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G = C42⋊C6order 96 = 25·3

1st semidirect product of C42 and C6 acting faithfully

metabelian, soluble, monomial

Aliases: C421C6, C23.1A4, C42⋊C31C2, C422C2⋊C3, C22.3(C2×A4), SmallGroup(96,71)

Series: Derived Chief Lower central Upper central

C1C42 — C42⋊C6
C1C22C42C42⋊C3 — C42⋊C6
C42 — C42⋊C6
C1

Generators and relations for C42⋊C6
 G = < a,b,c | a4=b4=c6=1, ab=ba, cac-1=b-1, cbc-1=a-1b >

3C2
4C2
16C3
6C4
6C22
6C4
16C6
3C2×C4
3C2×C4
4A4
3C22⋊C4
3C4⋊C4
4C2×A4

Character table of C42⋊C6

 class 12A2B3A3B4A4B4C6A6B
 size 134161666121616
ρ11111111111    trivial
ρ211-11111-1-1-1    linear of order 2
ρ311-1ζ3ζ3211-1ζ6ζ65    linear of order 6
ρ411-1ζ32ζ311-1ζ65ζ6    linear of order 6
ρ5111ζ3ζ32111ζ32ζ3    linear of order 3
ρ6111ζ32ζ3111ζ3ζ32    linear of order 3
ρ733300-1-1-100    orthogonal lifted from A4
ρ833-300-1-1100    orthogonal lifted from C2×A4
ρ96-2000-2i2i000    complex faithful
ρ106-20002i-2i000    complex faithful

Permutation representations of C42⋊C6
On 16 points - transitive group 16T184
Generators in S16
(1 13 4 8)(2 5 3 16)(6 7 9 15)(10 14 12 11)
(1 7 3 12)(2 10 4 15)(5 14 8 6)(9 16 11 13)
(2 3 4)(5 6 7 8 9 10)(11 12 13 14 15 16)

G:=sub<Sym(16)| (1,13,4,8)(2,5,3,16)(6,7,9,15)(10,14,12,11), (1,7,3,12)(2,10,4,15)(5,14,8,6)(9,16,11,13), (2,3,4)(5,6,7,8,9,10)(11,12,13,14,15,16)>;

G:=Group( (1,13,4,8)(2,5,3,16)(6,7,9,15)(10,14,12,11), (1,7,3,12)(2,10,4,15)(5,14,8,6)(9,16,11,13), (2,3,4)(5,6,7,8,9,10)(11,12,13,14,15,16) );

G=PermutationGroup([(1,13,4,8),(2,5,3,16),(6,7,9,15),(10,14,12,11)], [(1,7,3,12),(2,10,4,15),(5,14,8,6),(9,16,11,13)], [(2,3,4),(5,6,7,8,9,10),(11,12,13,14,15,16)])

G:=TransitiveGroup(16,184);

On 24 points - transitive group 24T120
Generators in S24
(1 12 6 9)(2 23)(3 18 5 15)(4 20)(7 14)(8 24 11 21)(10 17)(13 22 16 19)
(1 22)(2 14 4 17)(3 8 5 11)(6 19)(7 20 10 23)(9 13)(12 16)(15 21 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)

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

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

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

G:=TransitiveGroup(24,120);

On 24 points - transitive group 24T121
Generators in S24
(1 10 7 4)(2 21 8 15)(3 13)(5 24 11 18)(6 22)(9 19)(12 16)(14 23 20 17)
(1 14 7 20)(2 18)(3 12 9 6)(4 17 10 23)(5 21)(8 24)(11 15)(13 16 19 22)
(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)| (1,10,7,4)(2,21,8,15)(3,13)(5,24,11,18)(6,22)(9,19)(12,16)(14,23,20,17), (1,14,7,20)(2,18)(3,12,9,6)(4,17,10,23)(5,21)(8,24)(11,15)(13,16,19,22), (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( (1,10,7,4)(2,21,8,15)(3,13)(5,24,11,18)(6,22)(9,19)(12,16)(14,23,20,17), (1,14,7,20)(2,18)(3,12,9,6)(4,17,10,23)(5,21)(8,24)(11,15)(13,16,19,22), (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([(1,10,7,4),(2,21,8,15),(3,13),(5,24,11,18),(6,22),(9,19),(12,16),(14,23,20,17)], [(1,14,7,20),(2,18),(3,12,9,6),(4,17,10,23),(5,21),(8,24),(11,15),(13,16,19,22)], [(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(24,121);

On 24 points - transitive group 24T122
Generators in S24
(2 21 9 13)(3 22 10 14)(4 11)(5 16 12 24)(6 19 7 17)(15 23)
(1 18 8 20)(2 13 9 21)(3 10)(4 23 11 15)(5 16 12 24)(14 22)
(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)| (2,21,9,13)(3,22,10,14)(4,11)(5,16,12,24)(6,19,7,17)(15,23), (1,18,8,20)(2,13,9,21)(3,10)(4,23,11,15)(5,16,12,24)(14,22), (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( (2,21,9,13)(3,22,10,14)(4,11)(5,16,12,24)(6,19,7,17)(15,23), (1,18,8,20)(2,13,9,21)(3,10)(4,23,11,15)(5,16,12,24)(14,22), (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([(2,21,9,13),(3,22,10,14),(4,11),(5,16,12,24),(6,19,7,17),(15,23)], [(1,18,8,20),(2,13,9,21),(3,10),(4,23,11,15),(5,16,12,24),(14,22)], [(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(24,122);

C42⋊C6 is a maximal subgroup of   C24.6A4  C42⋊C3⋊S3  (C4×C20)⋊C6
C42⋊C6 is a maximal quotient of   C42⋊C12  C24.2A4  C23.19(C2×A4)  C42⋊C18  C42⋊C3⋊S3  (C4×C20)⋊C6

Matrix representation of C42⋊C6 in GL6(𝔽13)

11210080
11212031
11210081
2311850
508015
0120000
,
000100
000010
8551120
0121000
0120000
080015
,
001000
100000
010000
1122578
3210580
0001200

G:=sub<GL(6,GF(13))| [11,1,11,2,5,0,2,12,2,3,0,12,10,12,10,11,8,0,0,0,0,8,0,0,8,3,8,5,1,0,0,1,1,0,5,0],[0,0,8,0,0,0,0,0,5,12,12,8,0,0,5,1,0,0,1,0,1,0,0,0,0,1,12,0,0,1,0,0,0,0,0,5],[0,1,0,11,3,0,0,0,1,2,2,0,1,0,0,2,10,0,0,0,0,5,5,12,0,0,0,7,8,0,0,0,0,8,0,0] >;

C42⋊C6 in GAP, Magma, Sage, TeX

C_4^2\rtimes C_6
% in TeX

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

G:=SmallGroup(96,71);
// by ID

G=gap.SmallGroup(96,71);
# by ID

G:=PCGroup([6,-2,-3,-2,2,-2,2,542,116,230,1443,801,69,730,1307]);
// Polycyclic

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

Export

Subgroup lattice of C42⋊C6 in TeX
Character table of C42⋊C6 in TeX

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