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G = (C2×C62)⋊C4order 288 = 25·32

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

metabelian, soluble, monomial

Aliases: (C2×C62)⋊4C4, C23⋊(C32⋊C4), C62⋊C43C2, C323(C23⋊C4), C62.10(C2×C4), C2.10(C62⋊C4), (C2×C3⋊Dic3)⋊3C4, (C2×C3⋊S3).16D4, C22.4(C2×C32⋊C4), (C2×C327D4).3C2, (C3×C6).20(C22⋊C4), (C22×C3⋊S3).5C22, SmallGroup(288,434)

Series: Derived Chief Lower central Upper central

C1C62 — (C2×C62)⋊C4
C1C32C3×C6C2×C3⋊S3C22×C3⋊S3C62⋊C4 — (C2×C62)⋊C4
C32C3×C6C62 — (C2×C62)⋊C4
C1C2C22C23

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 [×4], C3 [×2], C4 [×3], C22, C22 [×5], S3 [×4], C6 [×10], C2×C4 [×3], D4 [×2], C23, C23, C32, Dic3 [×4], D6 [×8], C2×C6 [×10], C22⋊C4 [×2], C2×D4, C3⋊S3 [×2], C3×C6, C3×C6 [×2], C2×Dic3 [×2], C3⋊D4 [×8], C22×S3 [×2], C22×C6 [×2], C23⋊C4, C3⋊Dic3, C32⋊C4 [×2], C2×C3⋊S3 [×2], C2×C3⋊S3, C62, C62 [×2], C2×C3⋊D4 [×2], C2×C3⋊Dic3, C327D4 [×2], C2×C32⋊C4 [×2], C22×C3⋊S3, C2×C62, C62⋊C4 [×2], C2×C327D4, (C2×C62)⋊C4
Quotients: C1, C2 [×3], C4 [×2], C22, C2×C4, D4 [×2], 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 12)(8 10)(9 11)(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 12)(8 10)(9 11)(13 18 17 16 15 14)(19 24 23 22 21 20)
(1 19 8 13)(2 21 7 17)(3 23 9 15)(4 22 10 16)(5 24 12 14)(6 20 11 18)

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

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

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

G:=TransitiveGroup(24,584);

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

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

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

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

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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