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G = C4021(C2×C4)  order 320 = 26·5

11st semidirect product of C40 and C2×C4 acting via C2×C4/C2=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C86(C4×D5), C4021(C2×C4), C40⋊C27C4, C408C47C2, C2.D812D5, (C2×C8).67D10, C10.84(C4×D4), C4⋊C4.172D10, D20.23(C2×C4), C22.92(D4×D5), Dic1016(C2×C4), Dic53Q87C2, D206C4.7C2, D208C4.7C2, C20.44(C4○D4), C2.6(D8⋊D5), C10.Q1623C2, C55(SD16⋊C4), C10.44(C8⋊C22), (C2×C20).305C23, C20.110(C22×C4), (C2×C40).145C22, C4.12(Q82D5), C2.6(Q16⋊D5), (C2×Dic5).226D4, (C2×D20).89C22, C2.14(D208C4), C10.73(C8.C22), (C4×Dic5).43C22, (C2×Dic10).97C22, C4.45(C2×C4×D5), (C5×C2.D8)⋊9C2, (C2×C40⋊C2).7C2, (C2×C10).310(C2×D4), (C5×C4⋊C4).98C22, (C2×C52C8).74C22, (C2×C4).408(C22×D5), SmallGroup(320,516)

Series: Derived Chief Lower central Upper central

C1C20 — C4021(C2×C4)
C1C5C10C2×C10C2×C20C2×D20C2×C40⋊C2 — C4021(C2×C4)
C5C10C20 — C4021(C2×C4)
C1C22C2×C4C2.D8

Generators and relations for C4021(C2×C4)
 G = < a,b,c | a40=b2=c4=1, bab=a19, cac-1=a31, bc=cb >

Subgroups: 502 in 120 conjugacy classes, 49 normal (37 characteristic)
C1, C2 [×3], C2 [×2], C4 [×2], C4 [×6], C22, C22 [×4], C5, C8 [×2], C8, C2×C4, C2×C4 [×7], D4 [×3], Q8 [×3], C23, D5 [×2], C10 [×3], C42 [×2], C22⋊C4, C4⋊C4 [×2], C4⋊C4, C2×C8, C2×C8, SD16 [×4], C22×C4, C2×D4, C2×Q8, Dic5 [×4], C20 [×2], C20 [×2], D10 [×4], C2×C10, C8⋊C4, D4⋊C4, Q8⋊C4, C2.D8, C4×D4, C4×Q8, C2×SD16, C52C8, C40 [×2], Dic10 [×2], Dic10, C4×D5 [×2], D20 [×2], D20, C2×Dic5 [×2], C2×Dic5, C2×C20, C2×C20 [×2], C22×D5, SD16⋊C4, C40⋊C2 [×4], C2×C52C8, C4×Dic5, C4×Dic5, C10.D4, D10⋊C4, C5×C4⋊C4 [×2], C2×C40, C2×Dic10, C2×C4×D5, C2×D20, D206C4, C10.Q16, C408C4, C5×C2.D8, Dic53Q8, D208C4, C2×C40⋊C2, C4021(C2×C4)
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×2], C23, D5, C22×C4, C2×D4, C4○D4, D10 [×3], C4×D4, C8⋊C22, C8.C22, C4×D5 [×2], C22×D5, SD16⋊C4, C2×C4×D5, D4×D5, Q82D5, D208C4, D8⋊D5, Q16⋊D5, C4021(C2×C4)

Smallest permutation representation of C4021(C2×C4)
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)
(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 65)(42 44)(43 63)(45 61)(46 80)(47 59)(48 78)(49 57)(50 76)(51 55)(52 74)(54 72)(56 70)(58 68)(60 66)(62 64)(67 79)(69 77)(71 75)(81 83)(82 102)(84 100)(85 119)(86 98)(87 117)(88 96)(89 115)(90 94)(91 113)(93 111)(95 109)(97 107)(99 105)(101 103)(104 120)(106 118)(108 116)(110 114)(122 140)(123 159)(124 138)(125 157)(126 136)(127 155)(128 134)(129 153)(130 132)(131 151)(133 149)(135 147)(137 145)(139 143)(142 160)(144 158)(146 156)(148 154)(150 152)
(1 141 92 53)(2 132 93 44)(3 123 94 75)(4 154 95 66)(5 145 96 57)(6 136 97 48)(7 127 98 79)(8 158 99 70)(9 149 100 61)(10 140 101 52)(11 131 102 43)(12 122 103 74)(13 153 104 65)(14 144 105 56)(15 135 106 47)(16 126 107 78)(17 157 108 69)(18 148 109 60)(19 139 110 51)(20 130 111 42)(21 121 112 73)(22 152 113 64)(23 143 114 55)(24 134 115 46)(25 125 116 77)(26 156 117 68)(27 147 118 59)(28 138 119 50)(29 129 120 41)(30 160 81 72)(31 151 82 63)(32 142 83 54)(33 133 84 45)(34 124 85 76)(35 155 86 67)(36 146 87 58)(37 137 88 49)(38 128 89 80)(39 159 90 71)(40 150 91 62)

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

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

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

50 conjugacy classes

class 1 2A2B2C2D2E4A4B4C4D4E4F4G4H4I4J4K4L5A5B8A8B8C8D10A···10F20A20B20C20D20E···20L40A···40H
order12222244444444444455888810···102020202020···2040···40
size11112020224444101010102020224420202···244448···84···4

50 irreducible representations

dim111111111222222444444
type+++++++++++++-++
imageC1C2C2C2C2C2C2C2C4D4D5C4○D4D10D10C4×D5C8⋊C22C8.C22Q82D5D4×D5D8⋊D5Q16⋊D5
kernelC4021(C2×C4)D206C4C10.Q16C408C4C5×C2.D8Dic53Q8D208C4C2×C40⋊C2C40⋊C2C2×Dic5C2.D8C20C4⋊C4C2×C8C8C10C10C4C22C2C2
# reps111111118222428112244

Matrix representation of C4021(C2×C4) in GL6(𝔽41)

24250000
13170000
002313910
0010313110
0013600
0053600
,
100000
3400000
001000
00344000
0010400
00344071
,
3200000
1490000
0015142513
0027402816
00702627
0007141

G:=sub<GL(6,GF(41))| [24,13,0,0,0,0,25,17,0,0,0,0,0,0,2,10,1,5,0,0,31,31,36,36,0,0,39,31,0,0,0,0,10,10,0,0],[1,3,0,0,0,0,0,40,0,0,0,0,0,0,1,34,1,34,0,0,0,40,0,40,0,0,0,0,40,7,0,0,0,0,0,1],[32,14,0,0,0,0,0,9,0,0,0,0,0,0,15,27,7,0,0,0,14,40,0,7,0,0,25,28,26,14,0,0,13,16,27,1] >;

C4021(C2×C4) in GAP, Magma, Sage, TeX

C_{40}\rtimes_{21}(C_2\times C_4)
% in TeX

G:=Group("C40:21(C2xC4)");
// GroupNames label

G:=SmallGroup(320,516);
// by ID

G=gap.SmallGroup(320,516);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,253,120,219,58,1684,438,102,12550]);
// Polycyclic

G:=Group<a,b,c|a^40=b^2=c^4=1,b*a*b=a^19,c*a*c^-1=a^31,b*c=c*b>;
// generators/relations

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