Copied to
clipboard

G = C4×C40⋊C2order 320 = 26·5

Direct product of C4 and C40⋊C2

direct product, metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C4×C40⋊C2, C208SD16, C42.258D10, (C4×C8)⋊11D5, C811(C4×D5), C52(C4×SD16), (C4×C40)⋊16C2, C4033(C2×C4), C2.9(C4×D20), C406C428C2, (C4×D20).3C2, C10.36(C4×D4), (C2×C4).60D20, (C4×Dic10)⋊1C2, D20.26(C2×C4), C10.2(C4○D8), (C2×C8).285D10, (C2×C20).350D4, C10.3(C2×SD16), Dic1017(C2×C4), C22.27(C2×D20), D205C4.17C2, C20.216(C4○D4), C4.100(C4○D20), C2.1(D407C2), C20.44D443C2, (C2×C40).345C22, (C4×C20).325C22, C20.160(C22×C4), (C2×C20).720C23, (C2×D20).193C22, C4⋊Dic5.262C22, (C2×Dic10).212C22, C4.59(C2×C4×D5), C2.1(C2×C40⋊C2), (C2×C40⋊C2).10C2, (C2×C10).103(C2×D4), (C2×C4).663(C22×D5), SmallGroup(320,318)

Series: Derived Chief Lower central Upper central

C1C20 — C4×C40⋊C2
C1C5C10C2×C10C2×C20C2×D20C2×C40⋊C2 — C4×C40⋊C2
C5C10C20 — C4×C40⋊C2
C1C2×C4C42C4×C8

Generators and relations for C4×C40⋊C2
 G = < a,b,c | a4=b40=c2=1, ab=ba, ac=ca, cbc=b19 >

Subgroups: 518 in 122 conjugacy classes, 55 normal (39 characteristic)
C1, C2 [×3], C2 [×2], C4 [×2], C4 [×2], C4 [×5], C22, C22 [×4], C5, C8 [×2], C8, C2×C4 [×3], C2×C4 [×5], D4 [×3], Q8 [×3], C23, D5 [×2], C10 [×3], C42, C42, C22⋊C4, C4⋊C4 [×3], C2×C8 [×2], SD16 [×4], C22×C4, C2×D4, C2×Q8, Dic5 [×4], C20 [×2], C20 [×2], C20, D10 [×4], C2×C10, C4×C8, D4⋊C4, Q8⋊C4, C4.Q8, C4×D4, C4×Q8, C2×SD16, C40 [×2], C40, Dic10 [×2], Dic10, C4×D5 [×2], D20 [×2], D20, C2×Dic5 [×3], C2×C20 [×3], C22×D5, C4×SD16, C40⋊C2 [×4], C4×Dic5, C10.D4, C4⋊Dic5 [×2], D10⋊C4, C4×C20, C2×C40 [×2], C2×Dic10, C2×C4×D5, C2×D20, C20.44D4, C406C4, D205C4, C4×C40, C4×Dic10, C4×D20, C2×C40⋊C2, C4×C40⋊C2
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×2], C23, D5, SD16 [×2], C22×C4, C2×D4, C4○D4, D10 [×3], C4×D4, C2×SD16, C4○D8, C4×D5 [×2], D20 [×2], C22×D5, C4×SD16, C40⋊C2 [×2], C2×C4×D5, C2×D20, C4○D20, C4×D20, C2×C40⋊C2, D407C2, C4×C40⋊C2

Smallest permutation representation of C4×C40⋊C2
On 160 points
Generators in S160
(1 146 113 46)(2 147 114 47)(3 148 115 48)(4 149 116 49)(5 150 117 50)(6 151 118 51)(7 152 119 52)(8 153 120 53)(9 154 81 54)(10 155 82 55)(11 156 83 56)(12 157 84 57)(13 158 85 58)(14 159 86 59)(15 160 87 60)(16 121 88 61)(17 122 89 62)(18 123 90 63)(19 124 91 64)(20 125 92 65)(21 126 93 66)(22 127 94 67)(23 128 95 68)(24 129 96 69)(25 130 97 70)(26 131 98 71)(27 132 99 72)(28 133 100 73)(29 134 101 74)(30 135 102 75)(31 136 103 76)(32 137 104 77)(33 138 105 78)(34 139 106 79)(35 140 107 80)(36 141 108 41)(37 142 109 42)(38 143 110 43)(39 144 111 44)(40 145 112 45)
(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)
(2 20)(3 39)(4 18)(5 37)(6 16)(7 35)(8 14)(9 33)(10 12)(11 31)(13 29)(15 27)(17 25)(19 23)(22 40)(24 38)(26 36)(28 34)(30 32)(41 71)(42 50)(43 69)(44 48)(45 67)(47 65)(49 63)(51 61)(52 80)(53 59)(54 78)(55 57)(56 76)(58 74)(60 72)(62 70)(64 68)(73 79)(75 77)(81 105)(82 84)(83 103)(85 101)(86 120)(87 99)(88 118)(89 97)(90 116)(91 95)(92 114)(94 112)(96 110)(98 108)(100 106)(102 104)(107 119)(109 117)(111 115)(121 151)(122 130)(123 149)(124 128)(125 147)(127 145)(129 143)(131 141)(132 160)(133 139)(134 158)(135 137)(136 156)(138 154)(140 152)(142 150)(144 148)(153 159)(155 157)

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

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

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

92 conjugacy classes

class 1 2A2B2C2D2E4A4B4C4D4E4F4G4H4I···4N5A5B8A···8H10A···10F20A···20X40A···40AF
order122222444444444···4558···810···1020···2040···40
size111120201111222220···20222···22···22···22···2

92 irreducible representations

dim111111111222222222222
type+++++++++++++
imageC1C2C2C2C2C2C2C2C4D4D5SD16C4○D4D10D10C4○D8C4×D5D20C40⋊C2C4○D20D407C2
kernelC4×C40⋊C2C20.44D4C406C4D205C4C4×C40C4×Dic10C4×D20C2×C40⋊C2C40⋊C2C2×C20C4×C8C20C20C42C2×C8C10C8C2×C4C4C4C2
# reps11111111822422448816816

Matrix representation of C4×C40⋊C2 in GL4(𝔽41) generated by

32000
03200
00320
00032
,
6600
35100
003923
003331
,
1000
64000
0067
003635
G:=sub<GL(4,GF(41))| [32,0,0,0,0,32,0,0,0,0,32,0,0,0,0,32],[6,35,0,0,6,1,0,0,0,0,39,33,0,0,23,31],[1,6,0,0,0,40,0,0,0,0,6,36,0,0,7,35] >;

C4×C40⋊C2 in GAP, Magma, Sage, TeX

C_4\times C_{40}\rtimes C_2
% in TeX

G:=Group("C4xC40:C2");
// GroupNames label

G:=SmallGroup(320,318);
// by ID

G=gap.SmallGroup(320,318);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,253,120,58,1684,102,12550]);
// Polycyclic

G:=Group<a,b,c|a^4=b^40=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^19>;
// generators/relations

׿
×
𝔽