C2^4:C6 - GroupNames

G = C₂⁴⋊C₆ order 96 = 2⁵·3

1^st semidirect product of C₂⁴ and C₆ acting faithfully

metabelian, soluble, monomial

Aliases: C₂⁴⋊₁C₆, C₂³⋊₁A₄, C₂²≀C₂⋊C₃, C₂²⋊A₄⋊₁C₂, C₂².₂(C₂×A₄), SmallGroup(96,70)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂⁴ — C₂⁴⋊C₆

Chief series

C₁ — C₂² — C₂⁴ — C₂²⋊A₄ — C₂⁴⋊C₆

Lower central

C₂⁴ — C₂⁴⋊C₆

Upper central

C₁

Generators and relations for C₂⁴⋊C₆
G = < a,b,c,d,e | a²=b²=c²=d²=e⁶=1, ab=ba, ac=ca, ad=da, eae^-1=db=bd, bc=cb, ebe^-1=abcd, ede^-1=cd=dc, ece^-1=d >

C₁

₃C₂

₄C₂

₆C₂

₁₆C₃

C₂²

₂C₂²

₃C₂²

₆C₂²

₆C₄

₁₆C₆

C₂³

₃C₂×C₄

₃C₂³

₆C₂³

₆D₄

₆C₂³

₆D₄

₄A₄

₈A₄

C₂⁴

₃C₂×D₄

₃C₂²⋊C₄

₄C₂×A₄

C₂²≀C₂

C₂²⋊A₄

C₂⁴⋊C₆

Character table of C₂⁴⋊C₆

class	1	2A	2B	2C	2D	3A	3B	4	6A	6B
size	1	3	4	6	6	16	16	12	16	16
ρ₁	1	1	1	1	1	1	1	1	1	1	trivial
ρ₂	1	1	-1	1	1	1	1	-1	-1	-1	linear of order 2
ρ₃	1	1	1	1	1	ζ₃	ζ₃²	1	ζ₃²	ζ₃	linear of order 3
ρ₄	1	1	-1	1	1	ζ₃²	ζ₃	-1	ζ₆⁵	ζ₆	linear of order 6
ρ₅	1	1	-1	1	1	ζ₃	ζ₃²	-1	ζ₆	ζ₆⁵	linear of order 6
ρ₆	1	1	1	1	1	ζ₃²	ζ₃	1	ζ₃	ζ₃²	linear of order 3
ρ₇	3	3	3	-1	-1	0	0	-1	0	0	orthogonal lifted from A₄
ρ₈	3	3	-3	-1	-1	0	0	1	0	0	orthogonal lifted from C₂×A₄
ρ₉	6	-2	0	2	-2	0	0	0	0	0	orthogonal faithful
ρ₁₀	6	-2	0	-2	2	0	0	0	0	0	orthogonal faithful

Permutation representations of C₂⁴⋊C₆
►On 8 points - transitive group 8T33

Generators in S₈

(1 4)(2 5)(3 7)(6 8)

(1 6)(2 7)(3 5)(4 8)

(1 6)(2 3)(4 8)(5 7)

(1 8)(2 5)(3 7)(4 6)

(1 2)(3 4 5 6 7 8)

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

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

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

G:=TransitiveGroup(8,33);

►On 12 points - transitive group 12T58

Generators in S₁₂

(2 11)(6 9)

(2 11)(4 7)

(2 11)(3 12)(5 8)(6 9)

(1 10)(2 11)(4 7)(5 8)

(1 2 3 4 5 6)(7 8 9 10 11 12)

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

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

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

G:=TransitiveGroup(12,58);

►On 12 points - transitive group 12T59

Generators in S₁₂

(2 8)(3 12)(4 9)(6 11)

(1 10)(2 8)(5 7)(6 11)

(2 6)(3 4)(8 11)(9 12)

(1 5)(2 6)(7 10)(8 11)

(1 2 3)(4 5 6)(7 8 9 10 11 12)

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

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

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

G:=TransitiveGroup(12,59);

►On 16 points - transitive group 16T183

Generators in S₁₆

(1 6)(2 11)(3 9)(4 14)(5 12)(7 13)(8 10)(15 16)

(1 8)(2 5)(3 16)(4 13)(6 10)(7 14)(9 15)(11 12)

(1 3)(2 4)(5 13)(6 9)(7 12)(8 16)(10 15)(11 14)

(1 2)(3 4)(5 8)(6 11)(7 15)(9 14)(10 12)(13 16)

(2 3 4)(5 6 7 8 9 10)(11 12 13 14 15 16)

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

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

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

G:=TransitiveGroup(16,183);

►On 24 points - transitive group 24T181

Generators in S₂₄

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

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

(2 21)(3 22)(5 24)(6 19)(7 17)(9 13)(10 14)(12 16)

(1 20)(2 21)(4 23)(5 24)(8 18)(9 13)(11 15)(12 16)

(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)| (1,20)(2,13)(3,14)(4,23)(5,12)(6,7)(8,18)(9,21)(10,22)(11,15)(16,24)(17,19), (1,8)(2,9)(3,22)(4,15)(5,16)(6,19)(7,17)(10,14)(11,23)(12,24)(13,21)(18,20), (2,21)(3,22)(5,24)(6,19)(7,17)(9,13)(10,14)(12,16), (1,20)(2,21)(4,23)(5,24)(8,18)(9,13)(11,15)(12,16), (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( (1,20)(2,13)(3,14)(4,23)(5,12)(6,7)(8,18)(9,21)(10,22)(11,15)(16,24)(17,19), (1,8)(2,9)(3,22)(4,15)(5,16)(6,19)(7,17)(10,14)(11,23)(12,24)(13,21)(18,20), (2,21)(3,22)(5,24)(6,19)(7,17)(9,13)(10,14)(12,16), (1,20)(2,21)(4,23)(5,24)(8,18)(9,13)(11,15)(12,16), (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([[(1,20),(2,13),(3,14),(4,23),(5,12),(6,7),(8,18),(9,21),(10,22),(11,15),(16,24),(17,19)], [(1,8),(2,9),(3,22),(4,15),(5,16),(6,19),(7,17),(10,14),(11,23),(12,24),(13,21),(18,20)], [(2,21),(3,22),(5,24),(6,19),(7,17),(9,13),(10,14),(12,16)], [(1,20),(2,21),(4,23),(5,24),(8,18),(9,13),(11,15),(12,16)], [(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(24,181);

►On 24 points - transitive group 24T182

Generators in S₂₄

(2 21)(4 11)(6 17)(7 19)(9 13)(15 23)

(2 13)(4 23)(6 7)(9 21)(11 15)(17 19)

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

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

(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)(4,11)(6,17)(7,19)(9,13)(15,23), (2,13)(4,23)(6,7)(9,21)(11,15)(17,19), (1,8)(2,21)(3,14)(4,11)(5,24)(6,17)(7,19)(9,13)(10,22)(12,16)(15,23)(18,20), (1,20)(2,13)(3,10)(4,23)(5,16)(6,7)(8,18)(9,21)(11,15)(12,24)(14,22)(17,19), (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)(4,11)(6,17)(7,19)(9,13)(15,23), (2,13)(4,23)(6,7)(9,21)(11,15)(17,19), (1,8)(2,21)(3,14)(4,11)(5,24)(6,17)(7,19)(9,13)(10,22)(12,16)(15,23)(18,20), (1,20)(2,13)(3,10)(4,23)(5,16)(6,7)(8,18)(9,21)(11,15)(12,24)(14,22)(17,19), (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),(4,11),(6,17),(7,19),(9,13),(15,23)], [(2,13),(4,23),(6,7),(9,21),(11,15),(17,19)], [(1,8),(2,21),(3,14),(4,11),(5,24),(6,17),(7,19),(9,13),(10,22),(12,16),(15,23),(18,20)], [(1,20),(2,13),(3,10),(4,23),(5,16),(6,7),(8,18),(9,21),(11,15),(12,24),(14,22),(17,19)], [(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(24,182);

►On 24 points - transitive group 24T183

Generators in S₂₄

(1 8)(2 21)(3 10)(6 19)(7 17)(9 13)(14 22)(18 20)

(2 21)(3 10)(4 23)(5 12)(9 13)(11 15)(14 22)(16 24)

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

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

(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)| (1,8)(2,21)(3,10)(6,19)(7,17)(9,13)(14,22)(18,20), (2,21)(3,10)(4,23)(5,12)(9,13)(11,15)(14,22)(16,24), (1,8)(2,21)(3,14)(4,11)(5,24)(6,17)(7,19)(9,13)(10,22)(12,16)(15,23)(18,20), (1,20)(2,13)(3,10)(4,23)(5,16)(6,7)(8,18)(9,21)(11,15)(12,24)(14,22)(17,19), (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( (1,8)(2,21)(3,10)(6,19)(7,17)(9,13)(14,22)(18,20), (2,21)(3,10)(4,23)(5,12)(9,13)(11,15)(14,22)(16,24), (1,8)(2,21)(3,14)(4,11)(5,24)(6,17)(7,19)(9,13)(10,22)(12,16)(15,23)(18,20), (1,20)(2,13)(3,10)(4,23)(5,16)(6,7)(8,18)(9,21)(11,15)(12,24)(14,22)(17,19), (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([[(1,8),(2,21),(3,10),(6,19),(7,17),(9,13),(14,22),(18,20)], [(2,21),(3,10),(4,23),(5,12),(9,13),(11,15),(14,22),(16,24)], [(1,8),(2,21),(3,14),(4,11),(5,24),(6,17),(7,19),(9,13),(10,22),(12,16),(15,23),(18,20)], [(1,20),(2,13),(3,10),(4,23),(5,16),(6,7),(8,18),(9,21),(11,15),(12,24),(14,22),(17,19)], [(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(24,183);

►On 24 points - transitive group 24T184

Generators in S₂₄

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

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

(1 8)(2 9)(4 11)(5 12)(13 21)(15 23)(16 24)(18 20)

(1 8)(3 10)(4 11)(6 7)(14 22)(15 23)(17 19)(18 20)

(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)| (2,13)(3,22)(4,11)(5,24)(6,19)(7,17)(9,21)(10,14)(12,16)(15,23), (1,20)(2,21)(4,15)(5,24)(6,7)(8,18)(9,13)(11,23)(12,16)(17,19), (1,8)(2,9)(4,11)(5,12)(13,21)(15,23)(16,24)(18,20), (1,8)(3,10)(4,11)(6,7)(14,22)(15,23)(17,19)(18,20), (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( (2,13)(3,22)(4,11)(5,24)(6,19)(7,17)(9,21)(10,14)(12,16)(15,23), (1,20)(2,21)(4,15)(5,24)(6,7)(8,18)(9,13)(11,23)(12,16)(17,19), (1,8)(2,9)(4,11)(5,12)(13,21)(15,23)(16,24)(18,20), (1,8)(3,10)(4,11)(6,7)(14,22)(15,23)(17,19)(18,20), (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([[(2,13),(3,22),(4,11),(5,24),(6,19),(7,17),(9,21),(10,14),(12,16),(15,23)], [(1,20),(2,21),(4,15),(5,24),(6,7),(8,18),(9,13),(11,23),(12,16),(17,19)], [(1,8),(2,9),(4,11),(5,12),(13,21),(15,23),(16,24),(18,20)], [(1,8),(3,10),(4,11),(6,7),(14,22),(15,23),(17,19),(18,20)], [(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(24,184);

►On 24 points - transitive group 24T185

Generators in S₂₄

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

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

(1 7)(3 9)(4 12)(6 11)(13 16)(15 18)(19 22)(20 23)

(2 8)(3 9)(5 10)(6 11)(14 17)(15 18)(19 22)(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)

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

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

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

G:=TransitiveGroup(24,185);

►On 24 points - transitive group 24T186

Generators in S₂₄

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

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

(2 8)(3 9)(5 11)(6 12)(13 22)(15 24)(16 19)(18 21)

(1 7)(2 8)(4 10)(5 11)(14 23)(15 24)(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)

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

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

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

G:=TransitiveGroup(24,186);

C₂⁴⋊C₆ is a maximal subgroup of
C₂⁴⋊Dic₃ C₂⁴⋊D₆ C₂⁴⋊A₄ (C₂²×S₃)⋊A₄ A₄≀C₂ (C₂²×D₅)⋊A₄ F₁₆⋊C₂
C₂⁴⋊C₆ is a maximal quotient of
C₂⁴⋊C₁₂ C₂⁴.A₄ (C₂²×C₄).A₄ C₂≀A₄ 2₊¹⁺⁴.C₆ C₂⁴⋊C₁₈ (C₂²×S₃)⋊A₄ (C₂²×D₅)⋊A₄

Polynomial with Galois group C₂⁴⋊C₆ over ℚ

action	f(x)	Disc(f)
8T33	x⁸-x⁷-15x⁶+15x⁵+62x⁴-72x³-53x²+78x-19	2²·3⁶·5⁶·7⁴·251²
12T58	x¹²-23x¹⁰+158x⁸-378x⁶+372x⁴-160x²+25	2¹²·3⁶·5¹⁰·7⁶·13⁸
12T59	x¹²-4x¹¹-9x¹⁰+48x⁹+5x⁸-164x⁷+69x⁶+188x⁵-89x⁴-82x³+23x²+14x+1	2¹⁵·13⁸·47⁴·2029²

Matrix representation of C₂⁴⋊C₆ ►in GL₆(ℤ)

0	1	0	0	0	0
1	0	0	0	0	0
-1	-1	-1	0	0	0
0	0	0	-1	-1	-1
0	0	0	0	0	1
0	0	0	0	1	0

-1	-1	-1	0	0	0
0	0	1	0	0	0
0	1	0	0	0	0
0	0	0	0	0	1
0	0	0	-1	-1	-1
0	0	0	1	0	0

0	0	1	0	0	0
-1	-1	-1	0	0	0
1	0	0	0	0	0
0	0	0	0	0	1
0	0	0	-1	-1	-1
0	0	0	1	0	0

0	1	0	0	0	0
1	0	0	0	0	0
-1	-1	-1	0	0	0
0	0	0	0	1	0
0	0	0	1	0	0
0	0	0	-1	-1	-1

0	0	0	1	0	0
0	0	0	-1	-1	-1
0	0	0	0	1	0
1	0	0	0	0	0
-1	-1	-1	0	0	0
0	1	0	0	0	0

G:=sub<GL(6,Integers())| [0,1,-1,0,0,0,1,0,-1,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,-1,0,1,0,0,0,-1,1,0],[-1,0,0,0,0,0,-1,0,1,0,0,0,-1,1,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,-1,0,0,0,0,1,-1,0],[0,-1,1,0,0,0,0,-1,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,-1,0,0,0,0,1,-1,0],[0,1,-1,0,0,0,1,0,-1,0,0,0,0,0,-1,0,0,0,0,0,0,0,1,-1,0,0,0,1,0,-1,0,0,0,0,0,-1],[0,0,0,1,-1,0,0,0,0,0,-1,1,0,0,0,0,-1,0,1,-1,0,0,0,0,0,-1,1,0,0,0,0,-1,0,0,0,0] >;

C₂⁴⋊C₆ in GAP, Magma, Sage, TeX

C_2^4\rtimes C_6

% in TeX

G:=Group("C2^4:C6");

// GroupNames label

G:=SmallGroup(96,70);

// by ID

G=gap.SmallGroup(96,70);

# by ID

G:=PCGroup([6,-2,-3,-2,2,-2,2,542,116,1443,225,730,1307]);

// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^2=c^2=d^2=e^6=1,a*b=b*a,a*c=c*a,a*d=d*a,e*a*e^-1=d*b=b*d,b*c=c*b,e*b*e^-1=a*b*c*d,e*d*e^-1=c*d=d*c,e*c*e^-1=d>;

// generators/relations

Export

Subgroup lattice of C₂⁴⋊C₆ in TeX
Character table of C₂⁴⋊C₆ in TeX

G = C24⋊C6 order 96 = 25·3

1st semidirect product of C24 and C6 acting faithfully

G = C₂⁴⋊C₆ order 96 = 2⁵·3

1^st semidirect product of C₂⁴ and C₆ acting faithfully