direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×C4.D20, C42⋊40D10, (C2×C4).98D20, C4.43(C2×D20), (C2×C42)⋊10D5, (C4×C20)⋊51C22, C20.286(C2×D4), (C2×C20).389D4, C10.4(C22×D4), C2.6(C22×D20), C10⋊1(C4.4D4), (C2×C10).20C24, (C22×D20).8C2, C22.65(C2×D20), (C2×C20).781C23, (C22×Dic10)⋊4C2, (C22×C4).439D10, (C2×Dic5).4C23, D10⋊C4⋊39C22, (C22×D5).2C23, C22.63(C23×D5), (C2×Dic10)⋊46C22, (C2×D20).210C22, C22.69(C4○D20), (C23×D5).28C22, C23.316(C22×D5), (C22×C10).382C23, (C22×C20).504C22, (C22×Dic5).74C22, (C2×C4×C20)⋊9C2, C5⋊1(C2×C4.4D4), C2.9(C2×C4○D20), C10.7(C2×C4○D4), (C2×C10).171(C2×D4), (C2×D10⋊C4)⋊12C2, (C2×C10).97(C4○D4), (C2×C4).649(C22×D5), SmallGroup(320,1148)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 1374 in 330 conjugacy classes, 127 normal (15 characteristic)
C1, C2, C2 [×6], C2 [×4], C4 [×4], C4 [×8], C22, C22 [×6], C22 [×20], C5, C2×C4 [×10], C2×C4 [×12], D4 [×8], Q8 [×8], C23, C23 [×16], D5 [×4], C10, C10 [×6], C42 [×4], C22⋊C4 [×16], C22×C4, C22×C4 [×2], C22×C4 [×2], C2×D4 [×8], C2×Q8 [×8], C24 [×2], Dic5 [×4], C20 [×4], C20 [×4], D10 [×20], C2×C10, C2×C10 [×6], C2×C42, C2×C22⋊C4 [×4], C4.4D4 [×8], C22×D4, C22×Q8, Dic10 [×8], D20 [×8], C2×Dic5 [×4], C2×Dic5 [×4], C2×C20 [×10], C2×C20 [×4], C22×D5 [×4], C22×D5 [×12], C22×C10, C2×C4.4D4, D10⋊C4 [×16], C4×C20 [×4], C2×Dic10 [×4], C2×Dic10 [×4], C2×D20 [×4], C2×D20 [×4], C22×Dic5 [×2], C22×C20, C22×C20 [×2], C23×D5 [×2], C4.D20 [×8], C2×D10⋊C4 [×4], C2×C4×C20, C22×Dic10, C22×D20, C2×C4.D20
Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], D5, C2×D4 [×6], C4○D4 [×4], C24, D10 [×7], C4.4D4 [×4], C22×D4, C2×C4○D4 [×2], D20 [×4], C22×D5 [×7], C2×C4.4D4, C2×D20 [×6], C4○D20 [×4], C23×D5, C4.D20 [×4], C22×D20, C2×C4○D20 [×2], C2×C4.D20
Generators and relations
G = < a,b,c,d | a2=b4=c20=1, d2=b2, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1, dcd-1=b2c-1 >
►On 160 points
Generators in S160
(1 109)(2 110)(3 111)(4 112)(5 113)(6 114)(7 115)(8 116)(9 117)(10 118)(11 119)(12 120)(13 101)(14 102)(15 103)(16 104)(17 105)(18 106)(19 107)(20 108)(21 48)(22 49)(23 50)(24 51)(25 52)(26 53)(27 54)(28 55)(29 56)(30 57)(31 58)(32 59)(33 60)(34 41)(35 42)(36 43)(37 44)(38 45)(39 46)(40 47)(61 100)(62 81)(63 82)(64 83)(65 84)(66 85)(67 86)(68 87)(69 88)(70 89)(71 90)(72 91)(73 92)(74 93)(75 94)(76 95)(77 96)(78 97)(79 98)(80 99)(121 150)(122 151)(123 152)(124 153)(125 154)(126 155)(127 156)(128 157)(129 158)(130 159)(131 160)(132 141)(133 142)(134 143)(135 144)(136 145)(137 146)(138 147)(139 148)(140 149)
(1 132 44 72)(2 133 45 73)(3 134 46 74)(4 135 47 75)(5 136 48 76)(6 137 49 77)(7 138 50 78)(8 139 51 79)(9 140 52 80)(10 121 53 61)(11 122 54 62)(12 123 55 63)(13 124 56 64)(14 125 57 65)(15 126 58 66)(16 127 59 67)(17 128 60 68)(18 129 41 69)(19 130 42 70)(20 131 43 71)(21 95 113 145)(22 96 114 146)(23 97 115 147)(24 98 116 148)(25 99 117 149)(26 100 118 150)(27 81 119 151)(28 82 120 152)(29 83 101 153)(30 84 102 154)(31 85 103 155)(32 86 104 156)(33 87 105 157)(34 88 106 158)(35 89 107 159)(36 90 108 160)(37 91 109 141)(38 92 110 142)(39 93 111 143)(40 94 112 144)
(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 90 44 160)(2 159 45 89)(3 88 46 158)(4 157 47 87)(5 86 48 156)(6 155 49 85)(7 84 50 154)(8 153 51 83)(9 82 52 152)(10 151 53 81)(11 100 54 150)(12 149 55 99)(13 98 56 148)(14 147 57 97)(15 96 58 146)(16 145 59 95)(17 94 60 144)(18 143 41 93)(19 92 42 142)(20 141 43 91)(21 127 113 67)(22 66 114 126)(23 125 115 65)(24 64 116 124)(25 123 117 63)(26 62 118 122)(27 121 119 61)(28 80 120 140)(29 139 101 79)(30 78 102 138)(31 137 103 77)(32 76 104 136)(33 135 105 75)(34 74 106 134)(35 133 107 73)(36 72 108 132)(37 131 109 71)(38 70 110 130)(39 129 111 69)(40 68 112 128)
G:=sub<Sym(160)| (1,109)(2,110)(3,111)(4,112)(5,113)(6,114)(7,115)(8,116)(9,117)(10,118)(11,119)(12,120)(13,101)(14,102)(15,103)(16,104)(17,105)(18,106)(19,107)(20,108)(21,48)(22,49)(23,50)(24,51)(25,52)(26,53)(27,54)(28,55)(29,56)(30,57)(31,58)(32,59)(33,60)(34,41)(35,42)(36,43)(37,44)(38,45)(39,46)(40,47)(61,100)(62,81)(63,82)(64,83)(65,84)(66,85)(67,86)(68,87)(69,88)(70,89)(71,90)(72,91)(73,92)(74,93)(75,94)(76,95)(77,96)(78,97)(79,98)(80,99)(121,150)(122,151)(123,152)(124,153)(125,154)(126,155)(127,156)(128,157)(129,158)(130,159)(131,160)(132,141)(133,142)(134,143)(135,144)(136,145)(137,146)(138,147)(139,148)(140,149), (1,132,44,72)(2,133,45,73)(3,134,46,74)(4,135,47,75)(5,136,48,76)(6,137,49,77)(7,138,50,78)(8,139,51,79)(9,140,52,80)(10,121,53,61)(11,122,54,62)(12,123,55,63)(13,124,56,64)(14,125,57,65)(15,126,58,66)(16,127,59,67)(17,128,60,68)(18,129,41,69)(19,130,42,70)(20,131,43,71)(21,95,113,145)(22,96,114,146)(23,97,115,147)(24,98,116,148)(25,99,117,149)(26,100,118,150)(27,81,119,151)(28,82,120,152)(29,83,101,153)(30,84,102,154)(31,85,103,155)(32,86,104,156)(33,87,105,157)(34,88,106,158)(35,89,107,159)(36,90,108,160)(37,91,109,141)(38,92,110,142)(39,93,111,143)(40,94,112,144), (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), (1,90,44,160)(2,159,45,89)(3,88,46,158)(4,157,47,87)(5,86,48,156)(6,155,49,85)(7,84,50,154)(8,153,51,83)(9,82,52,152)(10,151,53,81)(11,100,54,150)(12,149,55,99)(13,98,56,148)(14,147,57,97)(15,96,58,146)(16,145,59,95)(17,94,60,144)(18,143,41,93)(19,92,42,142)(20,141,43,91)(21,127,113,67)(22,66,114,126)(23,125,115,65)(24,64,116,124)(25,123,117,63)(26,62,118,122)(27,121,119,61)(28,80,120,140)(29,139,101,79)(30,78,102,138)(31,137,103,77)(32,76,104,136)(33,135,105,75)(34,74,106,134)(35,133,107,73)(36,72,108,132)(37,131,109,71)(38,70,110,130)(39,129,111,69)(40,68,112,128)>;
G:=Group( (1,109)(2,110)(3,111)(4,112)(5,113)(6,114)(7,115)(8,116)(9,117)(10,118)(11,119)(12,120)(13,101)(14,102)(15,103)(16,104)(17,105)(18,106)(19,107)(20,108)(21,48)(22,49)(23,50)(24,51)(25,52)(26,53)(27,54)(28,55)(29,56)(30,57)(31,58)(32,59)(33,60)(34,41)(35,42)(36,43)(37,44)(38,45)(39,46)(40,47)(61,100)(62,81)(63,82)(64,83)(65,84)(66,85)(67,86)(68,87)(69,88)(70,89)(71,90)(72,91)(73,92)(74,93)(75,94)(76,95)(77,96)(78,97)(79,98)(80,99)(121,150)(122,151)(123,152)(124,153)(125,154)(126,155)(127,156)(128,157)(129,158)(130,159)(131,160)(132,141)(133,142)(134,143)(135,144)(136,145)(137,146)(138,147)(139,148)(140,149), (1,132,44,72)(2,133,45,73)(3,134,46,74)(4,135,47,75)(5,136,48,76)(6,137,49,77)(7,138,50,78)(8,139,51,79)(9,140,52,80)(10,121,53,61)(11,122,54,62)(12,123,55,63)(13,124,56,64)(14,125,57,65)(15,126,58,66)(16,127,59,67)(17,128,60,68)(18,129,41,69)(19,130,42,70)(20,131,43,71)(21,95,113,145)(22,96,114,146)(23,97,115,147)(24,98,116,148)(25,99,117,149)(26,100,118,150)(27,81,119,151)(28,82,120,152)(29,83,101,153)(30,84,102,154)(31,85,103,155)(32,86,104,156)(33,87,105,157)(34,88,106,158)(35,89,107,159)(36,90,108,160)(37,91,109,141)(38,92,110,142)(39,93,111,143)(40,94,112,144), (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), (1,90,44,160)(2,159,45,89)(3,88,46,158)(4,157,47,87)(5,86,48,156)(6,155,49,85)(7,84,50,154)(8,153,51,83)(9,82,52,152)(10,151,53,81)(11,100,54,150)(12,149,55,99)(13,98,56,148)(14,147,57,97)(15,96,58,146)(16,145,59,95)(17,94,60,144)(18,143,41,93)(19,92,42,142)(20,141,43,91)(21,127,113,67)(22,66,114,126)(23,125,115,65)(24,64,116,124)(25,123,117,63)(26,62,118,122)(27,121,119,61)(28,80,120,140)(29,139,101,79)(30,78,102,138)(31,137,103,77)(32,76,104,136)(33,135,105,75)(34,74,106,134)(35,133,107,73)(36,72,108,132)(37,131,109,71)(38,70,110,130)(39,129,111,69)(40,68,112,128) );
G=PermutationGroup([(1,109),(2,110),(3,111),(4,112),(5,113),(6,114),(7,115),(8,116),(9,117),(10,118),(11,119),(12,120),(13,101),(14,102),(15,103),(16,104),(17,105),(18,106),(19,107),(20,108),(21,48),(22,49),(23,50),(24,51),(25,52),(26,53),(27,54),(28,55),(29,56),(30,57),(31,58),(32,59),(33,60),(34,41),(35,42),(36,43),(37,44),(38,45),(39,46),(40,47),(61,100),(62,81),(63,82),(64,83),(65,84),(66,85),(67,86),(68,87),(69,88),(70,89),(71,90),(72,91),(73,92),(74,93),(75,94),(76,95),(77,96),(78,97),(79,98),(80,99),(121,150),(122,151),(123,152),(124,153),(125,154),(126,155),(127,156),(128,157),(129,158),(130,159),(131,160),(132,141),(133,142),(134,143),(135,144),(136,145),(137,146),(138,147),(139,148),(140,149)], [(1,132,44,72),(2,133,45,73),(3,134,46,74),(4,135,47,75),(5,136,48,76),(6,137,49,77),(7,138,50,78),(8,139,51,79),(9,140,52,80),(10,121,53,61),(11,122,54,62),(12,123,55,63),(13,124,56,64),(14,125,57,65),(15,126,58,66),(16,127,59,67),(17,128,60,68),(18,129,41,69),(19,130,42,70),(20,131,43,71),(21,95,113,145),(22,96,114,146),(23,97,115,147),(24,98,116,148),(25,99,117,149),(26,100,118,150),(27,81,119,151),(28,82,120,152),(29,83,101,153),(30,84,102,154),(31,85,103,155),(32,86,104,156),(33,87,105,157),(34,88,106,158),(35,89,107,159),(36,90,108,160),(37,91,109,141),(38,92,110,142),(39,93,111,143),(40,94,112,144)], [(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)], [(1,90,44,160),(2,159,45,89),(3,88,46,158),(4,157,47,87),(5,86,48,156),(6,155,49,85),(7,84,50,154),(8,153,51,83),(9,82,52,152),(10,151,53,81),(11,100,54,150),(12,149,55,99),(13,98,56,148),(14,147,57,97),(15,96,58,146),(16,145,59,95),(17,94,60,144),(18,143,41,93),(19,92,42,142),(20,141,43,91),(21,127,113,67),(22,66,114,126),(23,125,115,65),(24,64,116,124),(25,123,117,63),(26,62,118,122),(27,121,119,61),(28,80,120,140),(29,139,101,79),(30,78,102,138),(31,137,103,77),(32,76,104,136),(33,135,105,75),(34,74,106,134),(35,133,107,73),(36,72,108,132),(37,131,109,71),(38,70,110,130),(39,129,111,69),(40,68,112,128)])
Matrix representation ►G ⊆ GL5(𝔽41)
| 40 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 40 | 0 | 0 | 0 | 0 |
| 0 | 30 | 32 | 0 | 0 |
| 0 | 9 | 11 | 0 | 0 |
| 0 | 0 | 0 | 11 | 19 |
| 0 | 0 | 0 | 13 | 30 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 22 | 9 | 0 | 0 |
| 0 | 32 | 0 | 0 | 0 |
| 0 | 0 | 0 | 3 | 21 |
| 0 | 0 | 0 | 23 | 24 |
| 40 | 0 | 0 | 0 | 0 |
| 0 | 38 | 17 | 0 | 0 |
| 0 | 38 | 3 | 0 | 0 |
| 0 | 0 | 0 | 0 | 22 |
| 0 | 0 | 0 | 13 | 0 |
G:=sub<GL(5,GF(41))| [40,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[40,0,0,0,0,0,30,9,0,0,0,32,11,0,0,0,0,0,11,13,0,0,0,19,30],[1,0,0,0,0,0,22,32,0,0,0,9,0,0,0,0,0,0,3,23,0,0,0,21,24],[40,0,0,0,0,0,38,38,0,0,0,17,3,0,0,0,0,0,0,13,0,0,0,22,0] >;
92 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 4A | ··· | 4L | 4M | 4N | 4O | 4P | 5A | 5B | 10A | ··· | 10N | 20A | ··· | 20AV |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 10 | ··· | 10 | 20 | ··· | 20 |
| size | 1 | 1 | ··· | 1 | 20 | 20 | 20 | 20 | 2 | ··· | 2 | 20 | 20 | 20 | 20 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
92 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | D4 | D5 | C4○D4 | D10 | D10 | D20 | C4○D20 |
| kernel | C2×C4.D20 | C4.D20 | C2×D10⋊C4 | C2×C4×C20 | C22×Dic10 | C22×D20 | C2×C20 | C2×C42 | C2×C10 | C42 | C22×C4 | C2×C4 | C22 |
| # reps | 1 | 8 | 4 | 1 | 1 | 1 | 4 | 2 | 8 | 8 | 6 | 16 | 32 |
In GAP, Magma, Sage, TeX
C_2\times C_4.D_{20} % in TeX
G:=Group("C2xC4.D20"); // GroupNames label
G:=SmallGroup(320,1148);
// by ID
G=gap.SmallGroup(320,1148);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,758,100,675,136,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^4=c^20=1,d^2=b^2,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=b^2*c^-1>;
// generators/relations
×
⋊
ℤ
𝔽
○
≀