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G = C62.96D6order 432 = 24·33

44th non-split extension by C62 of D6 acting via D6/C3=C22

metabelian, supersoluble, monomial

Aliases: C62.96D6, C327D47S3, C339D45C2, C335Q88C2, C3326(C4○D4), C3⋊Dic3.22D6, C37(D6.3D6), C35(D6.4D6), C3218(C4○D12), (C32×C6).71C23, (C3×C62).34C22, C22.(C324D6), C3218(D42S3), (C2×C6).12S32, C6.100(C2×S32), C339(C2×C4)⋊5C2, (C2×C3⋊S3).20D6, (C2×C3⋊Dic3)⋊13S3, (C6×C3⋊Dic3)⋊11C2, (C3×C327D4)⋊5C2, (C6×C3⋊S3).31C22, C2.7(C2×C324D6), (C3×C6).121(C22×S3), (C3×C3⋊Dic3).25C22, SmallGroup(432,693)

Series: Derived Chief Lower central Upper central

C1C32×C6 — C62.96D6
C1C3C32C33C32×C6C6×C3⋊S3C339(C2×C4) — C62.96D6
C33C32×C6 — C62.96D6
C1C2C22

Generators and relations for C62.96D6
 G = < a,b,c,d | a6=b6=1, c6=d2=b3, ab=ba, cac-1=a-1, dad-1=ab3, cbc-1=dbd-1=b-1, dcd-1=c5 >

Subgroups: 1032 in 218 conjugacy classes, 47 normal (21 characteristic)
C1, C2, C2 [×3], C3, C3 [×2], C3 [×4], C4 [×4], C22, C22 [×2], S3 [×6], C6, C6 [×2], C6 [×15], C2×C4 [×3], D4 [×3], Q8, C32, C32 [×2], C32 [×4], Dic3 [×12], C12 [×4], D6 [×6], C2×C6, C2×C6 [×2], C2×C6 [×6], C4○D4, C3×S3 [×6], C3⋊S3 [×2], C3×C6, C3×C6 [×2], C3×C6 [×13], Dic6 [×3], C4×S3 [×4], D12, C2×Dic3 [×5], C3⋊D4 [×8], C2×C12, C3×D4 [×2], C33, C3×Dic3 [×12], C3⋊Dic3 [×2], C3⋊Dic3 [×2], S3×C6 [×6], C2×C3⋊S3 [×2], C62, C62 [×2], C62 [×4], C4○D12, D42S3 [×2], C3×C3⋊S3 [×2], C32×C6, C32×C6, S3×Dic3 [×4], C6.D6 [×2], D6⋊S3, C3⋊D12 [×2], C322Q8 [×3], C6×Dic3 [×3], C3×C3⋊D4 [×6], C2×C3⋊Dic3, C327D4 [×2], C3×C3⋊Dic3 [×2], C3×C3⋊Dic3 [×2], C6×C3⋊S3 [×2], C3×C62, D6.3D6 [×2], D6.4D6, C339(C2×C4) [×2], C339D4, C335Q8, C6×C3⋊Dic3, C3×C327D4 [×2], C62.96D6
Quotients: C1, C2 [×7], C22 [×7], S3 [×3], C23, D6 [×9], C4○D4, C22×S3 [×3], S32 [×3], C4○D12, D42S3 [×2], C2×S32 [×3], C324D6, D6.3D6 [×2], D6.4D6, C2×C324D6, C62.96D6

Permutation representations of C62.96D6
On 24 points - transitive group 24T1285
Generators in S24
(1 5 9)(2 10 6)(3 7 11)(4 12 8)(13 23 21 19 17 15)(14 16 18 20 22 24)
(1 3 5 7 9 11)(2 12 10 8 6 4)(13 23 21 19 17 15)(14 16 18 20 22 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 7 22)(2 21 8 15)(3 14 9 20)(4 19 10 13)(5 24 11 18)(6 17 12 23)

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

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

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

G:=TransitiveGroup(24,1285);

48 conjugacy classes

class 1 2A2B2C2D3A3B3C3D···3H4A4B4C4D4E6A···6E6F···6V6W6X12A12B12C12D12E12F
order122223333···3444446···66···666121212121212
size11218182224···4991818182···24···43636181818183636

48 irreducible representations

dim111111222222244444444
type++++++++++++-+-
imageC1C2C2C2C2C2S3S3D6D6D6C4○D4C4○D12S32D42S3C2×S32C324D6D6.3D6D6.4D6C2×C324D6C62.96D6
kernelC62.96D6C339(C2×C4)C339D4C335Q8C6×C3⋊Dic3C3×C327D4C2×C3⋊Dic3C327D4C3⋊Dic3C2×C3⋊S3C62C33C32C2×C6C32C6C22C3C3C2C1
# reps121112124232432324224

Matrix representation of C62.96D6 in GL4(𝔽7) generated by

3011
3622
1115
0004
,
1526
0302
3301
0005
,
0662
6650
1636
1135
,
3231
1010
4446
4350
G:=sub<GL(4,GF(7))| [3,3,1,0,0,6,1,0,1,2,1,0,1,2,5,4],[1,0,3,0,5,3,3,0,2,0,0,0,6,2,1,5],[0,6,1,1,6,6,6,1,6,5,3,3,2,0,6,5],[3,1,4,4,2,0,4,3,3,1,4,5,1,0,6,0] >;

C62.96D6 in GAP, Magma, Sage, TeX

C_6^2._{96}D_6
% in TeX

G:=Group("C6^2.96D6");
// GroupNames label

G:=SmallGroup(432,693);
// by ID

G=gap.SmallGroup(432,693);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,64,254,1124,571,2028,14118]);
// Polycyclic

G:=Group<a,b,c,d|a^6=b^6=1,c^6=d^2=b^3,a*b=b*a,c*a*c^-1=a^-1,d*a*d^-1=a*b^3,c*b*c^-1=d*b*d^-1=b^-1,d*c*d^-1=c^5>;
// generators/relations

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