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G = (C2×C62)⋊C4  order 288 = 25·32

4th semidirect product of C2×C62 and C4 acting faithfully

metabelian, soluble, monomial

Aliases: (C2×C62)⋊4C4, C23⋊(C32⋊C4), C62⋊C4⋊3C2, C32⋊3(C23⋊C4), C62.10(C2×C4), C2.10(C62⋊C4), (C2×C3⋊Dic3)⋊3C4, (C2×C3⋊S3).16D4, C22.4(C2×C32⋊C4), (C2×C32⋊7D4).3C2, (C3×C6).20(C22⋊C4), (C22×C3⋊S3).5C22, SmallGroup(288,434)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C62 — (C2×C62)⋊C4
Chief series
C1 — C32 — C3×C6 — C2×C3⋊S3 — C22×C3⋊S3 — C62⋊C4 — (C2×C62)⋊C4
Lower central
C32 — C3×C6 — C62 — (C2×C62)⋊C4
Upper central
C1 — C2 — C22 — C23

Generators and relations for (C2×C62)⋊C4
 G = < a,b,c,d | a2=b6=c6=d4=1, ab=ba, ac=ca, dad-1=ab3c3, bc=cb, dbd-1=b-1c, dcd-1=b4c >

Subgroups: 664 in 106 conjugacy classes, 16 normal (12 characteristic)
C1, C2, C2, C3, C4, C22, C22, S3, C6, C2×C4, D4, C23, C23, C32, Dic3, D6, C2×C6, C22⋊C4, C2×D4, C3⋊S3, C3×C6, C3×C6, C2×Dic3, C3⋊D4, C22×S3, C22×C6, C23⋊C4, C3⋊Dic3, C32⋊C4, C2×C3⋊S3, C2×C3⋊S3, C62, C62, C2×C3⋊D4, C2×C3⋊Dic3, C32⋊7D4, C2×C32⋊C4, C22×C3⋊S3, C2×C62, C62⋊C4, C2×C32⋊7D4, (C2×C62)⋊C4
Quotients: C1, C2, C4, C22, C2×C4, D4, C22⋊C4, C23⋊C4, C32⋊C4, C2×C32⋊C4, C62⋊C4, (C2×C62)⋊C4

Character table of (C2×C62)⋊C4

 class 12A2B2C2D2E3A3B4A4B4C4D4E6A6B6C6D6E6F6G6H6I6J6K6L6M6N
 size 1124181844363636363644444444444444
ρ1111111111111111111111111111    trivial
ρ211111111-11-1-1-111111111111111    linear of order 2
ρ3111-111111-1-1-11-1-11-1-1-1-1-1-111111    linear of order 2
ρ4111-11111-1-111-1-1-11-1-1-1-1-1-111111    linear of order 2
ρ5111-1-1-111-i1-iii-1-11-1-1-1-1-1-111111    linear of order 4
ρ6111-1-1-111i1i-i-i-1-11-1-1-1-1-1-111111    linear of order 4
ρ71111-1-111-i-1i-ii11111111111111    linear of order 4
ρ81111-1-111i-1-ii-i11111111111111    linear of order 4
ρ922-20-222200000002000000-2-22-2-2    orthogonal lifted from D4
ρ1022-202-22200000002000000-2-22-2-2    orthogonal lifted from D4
ρ114444001-200000111-2-21-21-2-21-21-2    orthogonal lifted from C32⋊C4
ρ12444-4001-200000-1-1122-12-12-21-21-2    orthogonal lifted from C2×C32⋊C4
ρ13444400-2100000-2-2-211-21-211-21-21    orthogonal lifted from C32⋊C4
ρ1444-40001-20000033100-30-302-1-2-12    orthogonal lifted from C62⋊C4
ρ1544-40001-200000-3-310030302-1-2-12    orthogonal lifted from C62⋊C4
ρ164-40000440000000-400000000-400    orthogonal lifted from C23⋊C4
ρ1744-4000-210000000-2-330-303-1212-1    orthogonal lifted from C62⋊C4
ρ1844-4000-210000000-23-3030-3-1212-1    orthogonal lifted from C62⋊C4
ρ19444-400-210000022-2-1-12-12-11-21-21    orthogonal lifted from C2×C32⋊C4
ρ204-400001-200000√-3-√-3-102√-3√-30-√-3-2√-30-3230    complex faithful
ρ214-40000-2100000-2√-32√-32-√-3√-30√-30-√-3-30-103    complex faithful
ρ224-400001-200000-√-3√-3-10-2√-3-√-30√-32√-30-3230    complex faithful
ρ234-40000-21000002√-3-2√-32√-3-√-30-√-30√-3-30-103    complex faithful
ρ244-400001-200000-√-3√-3-12√-30√-3-2√-3-√-30032-30    complex faithful
ρ254-40000-2100000002-√-3-√-32√-3√-3-2√-3√-330-10-3    complex faithful
ρ264-40000-2100000002√-3√-3-2√-3-√-32√-3-√-330-10-3    complex faithful
ρ274-400001-200000√-3-√-3-1-2√-30-√-32√-3√-30032-30    complex faithful

Permutation representations of (C2×C62)⋊C4
►On 24 points - transitive group 24T584
Generators in S24
(7 10)(8 11)(9 12)(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 4)(2 5)(3 6)(7 10)(8 11)(9 12)(13 18 17 16 15 14)(19 24 23 22 21 20)

(1 23 8 16)(2 19 7 14)(3 21 9 18)(4 20 11 13)(5 22 10 17)(6 24 12 15)
 
G:=sub<Sym(24)| (7,10)(8,11)(9,12)(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), (1,4)(2,5)(3,6)(7,10)(8,11)(9,12)(13,18,17,16,15,14)(19,24,23,22,21,20), (1,23,8,16)(2,19,7,14)(3,21,9,18)(4,20,11,13)(5,22,10,17)(6,24,12,15)>;
 
G:=Group( (7,10)(8,11)(9,12)(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), (1,4)(2,5)(3,6)(7,10)(8,11)(9,12)(13,18,17,16,15,14)(19,24,23,22,21,20), (1,23,8,16)(2,19,7,14)(3,21,9,18)(4,20,11,13)(5,22,10,17)(6,24,12,15) );
 
G=PermutationGroup([[(7,10),(8,11),(9,12),(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)], [(1,4),(2,5),(3,6),(7,10),(8,11),(9,12),(13,18,17,16,15,14),(19,24,23,22,21,20)], [(1,23,8,16),(2,19,7,14),(3,21,9,18),(4,20,11,13),(5,22,10,17),(6,24,12,15)]])
 
G:=TransitiveGroup(24,584);
 

►On 24 points - transitive group 24T620
Generators in S24
(1 7)(2 8)(3 9)(4 10)(5 11)(6 12)(13 19)(14 20)(15 21)(16 22)(17 23)(18 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 6 2 4 3 5)(7 12 8 10 9 11)(13 16)(14 17)(15 18)(19 22)(20 23)(21 24)

(1 13)(2 17 3 15)(4 16)(5 14 6 18)(7 22 10 19)(8 20 12 21)(9 24 11 23)
 
G:=sub<Sym(24)| (1,7)(2,8)(3,9)(4,10)(5,11)(6,12)(13,19)(14,20)(15,21)(16,22)(17,23)(18,24), (1,2,3)(4,5,6)(7,8,9)(10,11,12)(13,14,15,16,17,18)(19,20,21,22,23,24), (1,6,2,4,3,5)(7,12,8,10,9,11)(13,16)(14,17)(15,18)(19,22)(20,23)(21,24), (1,13)(2,17,3,15)(4,16)(5,14,6,18)(7,22,10,19)(8,20,12,21)(9,24,11,23)>;
 
G:=Group( (1,7)(2,8)(3,9)(4,10)(5,11)(6,12)(13,19)(14,20)(15,21)(16,22)(17,23)(18,24), (1,2,3)(4,5,6)(7,8,9)(10,11,12)(13,14,15,16,17,18)(19,20,21,22,23,24), (1,6,2,4,3,5)(7,12,8,10,9,11)(13,16)(14,17)(15,18)(19,22)(20,23)(21,24), (1,13)(2,17,3,15)(4,16)(5,14,6,18)(7,22,10,19)(8,20,12,21)(9,24,11,23) );
 
G=PermutationGroup([[(1,7),(2,8),(3,9),(4,10),(5,11),(6,12),(13,19),(14,20),(15,21),(16,22),(17,23),(18,24)], [(1,2,3),(4,5,6),(7,8,9),(10,11,12),(13,14,15,16,17,18),(19,20,21,22,23,24)], [(1,6,2,4,3,5),(7,12,8,10,9,11),(13,16),(14,17),(15,18),(19,22),(20,23),(21,24)], [(1,13),(2,17,3,15),(4,16),(5,14,6,18),(7,22,10,19),(8,20,12,21),(9,24,11,23)]])
 
G:=TransitiveGroup(24,620);
 

Matrix representation of (C2×C62)⋊C4 ►in GL4(𝔽7) generated by

6656
4013
1165
1632
,
2250
1333
1423
5524
,
3656
4413
1135
1636
,
2545
3063
4426
2513
G:=sub<GL(4,GF(7))| [6,4,1,1,6,0,1,6,5,1,6,3,6,3,5,2],[2,1,1,5,2,3,4,5,5,3,2,2,0,3,3,4],[3,4,1,1,6,4,1,6,5,1,3,3,6,3,5,6],[2,3,4,2,5,0,4,5,4,6,2,1,5,3,6,3] >;
 

(C2×C62)⋊C4 in GAP, Magma, Sage, TeX 

(C_2\times C_6^2)\rtimes C_4
 
% in TeX
 
G:=Group("(C2xC6^2):C4");
 
// GroupNames label
 
G:=SmallGroup(288,434);
 
// by ID
 
G=gap.SmallGroup(288,434);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,3,28,141,219,675,9413,691,12550,2372]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^2=b^6=c^6=d^4=1,a*b=b*a,a*c=c*a,d*a*d^-1=a*b^3*c^3,b*c=c*b,d*b*d^-1=b^-1*c,d*c*d^-1=b^4*c>;
 
// generators/relations
 

Export

Character table of (C2×C62)⋊C4 in TeX

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