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G = C62.32D4order 288 = 25·32

16th non-split extension by C62 of D4 acting via D4/C2=C22

metabelian, supersoluble, monomial

Aliases: C62.32D4, C23.9S32, C6.7(D6⋊C4), (C2×C6).57D12, C6.D42S3, C325(C23⋊C4), C62.35(C2×C4), (C22×C6).55D6, (C2×C62).8C22, C31(C23.6D6), C2.8(C6.D12), C22.3(C3⋊D12), C22.4(C6.D6), (C22×C3⋊S3)⋊1C4, (C2×C6).12(C4×S3), (C2×C3⋊Dic3)⋊1C4, (C3×C6.D4)⋊2C2, (C2×C6).13(C3⋊D4), (C2×C327D4).1C2, (C3×C6).40(C22⋊C4), SmallGroup(288,229)

Series: Derived Chief Lower central Upper central

C1C62 — C62.32D4
C1C3C32C3×C6C62C2×C62C3×C6.D4 — C62.32D4
C32C3×C6C62 — C62.32D4
C1C2C23

Generators and relations for C62.32D4
 G = < a,b,c,d | a6=b6=c4=1, d2=b3, ab=ba, cac-1=ab3, dad-1=a-1b3, cbc-1=dbd-1=b-1, dcd-1=a3b3c-1 >

Subgroups: 642 in 131 conjugacy classes, 32 normal (12 characteristic)
C1, C2, C2 [×4], C3 [×2], C3, C4 [×3], C22, C22 [×2], C22 [×3], S3 [×4], C6 [×2], C6 [×11], C2×C4 [×3], D4 [×2], C23, C23, C32, Dic3 [×6], C12 [×2], D6 [×8], C2×C6 [×6], C2×C6 [×7], C22⋊C4 [×2], C2×D4, C3⋊S3, C3×C6, C3×C6 [×3], C2×Dic3 [×5], C3⋊D4 [×8], C2×C12 [×2], C22×S3 [×3], C22×C6 [×2], C22×C6, C23⋊C4, C3×Dic3 [×2], C3⋊Dic3, C2×C3⋊S3 [×2], C62, C62 [×2], C62, C6.D4 [×2], C3×C22⋊C4 [×2], C2×C3⋊D4 [×3], C6×Dic3 [×2], C2×C3⋊Dic3, C327D4 [×2], C22×C3⋊S3, C2×C62, C23.6D6 [×2], C3×C6.D4 [×2], C2×C327D4, C62.32D4
Quotients: C1, C2 [×3], C4 [×2], C22, S3 [×2], C2×C4, D4 [×2], D6 [×2], C22⋊C4, C4×S3 [×2], D12 [×2], C3⋊D4 [×2], C23⋊C4, S32, D6⋊C4 [×2], C6.D6, C3⋊D12 [×2], C23.6D6 [×2], C6.D12, C62.32D4

Permutation representations of C62.32D4
On 24 points - transitive group 24T583
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 5 2 6 3 4)(7 11 9 10 8 12)(13 14 15 16 17 18)(19 24 23 22 21 20)
(1 13)(2 17)(3 15)(4 14)(5 18)(6 16)(7 19 10 22)(8 23 11 20)(9 21 12 24)
(1 19 6 22)(2 21 4 24)(3 23 5 20)(7 16 10 13)(8 18 11 15)(9 14 12 17)

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

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

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

G:=TransitiveGroup(24,583);

On 24 points - transitive group 24T615
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 16 3 18 5 14)(2 17 4 13 6 15)(7 22 11 20 9 24)(8 23 12 21 10 19)
(1 23 4 7)(2 11 5 21)(3 19 6 9)(8 13 24 16)(10 15 20 18)(12 17 22 14)
(1 4 18 15)(2 14 13 3)(5 6 16 17)(7 23 20 10)(8 9 21 22)(11 19 24 12)

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

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

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

G:=TransitiveGroup(24,615);

39 conjugacy classes

class 1 2A2B2C2D2E3A3B3C4A4B4C4D4E6A···6F6G···6Q12A···12H
order122222333444446···66···612···12
size112223622412121212362···24···412···12

39 irreducible representations

dim11111222222444444
type+++++++++++
imageC1C2C2C4C4S3D4D6C4×S3D12C3⋊D4C23⋊C4S32C6.D6C3⋊D12C23.6D6C62.32D4
kernelC62.32D4C3×C6.D4C2×C327D4C2×C3⋊Dic3C22×C3⋊S3C6.D4C62C22×C6C2×C6C2×C6C2×C6C32C23C22C22C3C1
# reps12122222444111244

Matrix representation of C62.32D4 in GL4(𝔽7) generated by

5232
1434
2233
0002
,
0251
0505
4416
0003
,
0361
1144
4353
3321
,
4526
5602
5544
3420
G:=sub<GL(4,GF(7))| [5,1,2,0,2,4,2,0,3,3,3,0,2,4,3,2],[0,0,4,0,2,5,4,0,5,0,1,0,1,5,6,3],[0,1,4,3,3,1,3,3,6,4,5,2,1,4,3,1],[4,5,5,3,5,6,5,4,2,0,4,2,6,2,4,0] >;

C62.32D4 in GAP, Magma, Sage, TeX

C_6^2._{32}D_4
% in TeX

G:=Group("C6^2.32D4");
// GroupNames label

G:=SmallGroup(288,229);
// by ID

G=gap.SmallGroup(288,229);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,28,141,92,219,675,346,1356,9414]);
// Polycyclic

G:=Group<a,b,c,d|a^6=b^6=c^4=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=a^3*b^3*c^-1>;
// generators/relations

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