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G = C6224D6order 432 = 24·33

5th semidirect product of C62 and D6 acting via D6/C3=C22

metabelian, supersoluble, monomial

Aliases: C6224D6, C3328(C2×D4), C3⋊Dic310D6, C3219(S3×D4), C327D48S3, C35(Dic3⋊D6), C339D412C2, (C3×C62)⋊6C22, (C32×C6).74C23, C222(C324D6), (C2×C6)⋊7S32, C35(S3×C3⋊D4), C6.103(C2×S32), (C2×C3⋊S3)⋊23D6, (C3×C3⋊S3)⋊12D4, C339(C2×C4)⋊6C2, C3⋊S35(C3⋊D4), (C22×C3⋊S3)⋊12S3, (C6×C3⋊S3)⋊20C22, (C3×C327D4)⋊6C2, C3215(C2×C3⋊D4), (C2×C324D6)⋊4C2, (C3×C3⋊Dic3)⋊7C22, (C3×C6).124(C22×S3), C2.10(C2×C324D6), (C2×C6×C3⋊S3)⋊7C2, SmallGroup(432,696)

Series: Derived Chief Lower central Upper central

C1C32×C6 — C6224D6
C1C3C32C33C32×C6C6×C3⋊S3C2×C324D6 — C6224D6
C33C32×C6 — C6224D6
C1C2C22

Generators and relations for C6224D6
 G = < a,b,c,d | a6=b6=c6=d2=1, ab=ba, cac-1=a-1b3, dad=ab3, cbc-1=dbd=b-1, dcd=c-1 >

Subgroups: 1640 in 290 conjugacy classes, 51 normal (21 characteristic)
C1, C2, C2 [×6], C3, C3 [×2], C3 [×4], C4 [×2], C22, C22 [×8], S3 [×16], C6, C6 [×2], C6 [×18], C2×C4, D4 [×4], C23 [×2], C32, C32 [×2], C32 [×4], Dic3 [×6], C12 [×2], D6 [×26], C2×C6, C2×C6 [×2], C2×C6 [×10], C2×D4, C3×S3 [×16], C3⋊S3 [×2], C3⋊S3 [×3], C3×C6, C3×C6 [×2], C3×C6 [×13], C4×S3 [×2], D12 [×2], C2×Dic3, C3⋊D4 [×10], C3×D4 [×2], C22×S3 [×6], C22×C6, C33, C3×Dic3 [×6], C3⋊Dic3 [×2], S32 [×6], S3×C6 [×20], C2×C3⋊S3 [×2], C2×C3⋊S3 [×2], C2×C3⋊S3 [×2], C62, C62 [×2], C62 [×4], S3×D4 [×2], C2×C3⋊D4, C3×C3⋊S3 [×2], C3×C3⋊S3 [×3], C32×C6, C32×C6, S3×Dic3 [×2], C6.D6, D6⋊S3 [×2], C3⋊D12 [×4], C3×C3⋊D4 [×6], C327D4 [×2], C2×S32 [×3], S3×C2×C6 [×3], C22×C3⋊S3, C3×C3⋊Dic3 [×2], C324D6 [×2], C6×C3⋊S3 [×2], C6×C3⋊S3 [×2], C6×C3⋊S3 [×2], C3×C62, S3×C3⋊D4 [×2], Dic3⋊D6, C339(C2×C4), C339D4 [×2], C3×C327D4 [×2], C2×C324D6, C2×C6×C3⋊S3, C6224D6
Quotients: C1, C2 [×7], C22 [×7], S3 [×3], D4 [×2], C23, D6 [×9], C2×D4, C3⋊D4 [×2], C22×S3 [×3], S32 [×3], S3×D4 [×2], C2×C3⋊D4, C2×S32 [×3], C324D6, S3×C3⋊D4 [×2], Dic3⋊D6, C2×C324D6, C6224D6

Permutation representations of C6224D6
On 24 points - transitive group 24T1284
Generators in S24
(1 2 3)(4 5 6)(7 8 9)(10 11 12)(13 14 15 16 17 18)(19 20 21 22 23 24)
(1 6 3 5 2 4)(7 12 8 10 9 11)(13 18 17 16 15 14)(19 20 21 22 23 24)
(1 20 2 22 3 24)(4 21 5 23 6 19)(7 18 8 14 9 16)(10 15 11 17 12 13)
(1 14)(2 18)(3 16)(4 13)(5 17)(6 15)(7 22)(8 20)(9 24)(10 19)(11 23)(12 21)

G:=sub<Sym(24)| (1,2,3)(4,5,6)(7,8,9)(10,11,12)(13,14,15,16,17,18)(19,20,21,22,23,24), (1,6,3,5,2,4)(7,12,8,10,9,11)(13,18,17,16,15,14)(19,20,21,22,23,24), (1,20,2,22,3,24)(4,21,5,23,6,19)(7,18,8,14,9,16)(10,15,11,17,12,13), (1,14)(2,18)(3,16)(4,13)(5,17)(6,15)(7,22)(8,20)(9,24)(10,19)(11,23)(12,21)>;

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), (1,6,3,5,2,4)(7,12,8,10,9,11)(13,18,17,16,15,14)(19,20,21,22,23,24), (1,20,2,22,3,24)(4,21,5,23,6,19)(7,18,8,14,9,16)(10,15,11,17,12,13), (1,14)(2,18)(3,16)(4,13)(5,17)(6,15)(7,22)(8,20)(9,24)(10,19)(11,23)(12,21) );

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)], [(1,6,3,5,2,4),(7,12,8,10,9,11),(13,18,17,16,15,14),(19,20,21,22,23,24)], [(1,20,2,22,3,24),(4,21,5,23,6,19),(7,18,8,14,9,16),(10,15,11,17,12,13)], [(1,14),(2,18),(3,16),(4,13),(5,17),(6,15),(7,22),(8,20),(9,24),(10,19),(11,23),(12,21)])

G:=TransitiveGroup(24,1284);

48 conjugacy classes

class 1 2A2B2C2D2E2F2G3A3B3C3D···3H4A4B6A···6E6F···6V6W6X6Y6Z6AA6AB12A12B
order122222223333···3446···66···66666661212
size112991818182224···418182···24···41818181836363636

48 irreducible representations

dim111111222222244444444
type++++++++++++++++
imageC1C2C2C2C2C2S3S3D4D6D6D6C3⋊D4S32S3×D4C2×S32C324D6S3×C3⋊D4Dic3⋊D6C2×C324D6C6224D6
kernelC6224D6C339(C2×C4)C339D4C3×C327D4C2×C324D6C2×C6×C3⋊S3C327D4C22×C3⋊S3C3×C3⋊S3C3⋊Dic3C2×C3⋊S3C62C3⋊S3C2×C6C32C6C22C3C3C2C1
# reps112211212243432324224

Matrix representation of C6224D6 in GL4(𝔽7) generated by

2042
5016
4416
0004
,
4632
6442
0030
0005
,
2424
1106
5214
2263
,
1454
3440
5214
3341
G:=sub<GL(4,GF(7))| [2,5,4,0,0,0,4,0,4,1,1,0,2,6,6,4],[4,6,0,0,6,4,0,0,3,4,3,0,2,2,0,5],[2,1,5,2,4,1,2,2,2,0,1,6,4,6,4,3],[1,3,5,3,4,4,2,3,5,4,1,4,4,0,4,1] >;

C6224D6 in GAP, Magma, Sage, TeX

C_6^2\rtimes_{24}D_6
% in TeX

G:=Group("C6^2:24D6");
// GroupNames label

G:=SmallGroup(432,696);
// by ID

G=gap.SmallGroup(432,696);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,135,1124,571,2028,14118]);
// 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*b^3,c*b*c^-1=d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations

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