Copied to
clipboard

## G = C2×D20⋊5C4order 320 = 26·5

### Direct product of C2 and D20⋊5C4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C20 — C2×D20⋊5C4
 Chief series C1 — C5 — C10 — C20 — C2×C20 — C2×D20 — C22×D20 — C2×D20⋊5C4
 Lower central C5 — C10 — C20 — C2×D20⋊5C4
 Upper central C1 — C23 — C22×C4 — C22×C8

Generators and relations for C2×D205C4
G = < a,b,c,d | a2=b20=c2=d4=1, ab=ba, ac=ca, ad=da, cbc=dbd-1=b-1, dcd-1=b3c >

Subgroups: 1054 in 202 conjugacy classes, 79 normal (27 characteristic)
C1, C2, C2, C2, C4, C4, C4, C22, C22, C22, C5, C8, C2×C4, C2×C4, C2×C4, D4, C23, C23, D5, C10, C10, C4⋊C4, C2×C8, C2×C8, C22×C4, C22×C4, C2×D4, C24, Dic5, C20, C20, D10, C2×C10, C2×C10, D4⋊C4, C2×C4⋊C4, C22×C8, C22×D4, C40, D20, D20, C2×Dic5, C2×C20, C2×C20, C22×D5, C22×C10, C2×D4⋊C4, C4⋊Dic5, C4⋊Dic5, C2×C40, C2×C40, C2×D20, C2×D20, C22×Dic5, C22×C20, C23×D5, D205C4, C2×C4⋊Dic5, C22×C40, C22×D20, C2×D205C4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, D5, C22⋊C4, D8, SD16, C22×C4, C2×D4, D10, D4⋊C4, C2×C22⋊C4, C2×D8, C2×SD16, C4×D5, D20, C5⋊D4, C22×D5, C2×D4⋊C4, C40⋊C2, D40, D10⋊C4, C2×C4×D5, C2×D20, C2×C5⋊D4, D205C4, C2×C40⋊C2, C2×D40, C2×D10⋊C4, C2×D205C4

Smallest permutation representation of C2×D205C4
On 160 points
Generators in S160
(1 46)(2 47)(3 48)(4 49)(5 50)(6 51)(7 52)(8 53)(9 54)(10 55)(11 56)(12 57)(13 58)(14 59)(15 60)(16 41)(17 42)(18 43)(19 44)(20 45)(21 110)(22 111)(23 112)(24 113)(25 114)(26 115)(27 116)(28 117)(29 118)(30 119)(31 120)(32 101)(33 102)(34 103)(35 104)(36 105)(37 106)(38 107)(39 108)(40 109)(61 88)(62 89)(63 90)(64 91)(65 92)(66 93)(67 94)(68 95)(69 96)(70 97)(71 98)(72 99)(73 100)(74 81)(75 82)(76 83)(77 84)(78 85)(79 86)(80 87)(121 157)(122 158)(123 159)(124 160)(125 141)(126 142)(127 143)(128 144)(129 145)(130 146)(131 147)(132 148)(133 149)(134 150)(135 151)(136 152)(137 153)(138 154)(139 155)(140 156)
(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 45)(2 44)(3 43)(4 42)(5 41)(6 60)(7 59)(8 58)(9 57)(10 56)(11 55)(12 54)(13 53)(14 52)(15 51)(16 50)(17 49)(18 48)(19 47)(20 46)(21 107)(22 106)(23 105)(24 104)(25 103)(26 102)(27 101)(28 120)(29 119)(30 118)(31 117)(32 116)(33 115)(34 114)(35 113)(36 112)(37 111)(38 110)(39 109)(40 108)(61 96)(62 95)(63 94)(64 93)(65 92)(66 91)(67 90)(68 89)(69 88)(70 87)(71 86)(72 85)(73 84)(74 83)(75 82)(76 81)(77 100)(78 99)(79 98)(80 97)(121 157)(122 156)(123 155)(124 154)(125 153)(126 152)(127 151)(128 150)(129 149)(130 148)(131 147)(132 146)(133 145)(134 144)(135 143)(136 142)(137 141)(138 160)(139 159)(140 158)
(1 139 119 90)(2 138 120 89)(3 137 101 88)(4 136 102 87)(5 135 103 86)(6 134 104 85)(7 133 105 84)(8 132 106 83)(9 131 107 82)(10 130 108 81)(11 129 109 100)(12 128 110 99)(13 127 111 98)(14 126 112 97)(15 125 113 96)(16 124 114 95)(17 123 115 94)(18 122 116 93)(19 121 117 92)(20 140 118 91)(21 72 57 144)(22 71 58 143)(23 70 59 142)(24 69 60 141)(25 68 41 160)(26 67 42 159)(27 66 43 158)(28 65 44 157)(29 64 45 156)(30 63 46 155)(31 62 47 154)(32 61 48 153)(33 80 49 152)(34 79 50 151)(35 78 51 150)(36 77 52 149)(37 76 53 148)(38 75 54 147)(39 74 55 146)(40 73 56 145)

G:=sub<Sym(160)| (1,46)(2,47)(3,48)(4,49)(5,50)(6,51)(7,52)(8,53)(9,54)(10,55)(11,56)(12,57)(13,58)(14,59)(15,60)(16,41)(17,42)(18,43)(19,44)(20,45)(21,110)(22,111)(23,112)(24,113)(25,114)(26,115)(27,116)(28,117)(29,118)(30,119)(31,120)(32,101)(33,102)(34,103)(35,104)(36,105)(37,106)(38,107)(39,108)(40,109)(61,88)(62,89)(63,90)(64,91)(65,92)(66,93)(67,94)(68,95)(69,96)(70,97)(71,98)(72,99)(73,100)(74,81)(75,82)(76,83)(77,84)(78,85)(79,86)(80,87)(121,157)(122,158)(123,159)(124,160)(125,141)(126,142)(127,143)(128,144)(129,145)(130,146)(131,147)(132,148)(133,149)(134,150)(135,151)(136,152)(137,153)(138,154)(139,155)(140,156), (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), (1,45)(2,44)(3,43)(4,42)(5,41)(6,60)(7,59)(8,58)(9,57)(10,56)(11,55)(12,54)(13,53)(14,52)(15,51)(16,50)(17,49)(18,48)(19,47)(20,46)(21,107)(22,106)(23,105)(24,104)(25,103)(26,102)(27,101)(28,120)(29,119)(30,118)(31,117)(32,116)(33,115)(34,114)(35,113)(36,112)(37,111)(38,110)(39,109)(40,108)(61,96)(62,95)(63,94)(64,93)(65,92)(66,91)(67,90)(68,89)(69,88)(70,87)(71,86)(72,85)(73,84)(74,83)(75,82)(76,81)(77,100)(78,99)(79,98)(80,97)(121,157)(122,156)(123,155)(124,154)(125,153)(126,152)(127,151)(128,150)(129,149)(130,148)(131,147)(132,146)(133,145)(134,144)(135,143)(136,142)(137,141)(138,160)(139,159)(140,158), (1,139,119,90)(2,138,120,89)(3,137,101,88)(4,136,102,87)(5,135,103,86)(6,134,104,85)(7,133,105,84)(8,132,106,83)(9,131,107,82)(10,130,108,81)(11,129,109,100)(12,128,110,99)(13,127,111,98)(14,126,112,97)(15,125,113,96)(16,124,114,95)(17,123,115,94)(18,122,116,93)(19,121,117,92)(20,140,118,91)(21,72,57,144)(22,71,58,143)(23,70,59,142)(24,69,60,141)(25,68,41,160)(26,67,42,159)(27,66,43,158)(28,65,44,157)(29,64,45,156)(30,63,46,155)(31,62,47,154)(32,61,48,153)(33,80,49,152)(34,79,50,151)(35,78,51,150)(36,77,52,149)(37,76,53,148)(38,75,54,147)(39,74,55,146)(40,73,56,145)>;

G:=Group( (1,46)(2,47)(3,48)(4,49)(5,50)(6,51)(7,52)(8,53)(9,54)(10,55)(11,56)(12,57)(13,58)(14,59)(15,60)(16,41)(17,42)(18,43)(19,44)(20,45)(21,110)(22,111)(23,112)(24,113)(25,114)(26,115)(27,116)(28,117)(29,118)(30,119)(31,120)(32,101)(33,102)(34,103)(35,104)(36,105)(37,106)(38,107)(39,108)(40,109)(61,88)(62,89)(63,90)(64,91)(65,92)(66,93)(67,94)(68,95)(69,96)(70,97)(71,98)(72,99)(73,100)(74,81)(75,82)(76,83)(77,84)(78,85)(79,86)(80,87)(121,157)(122,158)(123,159)(124,160)(125,141)(126,142)(127,143)(128,144)(129,145)(130,146)(131,147)(132,148)(133,149)(134,150)(135,151)(136,152)(137,153)(138,154)(139,155)(140,156), (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), (1,45)(2,44)(3,43)(4,42)(5,41)(6,60)(7,59)(8,58)(9,57)(10,56)(11,55)(12,54)(13,53)(14,52)(15,51)(16,50)(17,49)(18,48)(19,47)(20,46)(21,107)(22,106)(23,105)(24,104)(25,103)(26,102)(27,101)(28,120)(29,119)(30,118)(31,117)(32,116)(33,115)(34,114)(35,113)(36,112)(37,111)(38,110)(39,109)(40,108)(61,96)(62,95)(63,94)(64,93)(65,92)(66,91)(67,90)(68,89)(69,88)(70,87)(71,86)(72,85)(73,84)(74,83)(75,82)(76,81)(77,100)(78,99)(79,98)(80,97)(121,157)(122,156)(123,155)(124,154)(125,153)(126,152)(127,151)(128,150)(129,149)(130,148)(131,147)(132,146)(133,145)(134,144)(135,143)(136,142)(137,141)(138,160)(139,159)(140,158), (1,139,119,90)(2,138,120,89)(3,137,101,88)(4,136,102,87)(5,135,103,86)(6,134,104,85)(7,133,105,84)(8,132,106,83)(9,131,107,82)(10,130,108,81)(11,129,109,100)(12,128,110,99)(13,127,111,98)(14,126,112,97)(15,125,113,96)(16,124,114,95)(17,123,115,94)(18,122,116,93)(19,121,117,92)(20,140,118,91)(21,72,57,144)(22,71,58,143)(23,70,59,142)(24,69,60,141)(25,68,41,160)(26,67,42,159)(27,66,43,158)(28,65,44,157)(29,64,45,156)(30,63,46,155)(31,62,47,154)(32,61,48,153)(33,80,49,152)(34,79,50,151)(35,78,51,150)(36,77,52,149)(37,76,53,148)(38,75,54,147)(39,74,55,146)(40,73,56,145) );

G=PermutationGroup([[(1,46),(2,47),(3,48),(4,49),(5,50),(6,51),(7,52),(8,53),(9,54),(10,55),(11,56),(12,57),(13,58),(14,59),(15,60),(16,41),(17,42),(18,43),(19,44),(20,45),(21,110),(22,111),(23,112),(24,113),(25,114),(26,115),(27,116),(28,117),(29,118),(30,119),(31,120),(32,101),(33,102),(34,103),(35,104),(36,105),(37,106),(38,107),(39,108),(40,109),(61,88),(62,89),(63,90),(64,91),(65,92),(66,93),(67,94),(68,95),(69,96),(70,97),(71,98),(72,99),(73,100),(74,81),(75,82),(76,83),(77,84),(78,85),(79,86),(80,87),(121,157),(122,158),(123,159),(124,160),(125,141),(126,142),(127,143),(128,144),(129,145),(130,146),(131,147),(132,148),(133,149),(134,150),(135,151),(136,152),(137,153),(138,154),(139,155),(140,156)], [(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)], [(1,45),(2,44),(3,43),(4,42),(5,41),(6,60),(7,59),(8,58),(9,57),(10,56),(11,55),(12,54),(13,53),(14,52),(15,51),(16,50),(17,49),(18,48),(19,47),(20,46),(21,107),(22,106),(23,105),(24,104),(25,103),(26,102),(27,101),(28,120),(29,119),(30,118),(31,117),(32,116),(33,115),(34,114),(35,113),(36,112),(37,111),(38,110),(39,109),(40,108),(61,96),(62,95),(63,94),(64,93),(65,92),(66,91),(67,90),(68,89),(69,88),(70,87),(71,86),(72,85),(73,84),(74,83),(75,82),(76,81),(77,100),(78,99),(79,98),(80,97),(121,157),(122,156),(123,155),(124,154),(125,153),(126,152),(127,151),(128,150),(129,149),(130,148),(131,147),(132,146),(133,145),(134,144),(135,143),(136,142),(137,141),(138,160),(139,159),(140,158)], [(1,139,119,90),(2,138,120,89),(3,137,101,88),(4,136,102,87),(5,135,103,86),(6,134,104,85),(7,133,105,84),(8,132,106,83),(9,131,107,82),(10,130,108,81),(11,129,109,100),(12,128,110,99),(13,127,111,98),(14,126,112,97),(15,125,113,96),(16,124,114,95),(17,123,115,94),(18,122,116,93),(19,121,117,92),(20,140,118,91),(21,72,57,144),(22,71,58,143),(23,70,59,142),(24,69,60,141),(25,68,41,160),(26,67,42,159),(27,66,43,158),(28,65,44,157),(29,64,45,156),(30,63,46,155),(31,62,47,154),(32,61,48,153),(33,80,49,152),(34,79,50,151),(35,78,51,150),(36,77,52,149),(37,76,53,148),(38,75,54,147),(39,74,55,146),(40,73,56,145)]])

92 conjugacy classes

 class 1 2A ··· 2G 2H 2I 2J 2K 4A 4B 4C 4D 4E 4F 4G 4H 5A 5B 8A ··· 8H 10A ··· 10N 20A ··· 20P 40A ··· 40AF order 1 2 ··· 2 2 2 2 2 4 4 4 4 4 4 4 4 5 5 8 ··· 8 10 ··· 10 20 ··· 20 40 ··· 40 size 1 1 ··· 1 20 20 20 20 2 2 2 2 20 20 20 20 2 2 2 ··· 2 2 ··· 2 2 ··· 2 2 ··· 2

92 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 type + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C4 D4 D4 D5 D8 SD16 D10 D10 C4×D5 D20 C5⋊D4 D20 C40⋊C2 D40 kernel C2×D20⋊5C4 D20⋊5C4 C2×C4⋊Dic5 C22×C40 C22×D20 C2×D20 C2×C20 C22×C10 C22×C8 C2×C10 C2×C10 C2×C8 C22×C4 C2×C4 C2×C4 C2×C4 C23 C22 C22 # reps 1 4 1 1 1 8 3 1 2 4 4 4 2 8 4 8 4 16 16

Matrix representation of C2×D205C4 in GL4(𝔽41) generated by

 40 0 0 0 0 40 0 0 0 0 40 0 0 0 0 40
,
 1 0 0 0 0 1 0 0 0 0 11 39 0 0 16 27
,
 1 0 0 0 0 1 0 0 0 0 9 25 0 0 5 32
,
 32 0 0 0 0 1 0 0 0 0 39 3 0 0 40 2
G:=sub<GL(4,GF(41))| [40,0,0,0,0,40,0,0,0,0,40,0,0,0,0,40],[1,0,0,0,0,1,0,0,0,0,11,16,0,0,39,27],[1,0,0,0,0,1,0,0,0,0,9,5,0,0,25,32],[32,0,0,0,0,1,0,0,0,0,39,40,0,0,3,2] >;

C2×D205C4 in GAP, Magma, Sage, TeX

C_2\times D_{20}\rtimes_5C_4
% in TeX

G:=Group("C2xD20:5C4");
// GroupNames label

G:=SmallGroup(320,739);
// by ID

G=gap.SmallGroup(320,739);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,254,142,1123,136,12550]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^20=c^2=d^4=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c=d*b*d^-1=b^-1,d*c*d^-1=b^3*c>;
// generators/relations

׿
×
𝔽