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G = C3×C23.7D6order 288 = 25·32

Direct product of C3 and C23.7D6

direct product, metabelian, supersoluble, monomial

Aliases: C3×C23.7D6, C62.35D4, (C6×C12)⋊6C4, (C2×C12)⋊1C12, (C2×C62)⋊2C4, (C22×C6)⋊3C12, (C6×D4).24S3, (C6×D4).17C6, (C2×C12)⋊1Dic3, C23.7(S3×C6), C6.D42C6, C328(C23⋊C4), C62.98(C2×C4), (C22×C6)⋊3Dic3, C232(C3×Dic3), (C22×C6).25D6, C22.3(C6×Dic3), (C2×C62).10C22, C6.33(C6.D4), (C2×C4)⋊(C3×Dic3), C32(C3×C23⋊C4), (D4×C3×C6).11C2, (C2×C6).3(C3×D4), (C2×D4).3(C3×S3), (C2×C6).39(C2×C12), C6.15(C3×C22⋊C4), C22.2(C3×C3⋊D4), (C3×C6.D4)⋊4C2, (C2×C6).43(C3⋊D4), (C22×C6).17(C2×C6), (C2×C6).24(C2×Dic3), C2.5(C3×C6.D4), (C3×C6).66(C22⋊C4), SmallGroup(288,268)

Series: Derived Chief Lower central Upper central

C1C2×C6 — C3×C23.7D6
C1C3C6C2×C6C22×C6C2×C62C3×C6.D4 — C3×C23.7D6
C3C6C2×C6 — C3×C23.7D6
C1C6C22×C6C6×D4

Generators and relations for C3×C23.7D6
 G = < a,b,c,d,e,f | a3=b2=c2=d2=e6=1, f2=cb=bc, ab=ba, ac=ca, ad=da, ae=ea, af=fa, ebe-1=fbf-1=bd=db, fcf-1=cd=dc, ce=ec, de=ed, df=fd, fef-1=cde-1 >

Subgroups: 330 in 131 conjugacy classes, 42 normal (30 characteristic)
C1, C2, C2 [×4], C3 [×2], C3, C4 [×3], C22, C22 [×2], C22 [×3], C6 [×2], C6 [×15], C2×C4, C2×C4 [×2], D4 [×2], C23 [×2], C32, Dic3 [×2], C12 [×6], C2×C6 [×2], C2×C6 [×4], C2×C6 [×15], C22⋊C4 [×2], C2×D4, C3×C6, C3×C6 [×4], C2×Dic3 [×2], C2×C12 [×2], C2×C12 [×3], C3×D4 [×8], C22×C6 [×4], C22×C6 [×2], C23⋊C4, C3×Dic3 [×2], C3×C12, C62, C62 [×2], C62 [×3], C6.D4 [×2], C3×C22⋊C4 [×2], C6×D4 [×2], C6×D4, C6×Dic3 [×2], C6×C12, D4×C32 [×2], C2×C62 [×2], C23.7D6, C3×C23⋊C4, C3×C6.D4 [×2], D4×C3×C6, C3×C23.7D6
Quotients: C1, C2 [×3], C3, C4 [×2], C22, S3, C6 [×3], C2×C4, D4 [×2], Dic3 [×2], C12 [×2], D6, C2×C6, C22⋊C4, C3×S3, C2×Dic3, C3⋊D4 [×2], C2×C12, C3×D4 [×2], C23⋊C4, C3×Dic3 [×2], S3×C6, C6.D4, C3×C22⋊C4, C6×Dic3, C3×C3⋊D4 [×2], C23.7D6, C3×C23⋊C4, C3×C6.D4, C3×C23.7D6

Permutation representations of C3×C23.7D6
On 24 points - transitive group 24T586
Generators in S24
(1 3 5)(2 4 6)(7 11 9)(8 12 10)(13 15 17)(14 16 18)(19 23 21)(20 24 22)
(2 16)(4 18)(6 14)(7 21)(9 23)(11 19)
(7 21)(8 22)(9 23)(10 24)(11 19)(12 20)
(1 15)(2 16)(3 17)(4 18)(5 13)(6 14)(7 21)(8 22)(9 23)(10 24)(11 19)(12 20)
(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 23)(2 8 16 22)(3 21)(4 12 18 20)(5 19)(6 10 14 24)(7 17)(9 15)(11 13)

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

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

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

G:=TransitiveGroup(24,586);

On 24 points - transitive group 24T589
Generators in S24
(1 3 2)(4 5 6)(7 9 8)(10 11 12)(13 17 15)(14 18 16)(19 21 23)(20 22 24)
(1 24)(2 22)(3 20)(4 15)(5 13)(6 17)(7 19)(8 23)(9 21)(10 14)(11 18)(12 16)
(1 9)(2 7)(3 8)(19 22)(20 23)(21 24)
(1 9)(2 7)(3 8)(4 11)(5 12)(6 10)(13 16)(14 17)(15 18)(19 22)(20 23)(21 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 12 21 16)(2 11 19 18)(3 10 23 14)(4 22 15 7)(5 24 13 9)(6 20 17 8)

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

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

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

G:=TransitiveGroup(24,589);

63 conjugacy classes

class 1 2A2B2C2D2E3A3B3C3D3E4A4B4C4D4E6A6B6C···6Q6R···6AE12A···12H12I···12P
order1222223333344444666···66···612···1212···12
size11222411222412121212112···24···44···412···12

63 irreducible representations

dim11111111112222222222224444
type+++++--++
imageC1C2C2C3C4C4C6C6C12C12S3D4Dic3Dic3D6C3×S3C3⋊D4C3×D4C3×Dic3C3×Dic3S3×C6C3×C3⋊D4C23⋊C4C23.7D6C3×C23⋊C4C3×C23.7D6
kernelC3×C23.7D6C3×C6.D4D4×C3×C6C23.7D6C6×C12C2×C62C6.D4C6×D4C2×C12C22×C6C6×D4C62C2×C12C22×C6C22×C6C2×D4C2×C6C2×C6C2×C4C23C23C22C32C3C3C1
# reps12122242441211124422281224

Matrix representation of C3×C23.7D6 in GL4(𝔽7) generated by

2000
0200
0020
0002
,
5103
5136
0010
2560
,
0145
1035
0010
0006
,
6000
0600
0060
0006
,
2443
3125
2223
0002
,
0563
5502
2566
2253
G:=sub<GL(4,GF(7))| [2,0,0,0,0,2,0,0,0,0,2,0,0,0,0,2],[5,5,0,2,1,1,0,5,0,3,1,6,3,6,0,0],[0,1,0,0,1,0,0,0,4,3,1,0,5,5,0,6],[6,0,0,0,0,6,0,0,0,0,6,0,0,0,0,6],[2,3,2,0,4,1,2,0,4,2,2,0,3,5,3,2],[0,5,2,2,5,5,5,2,6,0,6,5,3,2,6,3] >;

C3×C23.7D6 in GAP, Magma, Sage, TeX

C_3\times C_2^3._7D_6
% in TeX

G:=Group("C3xC2^3.7D6");
// GroupNames label

G:=SmallGroup(288,268);
// by ID

G=gap.SmallGroup(288,268);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,-2,-3,84,365,850,2524,9414]);
// Polycyclic

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

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