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G = C6223D6order 432 = 24·33

4th semidirect product of C62 and D6 acting via D6/C3=C22

metabelian, supersoluble, monomial, rational

Aliases: C6223D6, (S3×C6)⋊6D6, C3326(C2×D4), C3⋊Dic316D6, (C3×Dic3)⋊5D6, C3218(S3×D4), C327D46S3, C33⋊C24D4, C33(Dic3⋊D6), C338D410C2, C337D410C2, (C3×C62)⋊5C22, (C32×C6).64C23, (C32×Dic3)⋊8C22, (C2×C6)⋊6S32, C33(D4×C3⋊S3), C6.74(C2×S32), D64(C2×C3⋊S3), (C2×C3⋊S3)⋊17D6, (C3×C3⋊D4)⋊4S3, C225(S3×C3⋊S3), C338(C2×C4)⋊9C2, C3⋊D42(C3⋊S3), (S3×C3×C6)⋊14C22, Dic32(C2×C3⋊S3), (C6×C3⋊S3)⋊11C22, (C32×C3⋊D4)⋊8C2, (C3×C327D4)⋊4C2, C6.27(C22×C3⋊S3), (C3×C3⋊Dic3)⋊6C22, (C3×C6).114(C22×S3), (C22×C33⋊C2)⋊3C2, (C2×C33⋊C2)⋊11C22, (C2×S3×C3⋊S3)⋊10C2, (C2×C6)⋊5(C2×C3⋊S3), C2.27(C2×S3×C3⋊S3), SmallGroup(432,686)

Series: Derived Chief Lower central Upper central

C1C32×C6 — C6223D6
C1C3C32C33C32×C6S3×C3×C6C2×S3×C3⋊S3 — C6223D6
C33C32×C6 — C6223D6
C1C2C22

Generators and relations for C6223D6
 G = < a,b,c,d | a6=b6=c6=d2=1, ab=ba, cac-1=a-1b3, dad=a-1, cbc-1=dbd=b-1, dcd=c-1 >

Subgroups: 3296 in 452 conjugacy classes, 70 normal (32 characteristic)
C1, C2, C2 [×6], C3, C3 [×4], C3 [×4], C4 [×2], C22, C22 [×8], S3 [×36], C6, C6 [×4], C6 [×22], C2×C4, D4 [×4], C23 [×2], C32, C32 [×4], C32 [×4], Dic3, Dic3 [×4], C12 [×5], D6, D6 [×58], C2×C6, C2×C6 [×4], C2×C6 [×9], C2×D4, C3×S3 [×8], C3⋊S3 [×32], C3×C6, C3×C6 [×4], C3×C6 [×18], C4×S3 [×5], D12 [×5], C3⋊D4, C3⋊D4 [×9], C3×D4 [×5], C22×S3 [×14], C33, C3×Dic3 [×4], C3×Dic3 [×4], C3⋊Dic3, C3×C12, S32 [×8], S3×C6 [×4], S3×C6 [×4], C2×C3⋊S3, C2×C3⋊S3 [×46], C62, C62 [×4], C62 [×5], S3×D4 [×5], S3×C32, C3×C3⋊S3, C33⋊C2 [×2], C33⋊C2, C32×C6, C32×C6, C6.D6 [×4], C3⋊D12 [×8], C3×C3⋊D4 [×4], C3×C3⋊D4 [×4], C4×C3⋊S3, C12⋊S3, C327D4, C327D4, D4×C32, C2×S32 [×4], C22×C3⋊S3 [×10], C32×Dic3, C3×C3⋊Dic3, S3×C3⋊S3 [×2], S3×C3×C6, C6×C3⋊S3, C2×C33⋊C2 [×2], C2×C33⋊C2 [×2], C3×C62, Dic3⋊D6 [×4], D4×C3⋊S3, C338(C2×C4), C337D4, C338D4, C32×C3⋊D4, C3×C327D4, C2×S3×C3⋊S3, C22×C33⋊C2, C6223D6
Quotients: C1, C2 [×7], C22 [×7], S3 [×5], D4 [×2], C23, D6 [×15], C2×D4, C3⋊S3, C22×S3 [×5], S32 [×4], C2×C3⋊S3 [×3], S3×D4 [×5], C2×S32 [×4], C22×C3⋊S3, S3×C3⋊S3, Dic3⋊D6 [×4], D4×C3⋊S3, C2×S3×C3⋊S3, C6223D6

Smallest permutation representation of C6223D6
On 36 points
Generators in S36
(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)
(1 13 9 10 4 16)(2 14 7 11 5 17)(3 15 8 12 6 18)(19 34 26 22 31 29)(20 35 27 23 32 30)(21 36 28 24 33 25)
(1 30 2 26 3 28)(4 35 5 31 6 33)(7 19 8 21 9 23)(10 27 11 29 12 25)(13 32 14 34 15 36)(16 20 17 22 18 24)
(1 2)(4 7)(5 9)(6 8)(10 11)(13 17)(14 16)(15 18)(19 33)(20 32)(21 31)(22 36)(23 35)(24 34)(25 29)(26 28)

G:=sub<Sym(36)| (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), (1,13,9,10,4,16)(2,14,7,11,5,17)(3,15,8,12,6,18)(19,34,26,22,31,29)(20,35,27,23,32,30)(21,36,28,24,33,25), (1,30,2,26,3,28)(4,35,5,31,6,33)(7,19,8,21,9,23)(10,27,11,29,12,25)(13,32,14,34,15,36)(16,20,17,22,18,24), (1,2)(4,7)(5,9)(6,8)(10,11)(13,17)(14,16)(15,18)(19,33)(20,32)(21,31)(22,36)(23,35)(24,34)(25,29)(26,28)>;

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), (1,13,9,10,4,16)(2,14,7,11,5,17)(3,15,8,12,6,18)(19,34,26,22,31,29)(20,35,27,23,32,30)(21,36,28,24,33,25), (1,30,2,26,3,28)(4,35,5,31,6,33)(7,19,8,21,9,23)(10,27,11,29,12,25)(13,32,14,34,15,36)(16,20,17,22,18,24), (1,2)(4,7)(5,9)(6,8)(10,11)(13,17)(14,16)(15,18)(19,33)(20,32)(21,31)(22,36)(23,35)(24,34)(25,29)(26,28) );

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)], [(1,13,9,10,4,16),(2,14,7,11,5,17),(3,15,8,12,6,18),(19,34,26,22,31,29),(20,35,27,23,32,30),(21,36,28,24,33,25)], [(1,30,2,26,3,28),(4,35,5,31,6,33),(7,19,8,21,9,23),(10,27,11,29,12,25),(13,32,14,34,15,36),(16,20,17,22,18,24)], [(1,2),(4,7),(5,9),(6,8),(10,11),(13,17),(14,16),(15,18),(19,33),(20,32),(21,31),(22,36),(23,35),(24,34),(25,29),(26,28)])

51 conjugacy classes

class 1 2A2B2C2D2E2F2G3A···3E3F3G3H3I4A4B6A···6E6F···6V6W6X6Y6Z6AA12A12B12C12D12E
order122222223···33333446···66···6666661212121212
size1126182727542···244446182···24···412121212361212121236

51 irreducible representations

dim11111111222222224444
type++++++++++++++++++++
imageC1C2C2C2C2C2C2C2S3S3D4D6D6D6D6D6S32S3×D4C2×S32Dic3⋊D6
kernelC6223D6C338(C2×C4)C337D4C338D4C32×C3⋊D4C3×C327D4C2×S3×C3⋊S3C22×C33⋊C2C3×C3⋊D4C327D4C33⋊C2C3×Dic3C3⋊Dic3S3×C6C2×C3⋊S3C62C2×C6C32C6C3
# reps11111111412414154548

Matrix representation of C6223D6 in GL8(𝔽13)

10000000
312000000
000120000
00110000
000011200
00001000
000000120
000000012
,
120000000
012000000
00010000
0012120000
000001200
000011200
00000010
00000001
,
18000000
012000000
00100000
0012120000
00000100
00001000
000000121
000000120
,
10000000
01000000
00100000
0012120000
00000100
00001000
000000120
000000121

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

C6223D6 in GAP, Magma, Sage, TeX

C_6^2\rtimes_{23}D_6
% in TeX

G:=Group("C6^2:23D6");
// GroupNames label

G:=SmallGroup(432,686);
// by ID

G=gap.SmallGroup(432,686);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,254,135,571,2028,14118]);
// Polycyclic

G:=Group<a,b,c,d|a^6=b^6=c^6=d^2=1,a*b=b*a,c*a*c^-1=a^-1*b^3,d*a*d=a^-1,c*b*c^-1=d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations

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