1,70,157,100,130,110,41,39),(2,77,146,83,131,117,50,22),(3,64,155,86,132,104,59,25),(4,71,144,89,133,111,48,28),(5,78,153,92,134,118,57,31),(6,65,142,95,135,105,46,34),(7,72,151,98,136,112,55,37),(8,79,160,81,137,119,44,40),(9,66,149,84,138,106,53,23),(10,73,158,87,139,113,42,26),(11,80,147,90,140,120,51,29),(12,67,156,93,121,107,60,32),(13,74,145,96,122,114,49,35),(14,61,154,99,123,101,58,38),(15,68,143,82,124,108,47,21),(16,75,152,85,125,115,56,24),(17,62,141,88,126,102,45,27),(18,69,150,91,127,109,54,30),(19,76,159,94,128,116,43,33),(20,63,148,97,129,103,52,36)], [(1,6),(2,17),(3,8),(4,19),(5,10),(7,12),(9,14),(11,16),(13,18),(15,20),(21,97),(22,88),(23,99),(24,90),(25,81),(26,92),(27,83),(28,94),(29,85),(30,96),(31,87),(32,98),(33,89),(34,100),(35,91),(36,82),(37,93),(38,84),(39,95),(40,86),(41,46),(42,57),(43,48),(44,59),(45,50),(47,52),(49,54),(51,56),(53,58),(55,60),(61,106),(62,117),(63,108),(64,119),(65,110),(66,101),(67,112),(68,103),(69,114),(70,105),(71,116),(72,107),(73,118),(74,109),(75,120),(76,111),(77,102),(78,113),(79,104),(80,115),(121,136),(122,127),(123,138),(124,129),(125,140),(126,131),(128,133),(130,135),(132,137),(134,139),(141,146),(142,157),(143,148),(144,159),(145,150),(147,152),(149,154),(151,156),(153,158),(155,160)])
Matrix representation ►G ⊆ 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] >;
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 |
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
×
⋊
ℤ
𝔽
○
≀