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## G = (C2×C62)⋊C4order 288 = 25·32

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

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 [×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 1 2A 2B 2C 2D 2E 3A 3B 4A 4B 4C 4D 4E 6A 6B 6C 6D 6E 6F 6G 6H 6I 6J 6K 6L 6M 6N size 1 1 2 4 18 18 4 4 36 36 36 36 36 4 4 4 4 4 4 4 4 4 4 4 4 4 4 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 1 1 1 1 1 1 1 -1 1 -1 -1 -1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 linear of order 2 ρ3 1 1 1 -1 1 1 1 1 1 -1 -1 -1 1 -1 -1 1 -1 -1 -1 -1 -1 -1 1 1 1 1 1 linear of order 2 ρ4 1 1 1 -1 1 1 1 1 -1 -1 1 1 -1 -1 -1 1 -1 -1 -1 -1 -1 -1 1 1 1 1 1 linear of order 2 ρ5 1 1 1 -1 -1 -1 1 1 -i 1 -i i i -1 -1 1 -1 -1 -1 -1 -1 -1 1 1 1 1 1 linear of order 4 ρ6 1 1 1 -1 -1 -1 1 1 i 1 i -i -i -1 -1 1 -1 -1 -1 -1 -1 -1 1 1 1 1 1 linear of order 4 ρ7 1 1 1 1 -1 -1 1 1 -i -1 i -i i 1 1 1 1 1 1 1 1 1 1 1 1 1 1 linear of order 4 ρ8 1 1 1 1 -1 -1 1 1 i -1 -i i -i 1 1 1 1 1 1 1 1 1 1 1 1 1 1 linear of order 4 ρ9 2 2 -2 0 -2 2 2 2 0 0 0 0 0 0 0 2 0 0 0 0 0 0 -2 -2 2 -2 -2 orthogonal lifted from D4 ρ10 2 2 -2 0 2 -2 2 2 0 0 0 0 0 0 0 2 0 0 0 0 0 0 -2 -2 2 -2 -2 orthogonal lifted from D4 ρ11 4 4 4 4 0 0 1 -2 0 0 0 0 0 1 1 1 -2 -2 1 -2 1 -2 -2 1 -2 1 -2 orthogonal lifted from C32⋊C4 ρ12 4 4 4 -4 0 0 1 -2 0 0 0 0 0 -1 -1 1 2 2 -1 2 -1 2 -2 1 -2 1 -2 orthogonal lifted from C2×C32⋊C4 ρ13 4 4 4 4 0 0 -2 1 0 0 0 0 0 -2 -2 -2 1 1 -2 1 -2 1 1 -2 1 -2 1 orthogonal lifted from C32⋊C4 ρ14 4 4 -4 0 0 0 1 -2 0 0 0 0 0 3 3 1 0 0 -3 0 -3 0 2 -1 -2 -1 2 orthogonal lifted from C62⋊C4 ρ15 4 4 -4 0 0 0 1 -2 0 0 0 0 0 -3 -3 1 0 0 3 0 3 0 2 -1 -2 -1 2 orthogonal lifted from C62⋊C4 ρ16 4 -4 0 0 0 0 4 4 0 0 0 0 0 0 0 -4 0 0 0 0 0 0 0 0 -4 0 0 orthogonal lifted from C23⋊C4 ρ17 4 4 -4 0 0 0 -2 1 0 0 0 0 0 0 0 -2 -3 3 0 -3 0 3 -1 2 1 2 -1 orthogonal lifted from C62⋊C4 ρ18 4 4 -4 0 0 0 -2 1 0 0 0 0 0 0 0 -2 3 -3 0 3 0 -3 -1 2 1 2 -1 orthogonal lifted from C62⋊C4 ρ19 4 4 4 -4 0 0 -2 1 0 0 0 0 0 2 2 -2 -1 -1 2 -1 2 -1 1 -2 1 -2 1 orthogonal lifted from C2×C32⋊C4 ρ20 4 -4 0 0 0 0 1 -2 0 0 0 0 0 √-3 -√-3 -1 0 2√-3 √-3 0 -√-3 -2√-3 0 -3 2 3 0 complex faithful ρ21 4 -4 0 0 0 0 -2 1 0 0 0 0 0 -2√-3 2√-3 2 -√-3 √-3 0 √-3 0 -√-3 -3 0 -1 0 3 complex faithful ρ22 4 -4 0 0 0 0 1 -2 0 0 0 0 0 -√-3 √-3 -1 0 -2√-3 -√-3 0 √-3 2√-3 0 -3 2 3 0 complex faithful ρ23 4 -4 0 0 0 0 -2 1 0 0 0 0 0 2√-3 -2√-3 2 √-3 -√-3 0 -√-3 0 √-3 -3 0 -1 0 3 complex faithful ρ24 4 -4 0 0 0 0 1 -2 0 0 0 0 0 -√-3 √-3 -1 2√-3 0 √-3 -2√-3 -√-3 0 0 3 2 -3 0 complex faithful ρ25 4 -4 0 0 0 0 -2 1 0 0 0 0 0 0 0 2 -√-3 -√-3 2√-3 √-3 -2√-3 √-3 3 0 -1 0 -3 complex faithful ρ26 4 -4 0 0 0 0 -2 1 0 0 0 0 0 0 0 2 √-3 √-3 -2√-3 -√-3 2√-3 -√-3 3 0 -1 0 -3 complex faithful ρ27 4 -4 0 0 0 0 1 -2 0 0 0 0 0 √-3 -√-3 -1 -2√-3 0 -√-3 2√-3 √-3 0 0 3 2 -3 0 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

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

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