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## G = C3×D12⋊6C22order 288 = 25·32

### Direct product of C3 and D12⋊6C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C12 — C3×D12⋊6C22
 Chief series C1 — C3 — C6 — C12 — C3×C12 — C3×D12 — C3×C4○D12 — C3×D12⋊6C22
 Lower central C3 — C6 — C12 — C3×D12⋊6C22
 Upper central C1 — C6 — C2×C12 — C6×D4

Generators and relations for C3×D126C22
G = < a,b,c,d,e | a3=b12=c2=d2=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, ebe=b7, dcd=b6c, ece=b3c, de=ed >

Subgroups: 394 in 163 conjugacy classes, 58 normal (38 characteristic)
C1, C2, C2 [×4], C3 [×2], C3, C4 [×2], C4, C22, C22 [×5], S3, C6 [×2], C6 [×14], C8 [×2], C2×C4, C2×C4, D4 [×2], D4 [×3], Q8, C23, C32, Dic3, C12 [×4], C12 [×3], D6, C2×C6 [×2], C2×C6 [×18], M4(2), D8 [×2], SD16 [×2], C2×D4, C4○D4, C3×S3, C3×C6, C3×C6 [×3], C3⋊C8 [×2], C24 [×2], Dic6, C4×S3, D12, C3⋊D4, C2×C12 [×2], C2×C12 [×2], C3×D4 [×4], C3×D4 [×8], C3×Q8, C22×C6 [×4], C8⋊C22, C3×Dic3, C3×C12 [×2], S3×C6, C62, C62 [×4], C4.Dic3, D4⋊S3 [×2], D4.S3 [×2], C3×M4(2), C3×D8 [×2], C3×SD16 [×2], C4○D12, C6×D4 [×2], C6×D4, C3×C4○D4, C3×C3⋊C8 [×2], C3×Dic6, S3×C12, C3×D12, C3×C3⋊D4, C6×C12, D4×C32 [×2], D4×C32, C2×C62, D126C22, C3×C8⋊C22, C3×C4.Dic3, C3×D4⋊S3 [×2], C3×D4.S3 [×2], C3×C4○D12, D4×C3×C6, C3×D126C22
Quotients: C1, C2 [×7], C3, C22 [×7], S3, C6 [×7], D4 [×2], C23, D6 [×3], C2×C6 [×7], C2×D4, C3×S3, C3⋊D4 [×2], C3×D4 [×2], C22×S3, C22×C6, C8⋊C22, S3×C6 [×3], C2×C3⋊D4, C6×D4, C3×C3⋊D4 [×2], S3×C2×C6, D126C22, C3×C8⋊C22, C6×C3⋊D4, C3×D126C22

Permutation representations of C3×D126C22
On 24 points - transitive group 24T628
Generators in S24
(1 9 5)(2 10 6)(3 11 7)(4 12 8)(13 17 21)(14 18 22)(15 19 23)(16 20 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 21)(2 20)(3 19)(4 18)(5 17)(6 16)(7 15)(8 14)(9 13)(10 24)(11 23)(12 22)
(1 7)(2 8)(3 9)(4 10)(5 11)(6 12)
(2 8)(4 10)(6 12)(13 22)(14 17)(15 24)(16 19)(18 21)(20 23)

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

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

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

G:=TransitiveGroup(24,628);

63 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3A 3B 3C 3D 3E 4A 4B 4C 6A 6B 6C ··· 6M 6N ··· 6AC 6AD 6AE 8A 8B 12A 12B 12C 12D 12E ··· 12J 12K 12L 24A 24B 24C 24D order 1 2 2 2 2 2 3 3 3 3 3 4 4 4 6 6 6 ··· 6 6 ··· 6 6 6 8 8 12 12 12 12 12 ··· 12 12 12 24 24 24 24 size 1 1 2 4 4 12 1 1 2 2 2 2 2 12 1 1 2 ··· 2 4 ··· 4 12 12 12 12 2 2 2 2 4 ··· 4 12 12 12 12 12 12

63 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C3 C6 C6 C6 C6 C6 S3 D4 D4 D6 D6 C3×S3 C3⋊D4 C3×D4 C3⋊D4 C3×D4 S3×C6 S3×C6 C3×C3⋊D4 C3×C3⋊D4 C8⋊C22 D12⋊6C22 C3×C8⋊C22 C3×D12⋊6C22 kernel C3×D12⋊6C22 C3×C4.Dic3 C3×D4⋊S3 C3×D4.S3 C3×C4○D12 D4×C3×C6 D12⋊6C22 C4.Dic3 D4⋊S3 D4.S3 C4○D12 C6×D4 C6×D4 C3×C12 C62 C2×C12 C3×D4 C2×D4 C12 C12 C2×C6 C2×C6 C2×C4 D4 C4 C22 C32 C3 C3 C1 # reps 1 1 2 2 1 1 2 2 4 4 2 2 1 1 1 1 2 2 2 2 2 2 2 4 4 4 1 2 2 4

Matrix representation of C3×D126C22 in GL4(𝔽7) generated by

 4 0 0 0 0 4 0 0 0 0 4 0 0 0 0 4
,
 1 0 6 2 2 1 0 1 5 5 5 4 4 3 5 0
,
 6 2 5 1 1 1 2 1 4 3 5 5 6 6 4 2
,
 0 1 4 5 1 0 3 5 0 0 1 0 0 0 0 6
,
 0 1 1 0 1 0 1 0 0 0 6 0 0 0 0 1
G:=sub<GL(4,GF(7))| [4,0,0,0,0,4,0,0,0,0,4,0,0,0,0,4],[1,2,5,4,0,1,5,3,6,0,5,5,2,1,4,0],[6,1,4,6,2,1,3,6,5,2,5,4,1,1,5,2],[0,1,0,0,1,0,0,0,4,3,1,0,5,5,0,6],[0,1,0,0,1,0,0,0,1,1,6,0,0,0,0,1] >;

C3×D126C22 in GAP, Magma, Sage, TeX

C_3\times D_{12}\rtimes_6C_2^2
% in TeX

G:=Group("C3xD12:6C2^2");
// GroupNames label

G:=SmallGroup(288,703);
// by ID

G=gap.SmallGroup(288,703);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-3,344,590,555,2524,648,102,9414]);
// Polycyclic

G:=Group<a,b,c,d,e|a^3=b^12=c^2=d^2=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=b^-1,b*d=d*b,e*b*e=b^7,d*c*d=b^6*c,e*c*e=b^3*c,d*e=e*d>;
// generators/relations

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