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G = C628D4order 288 = 25·32

5th semidirect product of C62 and D4 acting via D4/C2=C22

metabelian, supersoluble, monomial

Aliases: C628D4, C62.123C23, (C2×C6)⋊5D12, C23.35S32, C6.74(S3×D4), (C2×Dic3)⋊5D6, C6.88(C2×D12), C326C22≀C2, (C22×S3)⋊4D6, C32(C232D6), C34(D6⋊D4), (C22×C6).81D6, C6.D414S3, (C6×Dic3)⋊4C22, C6.D1211C2, C2.33(Dic3⋊D6), C225(C3⋊D12), (C2×C62).42C22, (C2×C3⋊S3)⋊15D4, (C2×C3⋊D4)⋊8S3, (S3×C2×C6)⋊3C22, (C6×C3⋊D4)⋊13C2, (C2×C6)⋊7(C3⋊D4), (C23×C3⋊S3)⋊1C2, C6.25(C2×C3⋊D4), C22.146(C2×S32), (C3×C6).169(C2×D4), (C2×C3⋊D12)⋊13C2, C2.26(C2×C3⋊D12), (C3×C6.D4)⋊10C2, (C2×C6).142(C22×S3), (C22×C3⋊S3).78C22, SmallGroup(288,629)

Series: Derived Chief Lower central Upper central

C1C62 — C628D4
C1C3C32C3×C6C62S3×C2×C6C2×C3⋊D12 — C628D4
C32C62 — C628D4
C1C22C23

Generators and relations for C628D4
 G = < a,b,c,d | a6=b6=c4=d2=1, ab=ba, cac-1=a-1b3, dad=a-1, bc=cb, dbd=b-1, dcd=c-1 >

Subgroups: 1634 in 327 conjugacy classes, 56 normal (26 characteristic)
C1, C2, C2 [×2], C2 [×7], C3 [×2], C3, C4 [×3], C22, C22 [×2], C22 [×21], S3 [×17], C6 [×2], C6 [×4], C6 [×12], C2×C4 [×3], D4 [×6], C23, C23 [×9], C32, Dic3 [×3], C12 [×3], D6 [×63], C2×C6 [×2], C2×C6 [×4], C2×C6 [×14], C22⋊C4 [×3], C2×D4 [×3], C24, C3×S3, C3⋊S3 [×4], C3×C6, C3×C6 [×2], C3×C6 [×2], D12 [×4], C2×Dic3, C2×Dic3 [×2], C3⋊D4 [×6], C2×C12 [×3], C3×D4 [×2], C22×S3, C22×S3 [×30], C22×C6 [×2], C22×C6 [×2], C22≀C2, C3×Dic3 [×3], S3×C6 [×3], C2×C3⋊S3 [×4], C2×C3⋊S3 [×12], C62, C62 [×2], C62 [×2], D6⋊C4 [×4], C6.D4, C3×C22⋊C4, C2×D12 [×2], C2×C3⋊D4, C2×C3⋊D4 [×2], C6×D4, S3×C23 [×3], C3⋊D12 [×4], C6×Dic3, C6×Dic3 [×2], C3×C3⋊D4 [×2], S3×C2×C6, C22×C3⋊S3 [×2], C22×C3⋊S3 [×6], C2×C62, D6⋊D4, C232D6, C6.D12 [×2], C3×C6.D4, C2×C3⋊D12 [×2], C6×C3⋊D4, C23×C3⋊S3, C628D4
Quotients: C1, C2 [×7], C22 [×7], S3 [×2], D4 [×6], C23, D6 [×6], C2×D4 [×3], D12 [×2], C3⋊D4 [×2], C22×S3 [×2], C22≀C2, S32, C2×D12, S3×D4 [×4], C2×C3⋊D4, C3⋊D12 [×2], C2×S32, D6⋊D4, C232D6, C2×C3⋊D12, Dic3⋊D6 [×2], C628D4

Permutation representations of C628D4
On 24 points - transitive group 24T672
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 4 3 6 2 5)(7 11 9 10 8 12)(13 18 17 16 15 14)(19 24 23 22 21 20)
(1 15 12 22)(2 17 10 24)(3 13 11 20)(4 14 7 21)(5 16 8 23)(6 18 9 19)
(1 2)(4 6)(7 9)(10 12)(13 20)(14 19)(15 24)(16 23)(17 22)(18 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), (1,4,3,6,2,5)(7,11,9,10,8,12)(13,18,17,16,15,14)(19,24,23,22,21,20), (1,15,12,22)(2,17,10,24)(3,13,11,20)(4,14,7,21)(5,16,8,23)(6,18,9,19), (1,2)(4,6)(7,9)(10,12)(13,20)(14,19)(15,24)(16,23)(17,22)(18,21)>;

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

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

G:=TransitiveGroup(24,672);

42 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I2J3A3B3C4A4B4C6A···6F6G···6Q6R6S12A···12F
order122222222223334446···66···66612···12
size11112212181818182241212122···24···4121212···12

42 irreducible representations

dim11111122222222244444
type+++++++++++++++++++
imageC1C2C2C2C2C2S3S3D4D4D6D6D6D12C3⋊D4S32S3×D4C3⋊D12C2×S32Dic3⋊D6
kernelC628D4C6.D12C3×C6.D4C2×C3⋊D12C6×C3⋊D4C23×C3⋊S3C6.D4C2×C3⋊D4C2×C3⋊S3C62C2×Dic3C22×S3C22×C6C2×C6C2×C6C23C6C22C22C2
# reps12121111423124414214

Matrix representation of C628D4 in GL6(𝔽13)

1200000
010000
0012000
0001200
0000121
0000120
,
1200000
0120000
0012100
0012000
000010
000001
,
010000
100000
0010600
007300
000092
0000114
,
100000
010000
000100
001000
0000012
0000120

G:=sub<GL(6,GF(13))| [12,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,12,0,0,0,0,1,0],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,12,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,10,7,0,0,0,0,6,3,0,0,0,0,0,0,9,11,0,0,0,0,2,4],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,12,0,0,0,0,12,0] >;

C628D4 in GAP, Magma, Sage, TeX

C_6^2\rtimes_8D_4
% in TeX

G:=Group("C6^2:8D4");
// GroupNames label

G:=SmallGroup(288,629);
// by ID

G=gap.SmallGroup(288,629);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,141,64,422,219,1356,9414]);
// Polycyclic

G:=Group<a,b,c,d|a^6=b^6=c^4=d^2=1,a*b=b*a,c*a*c^-1=a^-1*b^3,d*a*d=a^-1,b*c=c*b,d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations

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