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G = C401D6order 480 = 25·3·5

1st semidirect product of C40 and D6 acting via D6/C3=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C401D6, C247D10, D6.4D20, D12013C2, C1206C22, Dic108D6, D20.19D6, D6017C22, C60.93C23, Dic3.6D20, C3⋊C81D10, C83(S3×D5), (S3×D20)⋊9C2, C40⋊C21S3, C8⋊S31D5, C51(Q83D6), C10.3(S3×D4), C30.7(C2×D4), C6.3(C2×D20), C2.8(S3×D20), C3⋊D4010C2, C31(C8⋊D10), C153(C8⋊C22), (S3×C10).1D4, (C4×S3).1D10, D60⋊C28C2, (C5×Dic3).1D4, C15⋊SD1610C2, C12.66(C22×D5), (S3×C20).24C22, C20.143(C22×S3), (C3×D20).21C22, (C3×Dic10)⋊13C22, C4.92(C2×S3×D5), (C5×C8⋊S3)⋊1C2, (C3×C40⋊C2)⋊1C2, (C5×C3⋊C8)⋊15C22, SmallGroup(480,329)

Series: Derived Chief Lower central Upper central

C1C60 — C401D6
C1C5C15C30C60C3×D20S3×D20 — C401D6
C15C30C60 — C401D6
C1C2C4C8

Generators and relations for C401D6
 G = < a,b,c | a40=b6=c2=1, bab-1=a19, cac=a-1, cbc=b-1 >

Subgroups: 1068 in 136 conjugacy classes, 40 normal (all characteristic)
C1, C2, C2 [×4], C3, C4, C4 [×2], C22 [×6], C5, S3 [×3], C6, C6, C8, C8, C2×C4 [×2], D4 [×5], Q8, C23, D5 [×3], C10, C10, Dic3, C12, C12, D6, D6 [×4], C2×C6, C15, M4(2), D8 [×2], SD16 [×2], C2×D4, C4○D4, Dic5, C20, C20, D10 [×5], C2×C10, C3⋊C8, C24, C4×S3, C4×S3, D12 [×3], C3⋊D4, C3×D4, C3×Q8, C22×S3, C5×S3, C3×D5, D15 [×2], C30, C8⋊C22, C40, C40, Dic10, C4×D5, D20, D20 [×3], C5⋊D4, C2×C20, C22×D5, C8⋊S3, D24, D4⋊S3, Q82S3, C3×SD16, S3×D4, Q83S3, C5×Dic3, C3×Dic5, C60, S3×D5 [×2], C6×D5, S3×C10, D30 [×2], C40⋊C2, C40⋊C2, D40 [×2], C5×M4(2), C2×D20, C4○D20, Q83D6, C5×C3⋊C8, C120, D30.C2, C3⋊D20, C5⋊D12, C3×Dic10, C3×D20, S3×C20, D60 [×2], C2×S3×D5, C8⋊D10, C3⋊D40, C15⋊SD16, C3×C40⋊C2, C5×C8⋊S3, D120, D60⋊C2, S3×D20, C401D6
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], C23, D5, D6 [×3], C2×D4, D10 [×3], C22×S3, C8⋊C22, D20 [×2], C22×D5, S3×D4, S3×D5, C2×D20, Q83D6, C2×S3×D5, C8⋊D10, S3×D20, C401D6

Smallest permutation representation of C401D6
On 120 points
Generators in S120
(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)
(1 84 50)(2 103 51 20 85 69)(3 82 52 39 86 48)(4 101 53 18 87 67)(5 120 54 37 88 46)(6 99 55 16 89 65)(7 118 56 35 90 44)(8 97 57 14 91 63)(9 116 58 33 92 42)(10 95 59 12 93 61)(11 114 60 31 94 80)(13 112 62 29 96 78)(15 110 64 27 98 76)(17 108 66 25 100 74)(19 106 68 23 102 72)(21 104 70)(22 83 71 40 105 49)(24 81 73 38 107 47)(26 119 75 36 109 45)(28 117 77 34 111 43)(30 115 79 32 113 41)
(1 50)(2 49)(3 48)(4 47)(5 46)(6 45)(7 44)(8 43)(9 42)(10 41)(11 80)(12 79)(13 78)(14 77)(15 76)(16 75)(17 74)(18 73)(19 72)(20 71)(21 70)(22 69)(23 68)(24 67)(25 66)(26 65)(27 64)(28 63)(29 62)(30 61)(31 60)(32 59)(33 58)(34 57)(35 56)(36 55)(37 54)(38 53)(39 52)(40 51)(81 87)(82 86)(83 85)(88 120)(89 119)(90 118)(91 117)(92 116)(93 115)(94 114)(95 113)(96 112)(97 111)(98 110)(99 109)(100 108)(101 107)(102 106)(103 105)

G:=sub<Sym(120)| (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), (1,84,50)(2,103,51,20,85,69)(3,82,52,39,86,48)(4,101,53,18,87,67)(5,120,54,37,88,46)(6,99,55,16,89,65)(7,118,56,35,90,44)(8,97,57,14,91,63)(9,116,58,33,92,42)(10,95,59,12,93,61)(11,114,60,31,94,80)(13,112,62,29,96,78)(15,110,64,27,98,76)(17,108,66,25,100,74)(19,106,68,23,102,72)(21,104,70)(22,83,71,40,105,49)(24,81,73,38,107,47)(26,119,75,36,109,45)(28,117,77,34,111,43)(30,115,79,32,113,41), (1,50)(2,49)(3,48)(4,47)(5,46)(6,45)(7,44)(8,43)(9,42)(10,41)(11,80)(12,79)(13,78)(14,77)(15,76)(16,75)(17,74)(18,73)(19,72)(20,71)(21,70)(22,69)(23,68)(24,67)(25,66)(26,65)(27,64)(28,63)(29,62)(30,61)(31,60)(32,59)(33,58)(34,57)(35,56)(36,55)(37,54)(38,53)(39,52)(40,51)(81,87)(82,86)(83,85)(88,120)(89,119)(90,118)(91,117)(92,116)(93,115)(94,114)(95,113)(96,112)(97,111)(98,110)(99,109)(100,108)(101,107)(102,106)(103,105)>;

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), (1,84,50)(2,103,51,20,85,69)(3,82,52,39,86,48)(4,101,53,18,87,67)(5,120,54,37,88,46)(6,99,55,16,89,65)(7,118,56,35,90,44)(8,97,57,14,91,63)(9,116,58,33,92,42)(10,95,59,12,93,61)(11,114,60,31,94,80)(13,112,62,29,96,78)(15,110,64,27,98,76)(17,108,66,25,100,74)(19,106,68,23,102,72)(21,104,70)(22,83,71,40,105,49)(24,81,73,38,107,47)(26,119,75,36,109,45)(28,117,77,34,111,43)(30,115,79,32,113,41), (1,50)(2,49)(3,48)(4,47)(5,46)(6,45)(7,44)(8,43)(9,42)(10,41)(11,80)(12,79)(13,78)(14,77)(15,76)(16,75)(17,74)(18,73)(19,72)(20,71)(21,70)(22,69)(23,68)(24,67)(25,66)(26,65)(27,64)(28,63)(29,62)(30,61)(31,60)(32,59)(33,58)(34,57)(35,56)(36,55)(37,54)(38,53)(39,52)(40,51)(81,87)(82,86)(83,85)(88,120)(89,119)(90,118)(91,117)(92,116)(93,115)(94,114)(95,113)(96,112)(97,111)(98,110)(99,109)(100,108)(101,107)(102,106)(103,105) );

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)], [(1,84,50),(2,103,51,20,85,69),(3,82,52,39,86,48),(4,101,53,18,87,67),(5,120,54,37,88,46),(6,99,55,16,89,65),(7,118,56,35,90,44),(8,97,57,14,91,63),(9,116,58,33,92,42),(10,95,59,12,93,61),(11,114,60,31,94,80),(13,112,62,29,96,78),(15,110,64,27,98,76),(17,108,66,25,100,74),(19,106,68,23,102,72),(21,104,70),(22,83,71,40,105,49),(24,81,73,38,107,47),(26,119,75,36,109,45),(28,117,77,34,111,43),(30,115,79,32,113,41)], [(1,50),(2,49),(3,48),(4,47),(5,46),(6,45),(7,44),(8,43),(9,42),(10,41),(11,80),(12,79),(13,78),(14,77),(15,76),(16,75),(17,74),(18,73),(19,72),(20,71),(21,70),(22,69),(23,68),(24,67),(25,66),(26,65),(27,64),(28,63),(29,62),(30,61),(31,60),(32,59),(33,58),(34,57),(35,56),(36,55),(37,54),(38,53),(39,52),(40,51),(81,87),(82,86),(83,85),(88,120),(89,119),(90,118),(91,117),(92,116),(93,115),(94,114),(95,113),(96,112),(97,111),(98,110),(99,109),(100,108),(101,107),(102,106),(103,105)])

54 conjugacy classes

class 1 2A2B2C2D2E 3 4A4B4C5A5B6A6B8A8B10A10B10C10D12A12B15A15B20A20B20C20D20E20F24A24B30A30B40A40B40C40D40E40F40G40H60A60B60C60D120A···120H
order1222223444556688101010101212151520202020202024243030404040404040404060606060120···120
size11620606022620222404122212124404422221212444444441212121244444···4

54 irreducible representations

dim1111111122222222222244444444
type++++++++++++++++++++++++++++
imageC1C2C2C2C2C2C2C2S3D4D4D5D6D6D6D10D10D10D20D20C8⋊C22S3×D4S3×D5Q83D6C2×S3×D5C8⋊D10S3×D20C401D6
kernelC401D6C3⋊D40C15⋊SD16C3×C40⋊C2C5×C8⋊S3D120D60⋊C2S3×D20C40⋊C2C5×Dic3S3×C10C8⋊S3C40Dic10D20C3⋊C8C24C4×S3Dic3D6C15C10C8C5C4C3C2C1
# reps1111111111121112224411222448

Matrix representation of C401D6 in GL8(𝔽241)

152000000
189189000000
001520000
001891890000
000002730
0000108003
0000721600214
00001611531330
,
240024000000
5215210000
10000000
189240000000
00001000
000024024000
00001780240239
00000001
,
240024000000
5215210000
00100000
001892400000
00001000
000024024000
0000811812
000016900240

G:=sub<GL(8,GF(241))| [1,189,0,0,0,0,0,0,52,189,0,0,0,0,0,0,0,0,1,189,0,0,0,0,0,0,52,189,0,0,0,0,0,0,0,0,0,108,72,161,0,0,0,0,27,0,160,153,0,0,0,0,3,0,0,133,0,0,0,0,0,3,214,0],[240,52,1,189,0,0,0,0,0,1,0,240,0,0,0,0,240,52,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,240,178,0,0,0,0,0,0,240,0,0,0,0,0,0,0,0,240,0,0,0,0,0,0,0,239,1],[240,52,0,0,0,0,0,0,0,1,0,0,0,0,0,0,240,52,1,189,0,0,0,0,0,1,0,240,0,0,0,0,0,0,0,0,1,240,81,169,0,0,0,0,0,240,18,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,2,240] >;

C401D6 in GAP, Magma, Sage, TeX

C_{40}\rtimes_1D_6
% in TeX

G:=Group("C40:1D6");
// GroupNames label

G:=SmallGroup(480,329);
// by ID

G=gap.SmallGroup(480,329);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,254,219,58,675,80,1356,18822]);
// Polycyclic

G:=Group<a,b,c|a^40=b^6=c^2=1,b*a*b^-1=a^19,c*a*c=a^-1,c*b*c=b^-1>;
// generators/relations

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