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G = C408D6order 480 = 25·3·5

8th semidirect product of C40 and D6 acting via D6/C3=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C408D6, D406S3, D202D6, C2411D10, D30.2D4, Dic62D10, D12.2D10, C12015C22, Dic15.2D4, C60.142C23, C86(S3×D5), (C3×D40)⋊9C2, C24⋊C23D5, C52(D8⋊S3), C6.31(D4×D5), C40⋊S35C2, C15⋊D811C2, C32(D40⋊C2), C156(C8⋊C22), C10.31(S3×D4), C30.12(C2×D4), D20⋊S38C2, C20⋊D610C2, C153C820C22, C30.D410C2, C2.9(C20⋊D6), (C3×D20)⋊17C22, C20.71(C22×S3), C12.71(C22×D5), (C5×Dic6)⋊15C22, (C5×D12).26C22, (C4×D15).31C22, C4.115(C2×S3×D5), (C5×C24⋊C2)⋊5C2, SmallGroup(480,334)

Series: Derived Chief Lower central Upper central

C1C60 — C408D6
C1C5C15C30C60C3×D20C20⋊D6 — C408D6
C15C30C60 — C408D6
C1C2C4C8

Generators and relations for C408D6
 G = < a,b,c | a40=b6=c2=1, bab-1=a-1, cac=a29, cbc=b-1 >

Subgroups: 956 in 136 conjugacy classes, 38 normal (all characteristic)
C1, C2, C2 [×4], C3, C4, C4 [×2], C22 [×6], C5, S3 [×2], C6, C6 [×2], C8, C8, C2×C4 [×2], D4 [×5], Q8, C23, D5 [×3], C10, C10, Dic3 [×2], C12, D6 [×4], C2×C6 [×2], C15, M4(2), D8 [×2], SD16 [×2], C2×D4, C4○D4, Dic5, C20, C20, D10 [×5], C2×C10, C3⋊C8, C24, Dic6, C4×S3, D12, C2×Dic3, C3⋊D4 [×2], C3×D4 [×2], C22×S3, C5×S3, C3×D5 [×2], D15, C30, C8⋊C22, C52C8, C40, C4×D5 [×2], D20 [×2], D20, C5⋊D4, C5×D4, C5×Q8, C22×D5, C8⋊S3, C24⋊C2, D4⋊S3, D4.S3, C3×D8, S3×D4, D42S3, C5×Dic3, Dic15, C60, S3×D5 [×2], C6×D5 [×2], S3×C10, D30, C8⋊D5, D40, D4⋊D5, Q8⋊D5, C5×SD16, D4×D5, Q82D5, D8⋊S3, C153C8, C120, D5×Dic3, C15⋊D4, C3⋊D20, C3×D20 [×2], C5×Dic6, C5×D12, C4×D15, C2×S3×D5, D40⋊C2, C15⋊D8, C30.D4, C3×D40, C5×C24⋊C2, C40⋊S3, D20⋊S3, C20⋊D6, C408D6
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], C23, D5, D6 [×3], C2×D4, D10 [×3], C22×S3, C8⋊C22, C22×D5, S3×D4, S3×D5, D4×D5, D8⋊S3, C2×S3×D5, D40⋊C2, C20⋊D6, C408D6

Smallest permutation representation of C408D6
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)
(1 118 61)(2 117 62 40 119 60)(3 116 63 39 120 59)(4 115 64 38 81 58)(5 114 65 37 82 57)(6 113 66 36 83 56)(7 112 67 35 84 55)(8 111 68 34 85 54)(9 110 69 33 86 53)(10 109 70 32 87 52)(11 108 71 31 88 51)(12 107 72 30 89 50)(13 106 73 29 90 49)(14 105 74 28 91 48)(15 104 75 27 92 47)(16 103 76 26 93 46)(17 102 77 25 94 45)(18 101 78 24 95 44)(19 100 79 23 96 43)(20 99 80 22 97 42)(21 98 41)
(1 61)(2 50)(3 79)(4 68)(5 57)(6 46)(7 75)(8 64)(9 53)(10 42)(11 71)(12 60)(13 49)(14 78)(15 67)(16 56)(17 45)(18 74)(19 63)(20 52)(21 41)(22 70)(23 59)(24 48)(25 77)(26 66)(27 55)(28 44)(29 73)(30 62)(31 51)(32 80)(33 69)(34 58)(35 47)(36 76)(37 65)(38 54)(39 43)(40 72)(81 85)(82 114)(83 103)(84 92)(86 110)(87 99)(89 117)(90 106)(91 95)(93 113)(94 102)(96 120)(97 109)(100 116)(101 105)(104 112)(107 119)(111 115)

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

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

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

48 conjugacy classes

class 1 2A2B2C2D2E 3 4A4B4C5A5B6A6B6C8A8B10A10B10C10D 12 15A15B20A20B20C20D24A24B30A30B40A40B40C40D60A60B60C60D120A···120H
order122222344455666881010101012151520202020242430304040404060606060120···120
size111220203022123022240404602224244444424244444444444444···4

48 irreducible representations

dim11111111222222222444444444
type+++++++++++++++++++++++
imageC1C2C2C2C2C2C2C2S3D4D4D5D6D6D10D10D10C8⋊C22S3×D4S3×D5D4×D5D8⋊S3C2×S3×D5D40⋊C2C20⋊D6C408D6
kernelC408D6C15⋊D8C30.D4C3×D40C5×C24⋊C2C40⋊S3D20⋊S3C20⋊D6D40Dic15D30C24⋊C2C40D20C24Dic6D12C15C10C8C6C5C4C3C2C1
# reps11111111111212222112222448

Matrix representation of C408D6 in GL8(𝔽241)

240189000000
5252000000
002401890000
0052520000
000023729128220
00002122957149
000011212900
00007222400
,
240024000000
5215210000
10000000
189240000000
00001000
00005124000
00001131495251
00003692188189
,
240024000000
5215210000
00100000
001892400000
00001000
00005124000
000000189190
0000005352

G:=sub<GL(8,GF(241))| [240,52,0,0,0,0,0,0,189,52,0,0,0,0,0,0,0,0,240,52,0,0,0,0,0,0,189,52,0,0,0,0,0,0,0,0,237,212,112,72,0,0,0,0,29,29,129,224,0,0,0,0,128,57,0,0,0,0,0,0,220,149,0,0],[240,52,1,189,0,0,0,0,0,1,0,240,0,0,0,0,240,52,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,51,113,36,0,0,0,0,0,240,149,92,0,0,0,0,0,0,52,188,0,0,0,0,0,0,51,189],[240,52,0,0,0,0,0,0,0,1,0,0,0,0,0,0,240,52,1,189,0,0,0,0,0,1,0,240,0,0,0,0,0,0,0,0,1,51,0,0,0,0,0,0,0,240,0,0,0,0,0,0,0,0,189,53,0,0,0,0,0,0,190,52] >;

C408D6 in GAP, Magma, Sage, TeX

C_{40}\rtimes_8D_6
% in TeX

G:=Group("C40:8D6");
// GroupNames label

G:=SmallGroup(480,334);
// by ID

G=gap.SmallGroup(480,334);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,254,303,142,675,346,80,1356,18822]);
// Polycyclic

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

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