Copied to
clipboard

## G = C3×C23.7D6order 288 = 25·32

### Direct product of C3 and C23.7D6

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C6 — C3×C23.7D6
 Chief series C1 — C3 — C6 — C2×C6 — C22×C6 — C2×C62 — C3×C6.D4 — C3×C23.7D6
 Lower central C3 — C6 — C2×C6 — C3×C23.7D6
 Upper central C1 — C6 — C22×C6 — C6×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, C3, C3, C4, C22, C22, C22, C6, C6, C2×C4, C2×C4, D4, C23, C32, Dic3, C12, C2×C6, C2×C6, C2×C6, C22⋊C4, C2×D4, C3×C6, C3×C6, C2×Dic3, C2×C12, C2×C12, C3×D4, C22×C6, C22×C6, C23⋊C4, C3×Dic3, C3×C12, C62, C62, C62, C6.D4, C3×C22⋊C4, C6×D4, C6×D4, C6×Dic3, C6×C12, D4×C32, C2×C62, C23.7D6, C3×C23⋊C4, C3×C6.D4, D4×C3×C6, C3×C23.7D6
Quotients: C1, C2, C3, C4, C22, S3, C6, C2×C4, D4, Dic3, C12, D6, C2×C6, C22⋊C4, C3×S3, C2×Dic3, C3⋊D4, C2×C12, C3×D4, C23⋊C4, C3×Dic3, S3×C6, C6.D4, C3×C22⋊C4, C6×Dic3, C3×C3⋊D4, 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 15 17)(14 16 18)(19 23 21)(20 24 22)
(1 14)(2 18)(3 16)(4 24)(5 22)(6 20)(7 13)(8 17)(9 15)(10 19)(11 23)(12 21)
(1 8)(2 9)(3 7)(13 16)(14 17)(15 18)
(1 8)(2 9)(3 7)(4 12)(5 10)(6 11)(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 10 17 19)(2 12 15 21)(3 11 13 23)(4 18 24 9)(5 14 22 8)(6 16 20 7)

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

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

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

G:=TransitiveGroup(24,589);

63 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3A 3B 3C 3D 3E 4A 4B 4C 4D 4E 6A 6B 6C ··· 6Q 6R ··· 6AE 12A ··· 12H 12I ··· 12P order 1 2 2 2 2 2 3 3 3 3 3 4 4 4 4 4 6 6 6 ··· 6 6 ··· 6 12 ··· 12 12 ··· 12 size 1 1 2 2 2 4 1 1 2 2 2 4 12 12 12 12 1 1 2 ··· 2 4 ··· 4 4 ··· 4 12 ··· 12

63 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + + - - + + image C1 C2 C2 C3 C4 C4 C6 C6 C12 C12 S3 D4 Dic3 Dic3 D6 C3×S3 C3⋊D4 C3×D4 C3×Dic3 C3×Dic3 S3×C6 C3×C3⋊D4 C23⋊C4 C23.7D6 C3×C23⋊C4 C3×C23.7D6 kernel C3×C23.7D6 C3×C6.D4 D4×C3×C6 C23.7D6 C6×C12 C2×C62 C6.D4 C6×D4 C2×C12 C22×C6 C6×D4 C62 C2×C12 C22×C6 C22×C6 C2×D4 C2×C6 C2×C6 C2×C4 C23 C23 C22 C32 C3 C3 C1 # reps 1 2 1 2 2 2 4 2 4 4 1 2 1 1 1 2 4 4 2 2 2 8 1 2 2 4

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

 2 0 0 0 0 2 0 0 0 0 2 0 0 0 0 2
,
 5 1 0 3 5 1 3 6 0 0 1 0 2 5 6 0
,
 0 1 4 5 1 0 3 5 0 0 1 0 0 0 0 6
,
 6 0 0 0 0 6 0 0 0 0 6 0 0 0 0 6
,
 2 4 4 3 3 1 2 5 2 2 2 3 0 0 0 2
,
 0 5 6 3 5 5 0 2 2 5 6 6 2 2 5 3
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

׿
×
𝔽