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G = (C2×C62).C4  order 288 = 25·32

5th non-split extension by C2×C62 of C4 acting faithfully

metabelian, soluble, monomial

Aliases: (C2×C62).5C4, C23.(C32⋊C4), C3⋊Dic3.12D4, C62.12(C2×C4), C62.C4⋊4C2, C32⋊4(C4.D4), C2.11(C62⋊C4), (C22×C3⋊S3).3C4, C22.5(C2×C32⋊C4), (C2×C32⋊7D4).4C2, (C3×C6).22(C22⋊C4), (C2×C3⋊Dic3).8C22, SmallGroup(288,436)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C62 — (C2×C62).C4
Chief series
C1 — C32 — C3×C6 — C3⋊Dic3 — C2×C3⋊Dic3 — 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=1, d4=c3, ab=ba, ac=ca, dad-1=ab3c3, bc=cb, dbd-1=b-1c, dcd-1=b4c >

Subgroups: 584 in 100 conjugacy classes, 16 normal (12 characteristic)
C1, C2, C2, C3, C4, C22, C22, S3, C6, C8, C2×C4, D4, C23, C23, C32, Dic3, D6, C2×C6, M4(2), C2×D4, C3⋊S3, C3×C6, C3×C6, C2×Dic3, C3⋊D4, C22×S3, C22×C6, C4.D4, C3⋊Dic3, C2×C3⋊S3, C62, C62, C2×C3⋊D4, C32⋊2C8, C2×C3⋊Dic3, C32⋊7D4, C22×C3⋊S3, C2×C62, C62.C4, C2×C32⋊7D4, (C2×C62).C4
Quotients: C1, C2, C4, C22, C2×C4, D4, C22⋊C4, C4.D4, C32⋊C4, C2×C32⋊C4, C62⋊C4, (C2×C62).C4

Character table of (C2×C62).C4

 class 12A2B2C2D3A3B4A4B6A6B6C6D6E6F6G6H6I6J6K6L6M6N8A8B8C8D
 size 1124364418184444444444444436363636
ρ1111111111111111111111111111    trivial
ρ2111-1-1111111111-1-1-1-1-1-1-1-1111-1-1    linear of order 2
ρ311111111111111111111111-1-1-1-1    linear of order 2
ρ4111-1-1111111111-1-1-1-1-1-1-1-11-1-111    linear of order 2
ρ51111-111-1-111111111111111-ii-ii    linear of order 4
ρ6111-1111-1-111111-1-1-1-1-1-1-1-11-iii-i    linear of order 4
ρ71111-111-1-111111111111111i-ii-i    linear of order 4
ρ8111-1111-1-111111-1-1-1-1-1-1-1-11i-i-ii    linear of order 4
ρ922-200222-22-22-2-200000000-20000    orthogonal lifted from D4
ρ1022-20022-222-22-2-200000000-20000    orthogonal lifted from D4
ρ1144-4001-200-2212-1003-3030-3-10000    orthogonal lifted from C62⋊C4
ρ1244440-210011-21-211-2-21-21-2-20000    orthogonal lifted from C32⋊C4
ρ1344-400-21001-1-2-12-3-300303020000    orthogonal lifted from C62⋊C4
ρ1444-4001-200-2212-100-330-303-10000    orthogonal lifted from C62⋊C4
ρ15444-401-200-2-21-2122-1-12-12-110000    orthogonal lifted from C2×C32⋊C4
ρ16444401-200-2-21-21-2-211-21-2110000    orthogonal lifted from C32⋊C4
ρ17444-40-210011-21-2-1-122-12-12-20000    orthogonal lifted from C2×C32⋊C4
ρ1844-400-21001-1-2-123300-30-3020000    orthogonal lifted from C62⋊C4
ρ194-40004400-40-4000000000000000    orthogonal lifted from C4.D4
ρ204-40001-20020-10-300√-3√-3-2√-3-√-32√-3-√-330000    complex faithful
ρ214-4000-2100-132-30-√-3√-3-2√-30-√-32√-3√-3000000    complex faithful
ρ224-40001-20020-103-2√-32√-3√-3-√-30-√-30√-3-30000    complex faithful
ρ234-4000-2100-1-3230-√-3√-302√-3√-30-√-3-2√-300000    complex faithful
ρ244-40001-20020-1032√-3-2√-3-√-3√-30√-30-√-3-30000    complex faithful
ρ254-40001-20020-10-300-√-3-√-32√-3√-3-2√-3√-330000    complex faithful
ρ264-4000-2100-1-3230√-3-√-30-2√-3-√-30√-32√-300000    complex faithful
ρ274-4000-2100-132-30√-3-√-32√-30√-3-2√-3-√-3000000    complex faithful

Permutation representations of (C2×C62).C4
►On 24 points - transitive group 24T585
Generators in S24
(2 6)(3 7)(9 13)(12 16)(18 22)(19 23)

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

(1 11 21 5 15 17)(2 6)(3 19 9 7 23 13)(4 8)(10 14)(12 16)(18 22)(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)
 
G:=sub<Sym(24)| (2,6)(3,7)(9,13)(12,16)(18,22)(19,23), (1,21,15)(2,18,16,6,22,12)(3,9,23)(4,14,24,8,10,20)(5,17,11)(7,13,19), (1,11,21,5,15,17)(2,6)(3,19,9,7,23,13)(4,8)(10,14)(12,16)(18,22)(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)>;
 
G:=Group( (2,6)(3,7)(9,13)(12,16)(18,22)(19,23), (1,21,15)(2,18,16,6,22,12)(3,9,23)(4,14,24,8,10,20)(5,17,11)(7,13,19), (1,11,21,5,15,17)(2,6)(3,19,9,7,23,13)(4,8)(10,14)(12,16)(18,22)(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) );
 
G=PermutationGroup([[(2,6),(3,7),(9,13),(12,16),(18,22),(19,23)], [(1,21,15),(2,18,16,6,22,12),(3,9,23),(4,14,24,8,10,20),(5,17,11),(7,13,19)], [(1,11,21,5,15,17),(2,6),(3,19,9,7,23,13),(4,8),(10,14),(12,16),(18,22),(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)]])
 
G:=TransitiveGroup(24,585);
 

►On 24 points - transitive group 24T625
Generators in S24
(1 7)(2 4)(3 5)(6 8)(9 23)(10 20)(11 21)(12 18)(13 19)(14 24)(15 17)(16 22)

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

(1 9 21 5 13 17)(2 6)(3 19 15 7 23 11)(4 8)(10 14)(12 16)(18 22)(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)
 
G:=sub<Sym(24)| (1,7)(2,4)(3,5)(6,8)(9,23)(10,20)(11,21)(12,18)(13,19)(14,24)(15,17)(16,22), (1,21,13)(2,18,14,6,22,10)(3,15,23)(4,12,24,8,16,20)(5,17,9)(7,11,19), (1,9,21,5,13,17)(2,6)(3,19,15,7,23,11)(4,8)(10,14)(12,16)(18,22)(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)>;
 
G:=Group( (1,7)(2,4)(3,5)(6,8)(9,23)(10,20)(11,21)(12,18)(13,19)(14,24)(15,17)(16,22), (1,21,13)(2,18,14,6,22,10)(3,15,23)(4,12,24,8,16,20)(5,17,9)(7,11,19), (1,9,21,5,13,17)(2,6)(3,19,15,7,23,11)(4,8)(10,14)(12,16)(18,22)(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) );
 
G=PermutationGroup([[(1,7),(2,4),(3,5),(6,8),(9,23),(10,20),(11,21),(12,18),(13,19),(14,24),(15,17),(16,22)], [(1,21,13),(2,18,14,6,22,10),(3,15,23),(4,12,24,8,16,20),(5,17,9),(7,11,19)], [(1,9,21,5,13,17),(2,6),(3,19,15,7,23,11),(4,8),(10,14),(12,16),(18,22),(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)]])
 
G:=TransitiveGroup(24,625);
 

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

4526
0602
3331
0001
,
3532
3126
4406
0003
,
4534
3221
4416
0006
,
2156
0215
3424
3321
G:=sub<GL(4,GF(7))| [4,0,3,0,5,6,3,0,2,0,3,0,6,2,1,1],[3,3,4,0,5,1,4,0,3,2,0,0,2,6,6,3],[4,3,4,0,5,2,4,0,3,2,1,0,4,1,6,6],[2,0,3,3,1,2,4,3,5,1,2,2,6,5,4,1] >;
 

(C2×C62).C4 in GAP, Magma, Sage, TeX 

(C_2\times C_6^2).C_4
 
% in TeX
 
G:=Group("(C2xC6^2).C4");
 
// GroupNames label
 
G:=SmallGroup(288,436);
 
// by ID
 
G=gap.SmallGroup(288,436);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,3,28,141,219,100,675,9413,691,12550,2372]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^2=b^6=c^6=1,d^4=c^3,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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