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G = C62.90D6  order 432 = 24·33

38th non-split extension by C62 of D6 acting via D6/C3=C22

metabelian, supersoluble, monomial

Aliases: C62.90D6, (S3×C6).15D6, C33⋊7D4⋊4C2, C33⋊4Q8⋊8C2, C33⋊23(C4○D4), C33⋊15D4⋊1C2, C3⋊Dic3.51D6, C3⋊6(D6.3D6), (C3×Dic3).17D6, C3⋊3(C12.D6), C32⋊25(C4○D12), (C32×C6).53C23, (C3×C62).24C22, C32⋊16(D4⋊2S3), C33⋊5C4.11C22, (C32×Dic3).17C22, C6.63(C2×S32), (C2×C6).10S32, (C3×C3⋊D4)⋊1S3, D6.3(C2×C3⋊S3), C33⋊8(C2×C4)⋊7C2, C3⋊D4⋊3(C3⋊S3), C22.2(S3×C3⋊S3), (S3×C3⋊Dic3)⋊10C2, (C6×C3⋊Dic3)⋊10C2, (C2×C3⋊Dic3)⋊10S3, (C32×C3⋊D4)⋊5C2, C6.16(C22×C3⋊S3), (S3×C3×C6).16C22, Dic3.5(C2×C3⋊S3), (C3×C6).108(C22×S3), (C3×C3⋊Dic3).44C22, (C2×C33⋊C2).9C22, C2.18(C2×S3×C3⋊S3), (C2×C6).20(C2×C3⋊S3), SmallGroup(432,675)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C32×C6 — C62.90D6
Chief series
C1 — C3 — C32 — C33 — C32×C6 — S3×C3×C6 — S3×C3⋊Dic3 — C62.90D6
Lower central
C33 — C32×C6 — C62.90D6
Upper central
C1 — C2 — C22

Generators and relations for C62.90D6
 G = < a,b,c,d | a6=b6=c6=1, d2=b3, ab=ba, cac-1=ab3, dad-1=a-1b3, cbc-1=dbd-1=b-1, dcd-1=c-1 >

Subgroups: 1752 in 304 conjugacy classes, 68 normal (32 characteristic)
C1, C2, C2, C3, C3, C3, C4, C22, C22, S3, C6, C6, C6, C2×C4, D4, Q8, C32, C32, C32, Dic3, Dic3, C12, D6, D6, C2×C6, C2×C6, C2×C6, C4○D4, C3×S3, C3⋊S3, C3×C6, C3×C6, C3×C6, Dic6, C4×S3, D12, C2×Dic3, C3⋊D4, C3⋊D4, C2×C12, C3×D4, C33, C3×Dic3, C3×Dic3, C3⋊Dic3, C3⋊Dic3, C3×C12, S3×C6, C2×C3⋊S3, C62, C62, C62, C4○D12, D4⋊2S3, S3×C32, C33⋊C2, C32×C6, C32×C6, S3×Dic3, C6.D6, C3⋊D12, C32⋊2Q8, C6×Dic3, C3×C3⋊D4, C32⋊4Q8, C4×C3⋊S3, C2×C3⋊Dic3, C2×C3⋊Dic3, C32⋊7D4, D4×C32, C32×Dic3, C3×C3⋊Dic3, C33⋊5C4, S3×C3×C6, C2×C33⋊C2, C3×C62, D6.3D6, C12.D6, S3×C3⋊Dic3, C33⋊8(C2×C4), C33⋊7D4, C33⋊4Q8, C32×C3⋊D4, C6×C3⋊Dic3, C33⋊15D4, C62.90D6
Quotients: C1, C2, C22, S3, C23, D6, C4○D4, C3⋊S3, C22×S3, S32, C2×C3⋊S3, C4○D12, D4⋊2S3, C2×S32, C22×C3⋊S3, S3×C3⋊S3, D6.3D6, C12.D6, C2×S3×C3⋊S3, C62.90D6

Smallest permutation representation of C62.90D6
►On 72 points
Generators in S72
(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)

(1 20 31 36 27 24)(2 21 32 34 25 22)(3 19 33 35 26 23)(4 29 16 12 8 15)(5 30 17 10 9 13)(6 28 18 11 7 14)(37 53 47 40 50 44)(38 54 48 41 51 45)(39 49 43 42 52 46)(55 64 69 58 61 72)(56 65 70 59 62 67)(57 66 71 60 63 68)

(1 58 31 64 27 72)(2 56 32 62 25 70)(3 60 33 66 26 68)(4 46 8 49 16 42)(5 44 9 53 17 40)(6 48 7 51 18 38)(10 47 30 50 13 37)(11 45 28 54 14 41)(12 43 29 52 15 39)(19 71 35 57 23 63)(20 69 36 55 24 61)(21 67 34 59 22 65)

(1 45 36 48)(2 47 34 44)(3 43 35 46)(4 60 12 57)(5 56 10 59)(6 58 11 55)(7 72 28 69)(8 68 29 71)(9 70 30 67)(13 65 17 62)(14 61 18 64)(15 63 16 66)(19 49 26 52)(20 51 27 54)(21 53 25 50)(22 40 32 37)(23 42 33 39)(24 38 31 41)
 
G:=sub<Sym(72)| (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), (1,20,31,36,27,24)(2,21,32,34,25,22)(3,19,33,35,26,23)(4,29,16,12,8,15)(5,30,17,10,9,13)(6,28,18,11,7,14)(37,53,47,40,50,44)(38,54,48,41,51,45)(39,49,43,42,52,46)(55,64,69,58,61,72)(56,65,70,59,62,67)(57,66,71,60,63,68), (1,58,31,64,27,72)(2,56,32,62,25,70)(3,60,33,66,26,68)(4,46,8,49,16,42)(5,44,9,53,17,40)(6,48,7,51,18,38)(10,47,30,50,13,37)(11,45,28,54,14,41)(12,43,29,52,15,39)(19,71,35,57,23,63)(20,69,36,55,24,61)(21,67,34,59,22,65), (1,45,36,48)(2,47,34,44)(3,43,35,46)(4,60,12,57)(5,56,10,59)(6,58,11,55)(7,72,28,69)(8,68,29,71)(9,70,30,67)(13,65,17,62)(14,61,18,64)(15,63,16,66)(19,49,26,52)(20,51,27,54)(21,53,25,50)(22,40,32,37)(23,42,33,39)(24,38,31,41)>;
 
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), (1,20,31,36,27,24)(2,21,32,34,25,22)(3,19,33,35,26,23)(4,29,16,12,8,15)(5,30,17,10,9,13)(6,28,18,11,7,14)(37,53,47,40,50,44)(38,54,48,41,51,45)(39,49,43,42,52,46)(55,64,69,58,61,72)(56,65,70,59,62,67)(57,66,71,60,63,68), (1,58,31,64,27,72)(2,56,32,62,25,70)(3,60,33,66,26,68)(4,46,8,49,16,42)(5,44,9,53,17,40)(6,48,7,51,18,38)(10,47,30,50,13,37)(11,45,28,54,14,41)(12,43,29,52,15,39)(19,71,35,57,23,63)(20,69,36,55,24,61)(21,67,34,59,22,65), (1,45,36,48)(2,47,34,44)(3,43,35,46)(4,60,12,57)(5,56,10,59)(6,58,11,55)(7,72,28,69)(8,68,29,71)(9,70,30,67)(13,65,17,62)(14,61,18,64)(15,63,16,66)(19,49,26,52)(20,51,27,54)(21,53,25,50)(22,40,32,37)(23,42,33,39)(24,38,31,41) );
 
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)], [(1,20,31,36,27,24),(2,21,32,34,25,22),(3,19,33,35,26,23),(4,29,16,12,8,15),(5,30,17,10,9,13),(6,28,18,11,7,14),(37,53,47,40,50,44),(38,54,48,41,51,45),(39,49,43,42,52,46),(55,64,69,58,61,72),(56,65,70,59,62,67),(57,66,71,60,63,68)], [(1,58,31,64,27,72),(2,56,32,62,25,70),(3,60,33,66,26,68),(4,46,8,49,16,42),(5,44,9,53,17,40),(6,48,7,51,18,38),(10,47,30,50,13,37),(11,45,28,54,14,41),(12,43,29,52,15,39),(19,71,35,57,23,63),(20,69,36,55,24,61),(21,67,34,59,22,65)], [(1,45,36,48),(2,47,34,44),(3,43,35,46),(4,60,12,57),(5,56,10,59),(6,58,11,55),(7,72,28,69),(8,68,29,71),(9,70,30,67),(13,65,17,62),(14,61,18,64),(15,63,16,66),(19,49,26,52),(20,51,27,54),(21,53,25,50),(22,40,32,37),(23,42,33,39),(24,38,31,41)]])
 

54 conjugacy classes

class 1 2A2B2C2D3A···3E3F3G3H3I4A4B4C4D4E6A···6G6H···6W6X6Y6Z6AA12A12B12C12D12E12F12G12H
order122223···33333444446···66···666661212121212121212
size1126542···2444469918542···24···4121212121212121218181818

54 irreducible representations

dim11111111222222224444
type+++++++++++++++-+
imageC1C2C2C2C2C2C2C2S3S3D6D6D6D6C4○D4C4○D12S32D4⋊2S3C2×S32D6.3D6
kernelC62.90D6S3×C3⋊Dic3C33⋊8(C2×C4)C33⋊7D4C33⋊4Q8C32×C3⋊D4C6×C3⋊Dic3C33⋊15D4C3×C3⋊D4C2×C3⋊Dic3C3×Dic3C3⋊Dic3S3×C6C62C33C32C2×C6C32C6C3
# reps11111111414245244448

Matrix representation of C62.90D6 ►in GL8(𝔽13)

10000000
012000000
00230000
0012120000
000012000
000001200
00000001
0000001212
,
120000000
012000000
00100000
00010000
000012100
000012000
00000010
00000001
,
01000000
10000000
00100000
00010000
000001200
000012000
00000001
0000001212
,
05000000
50000000
00440000
00690000
000001200
000012000
00000001
00000010

G:=sub<GL(8,GF(13))| [1,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,2,12,0,0,0,0,0,0,3,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,1,12],[12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,12,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,1,12],[0,5,0,0,0,0,0,0,5,0,0,0,0,0,0,0,0,0,4,6,0,0,0,0,0,0,4,9,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0] >;
 

C62.90D6 in GAP, Magma, Sage, TeX 

C_6^2._{90}D_6
 
% in TeX
 
G:=Group("C6^2.90D6");
 
// GroupNames label
 
G:=SmallGroup(432,675);
 
// by ID
 
G=gap.SmallGroup(432,675);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,64,254,571,2028,14118]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^6=b^6=c^6=1,d^2=b^3,a*b=b*a,c*a*c^-1=a*b^3,d*a*d^-1=a^-1*b^3,c*b*c^-1=d*b*d^-1=b^-1,d*c*d^-1=c^-1>;
 
// generators/relations
 

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