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G = C4.4S3≀C2  order 288 = 25·32

4th non-split extension by C4 of S3≀C2 acting via S3≀C2/C32⋊C4=C2

non-abelian, soluble, monomial

Aliases: C4.4S3≀C2, (C3×C12).9D4, C32⋊(C4.4D4), D6⋊D6.5C2, C3⋊Dic3.27D4, Dic3.D6⋊7C2, C6.D6.2C22, S32⋊C4⋊1C2, C2.5(C2×S3≀C2), (C4×C32⋊C4)⋊3C2, (C3×C6).2(C2×D4), (C2×S32).1C22, C3⋊S3.2(C4○D4), (C2×C3⋊S3).2C23, (C4×C3⋊S3).28C22, (C2×C32⋊C4).12C22, SmallGroup(288,869)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C32 — C2×C3⋊S3 — C4.4S3≀C2
Chief series
C1 — C32 — C3⋊S3 — C2×C3⋊S3 — C2×S32 — S32⋊C4 — C4.4S3≀C2
Lower central
C32 — C2×C3⋊S3 — C4.4S3≀C2
Upper central
C1 — C2 — C4

Generators and relations for C4.4S3≀C2
 G = < a,b,c,d,e | a4=b3=c3=d4=e2=1, ab=ba, ac=ca, ad=da, eae=a-1, bc=cb, dbd-1=c, ebe=dcd-1=b-1, ce=ec, ede=a2d-1 >

Subgroups: 696 in 122 conjugacy classes, 25 normal (13 characteristic)
C1, C2, C2, C3, C4, C4, C22, S3, C6, C2×C4, D4, Q8, C23, C32, Dic3, C12, D6, C2×C6, C42, C22⋊C4, C2×D4, C2×Q8, C3×S3, C3⋊S3, C3×C6, Dic6, C4×S3, D12, C3⋊D4, C3×D4, C3×Q8, C22×S3, C4.4D4, C3×Dic3, C3⋊Dic3, C3×C12, C32⋊C4, S32, S3×C6, C2×C3⋊S3, S3×D4, S3×Q8, C6.D6, D6⋊S3, C32⋊2Q8, C3×Dic6, C3×D12, C4×C3⋊S3, C2×C32⋊C4, C2×S32, S32⋊C4, C4×C32⋊C4, Dic3.D6, D6⋊D6, C4.4S3≀C2
Quotients: C1, C2, C22, D4, C23, C2×D4, C4○D4, C4.4D4, S3≀C2, C2×S3≀C2, C4.4S3≀C2

Character table of C4.4S3≀C2

 class 12A2B2C2D2E3A3B4A4B4C4D4E4F4G4H6A6B6C6D12A12B12C12D
 size 1199121244212121818181818442424882424
ρ1111111111111111111111111    trivial
ρ2111111111-1-11-1-1-1-1111111-1-1    linear of order 2
ρ311111-111-11-1-1-1-111111-1-1-11-1    linear of order 2
ρ411111-111-1-11-111-1-1111-1-1-1-11    linear of order 2
ρ51111-1111-1-11-1-1-11111-11-1-1-11    linear of order 2
ρ61111-1111-11-1-111-1-111-11-1-11-1    linear of order 2
ρ71111-1-1111-1-11111111-1-111-1-1    linear of order 2
ρ81111-1-1111111-1-1-1-111-1-11111    linear of order 2
ρ922-2-20022-200200002200-2-200    orthogonal lifted from D4
ρ1022-2-20022200-2000022002200    orthogonal lifted from D4
ρ112-22-200220000-2i2i00-2-2000000    complex lifted from C4○D4
ρ122-2-2200220000002i-2i-2-2000000    complex lifted from C4○D4
ρ132-22-2002200002i-2i00-2-2000000    complex lifted from C4○D4
ρ142-2-220022000000-2i2i-2-2000000    complex lifted from C4○D4
ρ154400-22-21-400000001-21-1-1200    orthogonal lifted from C2×S3≀C2
ρ164400001-2-4-2200000-21002-11-1    orthogonal lifted from C2×S3≀C2
ρ1744002-2-21-400000001-2-11-1200    orthogonal lifted from C2×S3≀C2
ρ184400001-242200000-2100-21-1-1    orthogonal lifted from S3≀C2
ρ194400-2-2-21400000001-2111-200    orthogonal lifted from S3≀C2
ρ204400001-24-2-200000-2100-2111    orthogonal lifted from S3≀C2
ρ21440022-21400000001-2-1-11-200    orthogonal lifted from S3≀C2
ρ224400001-2-42-200000-21002-1-11    orthogonal lifted from C2×S3≀C2
ρ238-80000-4200000000-24000000    orthogonal faithful
ρ248-800002-4000000004-2000000    symplectic faithful, Schur index 2

Permutation representations of C4.4S3≀C2
►On 24 points - transitive group 24T642
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)

(5 21 10)(6 22 11)(7 23 12)(8 24 9)

(1 14 19)(2 15 20)(3 16 17)(4 13 18)

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

(1 4)(2 3)(5 11)(6 10)(7 9)(8 12)(13 14)(15 16)(17 20)(18 19)(21 22)(23 24)
 
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), (5,21,10)(6,22,11)(7,23,12)(8,24,9), (1,14,19)(2,15,20)(3,16,17)(4,13,18), (1,21)(2,22)(3,23)(4,24)(5,14,10,19)(6,15,11,20)(7,16,12,17)(8,13,9,18), (1,4)(2,3)(5,11)(6,10)(7,9)(8,12)(13,14)(15,16)(17,20)(18,19)(21,22)(23,24)>;
 
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), (5,21,10)(6,22,11)(7,23,12)(8,24,9), (1,14,19)(2,15,20)(3,16,17)(4,13,18), (1,21)(2,22)(3,23)(4,24)(5,14,10,19)(6,15,11,20)(7,16,12,17)(8,13,9,18), (1,4)(2,3)(5,11)(6,10)(7,9)(8,12)(13,14)(15,16)(17,20)(18,19)(21,22)(23,24) );
 
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)], [(5,21,10),(6,22,11),(7,23,12),(8,24,9)], [(1,14,19),(2,15,20),(3,16,17),(4,13,18)], [(1,21),(2,22),(3,23),(4,24),(5,14,10,19),(6,15,11,20),(7,16,12,17),(8,13,9,18)], [(1,4),(2,3),(5,11),(6,10),(7,9),(8,12),(13,14),(15,16),(17,20),(18,19),(21,22),(23,24)]])
 
G:=TransitiveGroup(24,642);
 

Matrix representation of C4.4S3≀C2 ►in GL6(𝔽13)

0120000
100000
001000
000100
000010
000001
,
100000
010000
001000
000100
0000121
0000120
,
100000
010000
0012100
0012000
000010
000001
,
050000
800000
000001
000010
001000
000100
,
010000
100000
001000
000100
000001
000010

G:=sub<GL(6,GF(13))| [0,1,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,12,0,0,0,0,1,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,12,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,8,0,0,0,0,5,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,1,0,0,0],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;
 

C4.4S3≀C2 in GAP, Magma, Sage, TeX 

C_4._4S_3\wr C_2
 
% in TeX
 
G:=Group("C4.4S3wrC2");
 
// GroupNames label
 
G:=SmallGroup(288,869);
 
// by ID
 
G=gap.SmallGroup(288,869);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,3,141,120,422,219,100,2693,2028,362,797,1203]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^4=b^3=c^3=d^4=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,e*a*e=a^-1,b*c=c*b,d*b*d^-1=c,e*b*e=d*c*d^-1=b^-1,c*e=e*c,e*d*e=a^2*d^-1>;
 
// generators/relations
 

Export

Character table of C4.4S3≀C2 in TeX

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