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G = C40⋊D6order 480 = 25·3·5

17th semidirect product of C40 and D6 acting via D6/C3=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C4017D6, C2420D10, C12024C22, D153M4(2), C60.169C23, C3⋊C820D10, C815(S3×D5), C8⋊D55S3, C8⋊S35D5, C52C820D6, D6.6(C4×D5), C56(S3×M4(2)), C32(D5×M4(2)), (C8×D15)⋊13C2, (C4×D5).55D6, D152C88C2, (C4×S3).31D10, D10.17(C4×S3), D30.27(C2×C4), D30.C2.3C4, C1510(C2×M4(2)), (S3×Dic5).3C4, (D5×Dic3).3C4, C153C841C22, D6.Dic510C2, C30.35(C22×C4), Dic3.10(C4×D5), Dic5.22(C4×S3), C20.32D610C2, (S3×C20).31C22, C20.166(C22×S3), Dic15.34(C2×C4), (C4×D15).61C22, (D5×C12).55C22, C12.166(C22×D5), C6.4(C2×C4×D5), C2.7(C4×S3×D5), (C2×S3×D5).3C4, (C4×S3×D5).8C2, C10.35(S3×C2×C4), C4.139(C2×S3×D5), (C5×C8⋊S3)⋊6C2, (C3×C8⋊D5)⋊8C2, (C5×C3⋊C8)⋊20C22, (C6×D5).2(C2×C4), (S3×C10).17(C2×C4), (C3×C52C8)⋊20C22, (C3×Dic5).2(C2×C4), (C5×Dic3).18(C2×C4), SmallGroup(480,322)

Series: Derived Chief Lower central Upper central

C1C30 — C40⋊D6
C1C5C15C30C60D5×C12C4×S3×D5 — C40⋊D6
C15C30 — C40⋊D6
C1C4C8

Generators and relations for C40⋊D6
 G = < a,b,c | a40=b6=c2=1, bab-1=a29, cac=a9, cbc=b-1 >

Subgroups: 636 in 136 conjugacy classes, 52 normal (50 characteristic)
C1, C2, C2 [×4], C3, C4, C4 [×3], C22 [×5], C5, S3 [×3], C6, C6, C8, C8 [×3], C2×C4 [×6], C23, D5 [×3], C10, C10, Dic3, Dic3, C12, C12, D6, D6 [×3], C2×C6, C15, C2×C8 [×2], M4(2) [×4], C22×C4, Dic5, Dic5, C20, C20, D10, D10 [×3], C2×C10, C3⋊C8, C3⋊C8, C24, C24, C4×S3, C4×S3 [×3], C2×Dic3, C2×C12, C22×S3, C5×S3, C3×D5, D15 [×2], C30, C2×M4(2), C52C8, C52C8, C40, C40, C4×D5, C4×D5 [×3], C2×Dic5, C2×C20, C22×D5, S3×C8 [×2], C8⋊S3, C8⋊S3, C4.Dic3, C3×M4(2), S3×C2×C4, C5×Dic3, C3×Dic5, Dic15, C60, S3×D5 [×2], C6×D5, S3×C10, D30, C8×D5 [×2], C8⋊D5, C8⋊D5, C4.Dic5, C5×M4(2), C2×C4×D5, S3×M4(2), C5×C3⋊C8, C3×C52C8, C153C8, C120, D5×Dic3, S3×Dic5, D30.C2, D5×C12, S3×C20, C4×D15, C2×S3×D5, D5×M4(2), D152C8, C20.32D6, D6.Dic5, C3×C8⋊D5, C5×C8⋊S3, C8×D15, C4×S3×D5, C40⋊D6
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3, C2×C4 [×6], C23, D5, D6 [×3], M4(2) [×2], C22×C4, D10 [×3], C4×S3 [×2], C22×S3, C2×M4(2), C4×D5 [×2], C22×D5, S3×C2×C4, S3×D5, C2×C4×D5, S3×M4(2), C2×S3×D5, D5×M4(2), C4×S3×D5, C40⋊D6

Smallest permutation representation of C40⋊D6
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 87 66)(2 116 67 30 88 55)(3 105 68 19 89 44)(4 94 69 8 90 73)(5 83 70 37 91 62)(6 112 71 26 92 51)(7 101 72 15 93 80)(9 119 74 33 95 58)(10 108 75 22 96 47)(11 97 76)(12 86 77 40 98 65)(13 115 78 29 99 54)(14 104 79 18 100 43)(16 82 41 36 102 61)(17 111 42 25 103 50)(20 118 45 32 106 57)(21 107 46)(23 85 48 39 109 64)(24 114 49 28 110 53)(27 81 52 35 113 60)(31 117 56)(34 84 59 38 120 63)
(1 46)(2 55)(3 64)(4 73)(5 42)(6 51)(7 60)(8 69)(9 78)(10 47)(11 56)(12 65)(13 74)(14 43)(15 52)(16 61)(17 70)(18 79)(19 48)(20 57)(21 66)(22 75)(23 44)(24 53)(25 62)(26 71)(27 80)(28 49)(29 58)(30 67)(31 76)(32 45)(33 54)(34 63)(35 72)(36 41)(37 50)(38 59)(39 68)(40 77)(81 93)(82 102)(83 111)(84 120)(85 89)(86 98)(87 107)(88 116)(90 94)(91 103)(92 112)(95 99)(96 108)(97 117)(100 104)(101 113)(105 109)(106 118)(110 114)(115 119)

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

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

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

66 conjugacy classes

class 1 2A2B2C2D2E 3 4A4B4C4D4E4F5A5B6A6B8A8B8C8D8E8F8G8H10A10B10C10D12A12B12C15A15B20A20B20C20D20E20F24A24B24C24D30A30B40A40B40C40D40E40F40G40H60A60B60C60D120A···120H
order1222223444444556688888888101010101212121515202020202020242424243030404040404040404060606060120···120
size116101515211610151522220226610103030221212222044222212124420204444441212121244444···4

66 irreducible representations

dim1111111111112222222222222444444
type++++++++++++++++++
imageC1C2C2C2C2C2C2C2C4C4C4C4S3D5D6D6D6M4(2)D10D10D10C4×S3C4×S3C4×D5C4×D5S3×D5S3×M4(2)C2×S3×D5D5×M4(2)C4×S3×D5C40⋊D6
kernelC40⋊D6D152C8C20.32D6D6.Dic5C3×C8⋊D5C5×C8⋊S3C8×D15C4×S3×D5D5×Dic3S3×Dic5D30.C2C2×S3×D5C8⋊D5C8⋊S3C52C8C40C4×D5D15C3⋊C8C24C4×S3Dic5D10Dic3D6C8C5C4C3C2C1
# reps1111111122221211142222244222448

Matrix representation of C40⋊D6 in GL6(𝔽241)

24000000
02400000
002405100
0019019000
0000240177
00001781
,
239490000
17710000
001000
005124000
000010
0000128240
,
239490000
17720000
001000
005124000
00002400
00000240

G:=sub<GL(6,GF(241))| [240,0,0,0,0,0,0,240,0,0,0,0,0,0,240,190,0,0,0,0,51,190,0,0,0,0,0,0,240,178,0,0,0,0,177,1],[239,177,0,0,0,0,49,1,0,0,0,0,0,0,1,51,0,0,0,0,0,240,0,0,0,0,0,0,1,128,0,0,0,0,0,240],[239,177,0,0,0,0,49,2,0,0,0,0,0,0,1,51,0,0,0,0,0,240,0,0,0,0,0,0,240,0,0,0,0,0,0,240] >;

C40⋊D6 in GAP, Magma, Sage, TeX

C_{40}\rtimes D_6
% in TeX

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

G:=SmallGroup(480,322);
// by ID

G=gap.SmallGroup(480,322);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,422,219,58,80,1356,18822]);
// Polycyclic

G:=Group<a,b,c|a^40=b^6=c^2=1,b*a*b^-1=a^29,c*a*c=a^9,c*b*c=b^-1>;
// generators/relations

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