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G = C62⋊C4order 144 = 24·32

1st semidirect product of C62 and C4 acting faithfully

metabelian, soluble, monomial

Aliases: C621C4, C3⋊S3.5D4, C22⋊(C32⋊C4), C322(C22⋊C4), (C2×C3⋊S3)⋊3C4, (C2×C32⋊C4)⋊2C2, (C3×C6).7(C2×C4), C2.7(C2×C32⋊C4), (C22×C3⋊S3).3C2, (C2×C3⋊S3).10C22, SmallGroup(144,136)

Series: Derived Chief Lower central Upper central

C1C3×C6 — C62⋊C4
C1C32C3⋊S3C2×C3⋊S3C2×C32⋊C4 — C62⋊C4
C32C3×C6 — C62⋊C4
C1C2C22

Generators and relations for C62⋊C4
 G = < a,b,c | a6=b6=c4=1, ab=ba, cac-1=a-1b, cbc-1=a4b >

Subgroups: 342 in 66 conjugacy classes, 14 normal (10 characteristic)
C1, C2, C2 [×4], C3 [×2], C4 [×2], C22, C22 [×4], S3 [×8], C6 [×6], C2×C4 [×2], C23, C32, D6 [×12], C2×C6 [×2], C22⋊C4, C3⋊S3 [×2], C3⋊S3, C3×C6, C3×C6, C22×S3 [×2], C32⋊C4 [×2], C2×C3⋊S3 [×2], C2×C3⋊S3 [×2], C62, C2×C32⋊C4 [×2], C22×C3⋊S3, C62⋊C4
Quotients: C1, C2 [×3], C4 [×2], C22, C2×C4, D4 [×2], C22⋊C4, C32⋊C4, C2×C32⋊C4, C62⋊C4

Character table of C62⋊C4

 class 12A2B2C2D2E3A3B4A4B4C4D6A6B6C6D6E6F
 size 11299184418181818444444
ρ1111111111111111111    trivial
ρ211-111-1111-11-1-111-1-1-1    linear of order 2
ρ311111111-1-1-1-1111111    linear of order 2
ρ411-111-111-11-11-111-1-1-1    linear of order 2
ρ5111-1-1-111ii-i-i111111    linear of order 4
ρ611-1-1-1111i-i-ii-111-1-1-1    linear of order 4
ρ7111-1-1-111-i-iii111111    linear of order 4
ρ811-1-1-1111-iii-i-111-1-1-1    linear of order 4
ρ92-202-202200000-2-2000    orthogonal lifted from D4
ρ102-20-2202200000-2-2000    orthogonal lifted from D4
ρ114-40000-21000032-1-300    orthogonal faithful
ρ124-40000-210000-32-1300    orthogonal faithful
ρ134-400001-200000-120-33    orthogonal faithful
ρ144440001-20000-21-2-211    orthogonal lifted from C32⋊C4
ρ154-400001-200000-1203-3    orthogonal faithful
ρ1644-40001-2000021-22-1-1    orthogonal lifted from C2×C32⋊C4
ρ17444000-2100001-211-2-2    orthogonal lifted from C32⋊C4
ρ1844-4000-210000-1-21-122    orthogonal lifted from C2×C32⋊C4

Permutation representations of C62⋊C4
On 12 points - transitive group 12T82
Generators in S12
(1 2 3)(4 5 6)(7 8 9 10 11 12)
(1 4 2 5 3 6)(7 10)(8 11)(9 12)
(1 10)(2 8 3 12)(4 9 6 11)(5 7)

G:=sub<Sym(12)| (1,2,3)(4,5,6)(7,8,9,10,11,12), (1,4,2,5,3,6)(7,10)(8,11)(9,12), (1,10)(2,8,3,12)(4,9,6,11)(5,7)>;

G:=Group( (1,2,3)(4,5,6)(7,8,9,10,11,12), (1,4,2,5,3,6)(7,10)(8,11)(9,12), (1,10)(2,8,3,12)(4,9,6,11)(5,7) );

G=PermutationGroup([(1,2,3),(4,5,6),(7,8,9,10,11,12)], [(1,4,2,5,3,6),(7,10),(8,11),(9,12)], [(1,10),(2,8,3,12),(4,9,6,11),(5,7)])

G:=TransitiveGroup(12,82);

On 24 points - transitive group 24T272
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)
(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 7 13)(2 19 9 17)(3 21 8 15)(4 20 10 16)(5 22 12 14)(6 24 11 18)

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

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

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

G:=TransitiveGroup(24,272);

On 24 points - transitive group 24T273
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)
(1 14)(2 15)(3 16)(4 17)(5 18)(6 13)(7 22 11 20 9 24)(8 23 12 21 10 19)
(1 22 14 9)(2 8 13 23)(3 20 18 11)(4 12 17 19)(5 24 16 7)(6 10 15 21)

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

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

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

G:=TransitiveGroup(24,273);

On 24 points - transitive group 24T274
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)
(1 6 11 10 4 7)(2 5 12 9 3 8)(13 24 15 20 17 22)(14 19 16 21 18 23)
(1 13 3 21)(2 23 4 17)(5 14 10 20)(6 24 9 16)(7 22 8 18)(11 15 12 19)

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

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

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

G:=TransitiveGroup(24,274);

C62⋊C4 is a maximal subgroup of
C62.2D4  (C6×C12)⋊C4  (C2×C62)⋊C4  C62.9D4  D6≀C2  C62⋊Q8  (C6×C12)⋊5C4  D4×C32⋊C4  D6⋊(C32⋊C4)  C6211Dic3  C62⋊Dic3
C62⋊C4 is a maximal quotient of
(C6×C12)⋊C4  C62.6(C2×C4)  C3⋊Dic3.D4  (C6×C12)⋊2C4  C3⋊S3.5D8  C326C4≀C2  C3⋊S3.5Q16  C327C4≀C2  (C2×C62)⋊C4  C623C8  (C2×C62).C4  C22⋊(He3⋊C4)  D6⋊(C32⋊C4)  C6211Dic3

Polynomial with Galois group C62⋊C4 over ℚ
actionf(x)Disc(f)
12T82x12-6x11+20x10-45x9+72x8-84x7+67x6-30x5+9x3-4x+1-39·138·172

Matrix representation of C62⋊C4 in GL4(ℤ) generated by

-1-100
1000
00-10
000-1
,
1100
-1000
0011
00-10
,
0010
0001
1000
-1-100
G:=sub<GL(4,Integers())| [-1,1,0,0,-1,0,0,0,0,0,-1,0,0,0,0,-1],[1,-1,0,0,1,0,0,0,0,0,1,-1,0,0,1,0],[0,0,1,-1,0,0,0,-1,1,0,0,0,0,1,0,0] >;

C62⋊C4 in GAP, Magma, Sage, TeX

C_6^2\rtimes C_4
% in TeX

G:=Group("C6^2:C4");
// GroupNames label

G:=SmallGroup(144,136);
// by ID

G=gap.SmallGroup(144,136);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-3,3,24,121,3364,256,4613,881]);
// Polycyclic

G:=Group<a,b,c|a^6=b^6=c^4=1,a*b=b*a,c*a*c^-1=a^-1*b,c*b*c^-1=a^4*b>;
// generators/relations

Export

Character table of C62⋊C4 in TeX

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