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## G = C2×A4⋊C4order 96 = 25·3

### Direct product of C2 and A4⋊C4

Aliases: C2×A4⋊C4, C24.S3, C23⋊Dic3, C22.6S4, C23.4D6, (C2×A4)⋊C4, A42(C2×C4), C2.2(C2×S4), (C22×A4).C2, C22⋊(C2×Dic3), (C2×A4).4C22, SmallGroup(96,194)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — A4 — C2×A4⋊C4
 Chief series C1 — C22 — A4 — C2×A4 — A4⋊C4 — C2×A4⋊C4
 Lower central A4 — C2×A4⋊C4
 Upper central C1 — C22

Generators and relations for C2×A4⋊C4
G = < a,b,c,d,e | a2=b2=c2=d3=e4=1, ab=ba, ac=ca, ad=da, ae=ea, dbd-1=ebe-1=bc=cb, dcd-1=b, ce=ec, ede-1=d-1 >

Subgroups: 188 in 63 conjugacy classes, 18 normal (10 characteristic)
C1, C2, C2 [×2], C2 [×4], C3, C4 [×4], C22 [×2], C22 [×10], C6 [×3], C2×C4 [×8], C23, C23 [×2], C23 [×4], Dic3 [×2], A4, C2×C6, C22⋊C4 [×4], C22×C4 [×2], C24, C2×Dic3, C2×A4, C2×A4 [×2], C2×C22⋊C4, A4⋊C4 [×2], C22×A4, C2×A4⋊C4
Quotients: C1, C2 [×3], C4 [×2], C22, S3, C2×C4, Dic3 [×2], D6, C2×Dic3, S4, A4⋊C4 [×2], C2×S4, C2×A4⋊C4

Character table of C2×A4⋊C4

 class 1 2A 2B 2C 2D 2E 2F 2G 3 4A 4B 4C 4D 4E 4F 4G 4H 6A 6B 6C size 1 1 1 1 3 3 3 3 8 6 6 6 6 6 6 6 6 8 8 8 ρ1 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 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 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 linear of order 2 ρ5 1 -1 1 -1 -1 1 -1 1 1 -i i i -i -i i i -i -1 -1 1 linear of order 4 ρ6 1 -1 1 -1 -1 1 -1 1 1 i -i -i i i -i -i i -1 -1 1 linear of order 4 ρ7 1 -1 -1 1 1 -1 -1 1 1 -i -i -i i i i i -i 1 -1 -1 linear of order 4 ρ8 1 -1 -1 1 1 -1 -1 1 1 i i i -i -i -i -i i 1 -1 -1 linear of order 4 ρ9 2 2 -2 -2 -2 -2 2 2 -1 0 0 0 0 0 0 0 0 1 -1 1 orthogonal lifted from D6 ρ10 2 2 2 2 2 2 2 2 -1 0 0 0 0 0 0 0 0 -1 -1 -1 orthogonal lifted from S3 ρ11 2 -2 2 -2 -2 2 -2 2 -1 0 0 0 0 0 0 0 0 1 1 -1 symplectic lifted from Dic3, Schur index 2 ρ12 2 -2 -2 2 2 -2 -2 2 -1 0 0 0 0 0 0 0 0 -1 1 1 symplectic lifted from Dic3, Schur index 2 ρ13 3 3 3 3 -1 -1 -1 -1 0 -1 1 -1 1 -1 1 -1 1 0 0 0 orthogonal lifted from S4 ρ14 3 3 -3 -3 1 1 -1 -1 0 -1 -1 1 -1 1 1 -1 1 0 0 0 orthogonal lifted from C2×S4 ρ15 3 3 3 3 -1 -1 -1 -1 0 1 -1 1 -1 1 -1 1 -1 0 0 0 orthogonal lifted from S4 ρ16 3 3 -3 -3 1 1 -1 -1 0 1 1 -1 1 -1 -1 1 -1 0 0 0 orthogonal lifted from C2×S4 ρ17 3 -3 3 -3 1 -1 1 -1 0 -i -i i i -i -i i i 0 0 0 complex lifted from A4⋊C4 ρ18 3 -3 3 -3 1 -1 1 -1 0 i i -i -i i i -i -i 0 0 0 complex lifted from A4⋊C4 ρ19 3 -3 -3 3 -1 1 1 -1 0 i -i i i -i i -i -i 0 0 0 complex lifted from A4⋊C4 ρ20 3 -3 -3 3 -1 1 1 -1 0 -i i -i -i i -i i i 0 0 0 complex lifted from A4⋊C4

Permutation representations of C2×A4⋊C4
On 24 points - transitive group 24T123
Generators in S24
(1 11)(2 12)(3 9)(4 10)(5 16)(6 13)(7 14)(8 15)(17 21)(18 22)(19 23)(20 24)
(1 3)(2 12)(4 10)(5 14)(6 8)(7 16)(9 11)(13 15)(17 21)(18 24)(19 23)(20 22)
(1 9)(2 10)(3 11)(4 12)(5 16)(6 13)(7 14)(8 15)(17 19)(18 20)(21 23)(22 24)
(1 19 7)(2 8 20)(3 17 5)(4 6 18)(9 21 16)(10 13 22)(11 23 14)(12 15 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)| (1,11)(2,12)(3,9)(4,10)(5,16)(6,13)(7,14)(8,15)(17,21)(18,22)(19,23)(20,24), (1,3)(2,12)(4,10)(5,14)(6,8)(7,16)(9,11)(13,15)(17,21)(18,24)(19,23)(20,22), (1,9)(2,10)(3,11)(4,12)(5,16)(6,13)(7,14)(8,15)(17,19)(18,20)(21,23)(22,24), (1,19,7)(2,8,20)(3,17,5)(4,6,18)(9,21,16)(10,13,22)(11,23,14)(12,15,24), (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( (1,11)(2,12)(3,9)(4,10)(5,16)(6,13)(7,14)(8,15)(17,21)(18,22)(19,23)(20,24), (1,3)(2,12)(4,10)(5,14)(6,8)(7,16)(9,11)(13,15)(17,21)(18,24)(19,23)(20,22), (1,9)(2,10)(3,11)(4,12)(5,16)(6,13)(7,14)(8,15)(17,19)(18,20)(21,23)(22,24), (1,19,7)(2,8,20)(3,17,5)(4,6,18)(9,21,16)(10,13,22)(11,23,14)(12,15,24), (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([(1,11),(2,12),(3,9),(4,10),(5,16),(6,13),(7,14),(8,15),(17,21),(18,22),(19,23),(20,24)], [(1,3),(2,12),(4,10),(5,14),(6,8),(7,16),(9,11),(13,15),(17,21),(18,24),(19,23),(20,22)], [(1,9),(2,10),(3,11),(4,12),(5,16),(6,13),(7,14),(8,15),(17,19),(18,20),(21,23),(22,24)], [(1,19,7),(2,8,20),(3,17,5),(4,6,18),(9,21,16),(10,13,22),(11,23,14),(12,15,24)], [(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(24,123);

On 24 points - transitive group 24T124
Generators in S24
(1 6)(2 7)(3 8)(4 5)(9 14)(10 15)(11 16)(12 13)(17 22)(18 23)(19 24)(20 21)
(2 7)(4 5)(9 14)(10 15)(11 16)(12 13)(18 23)(20 21)
(1 6)(2 7)(3 8)(4 5)(17 22)(18 23)(19 24)(20 21)
(1 9 23)(2 24 10)(3 11 21)(4 22 12)(5 17 13)(6 14 18)(7 19 15)(8 16 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)| (1,6)(2,7)(3,8)(4,5)(9,14)(10,15)(11,16)(12,13)(17,22)(18,23)(19,24)(20,21), (2,7)(4,5)(9,14)(10,15)(11,16)(12,13)(18,23)(20,21), (1,6)(2,7)(3,8)(4,5)(17,22)(18,23)(19,24)(20,21), (1,9,23)(2,24,10)(3,11,21)(4,22,12)(5,17,13)(6,14,18)(7,19,15)(8,16,20), (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( (1,6)(2,7)(3,8)(4,5)(9,14)(10,15)(11,16)(12,13)(17,22)(18,23)(19,24)(20,21), (2,7)(4,5)(9,14)(10,15)(11,16)(12,13)(18,23)(20,21), (1,6)(2,7)(3,8)(4,5)(17,22)(18,23)(19,24)(20,21), (1,9,23)(2,24,10)(3,11,21)(4,22,12)(5,17,13)(6,14,18)(7,19,15)(8,16,20), (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([(1,6),(2,7),(3,8),(4,5),(9,14),(10,15),(11,16),(12,13),(17,22),(18,23),(19,24),(20,21)], [(2,7),(4,5),(9,14),(10,15),(11,16),(12,13),(18,23),(20,21)], [(1,6),(2,7),(3,8),(4,5),(17,22),(18,23),(19,24),(20,21)], [(1,9,23),(2,24,10),(3,11,21),(4,22,12),(5,17,13),(6,14,18),(7,19,15),(8,16,20)], [(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(24,124);

On 24 points - transitive group 24T132
Generators in S24
(1 14)(2 15)(3 16)(4 13)(5 17)(6 18)(7 19)(8 20)(9 21)(10 22)(11 23)(12 24)
(1 4)(2 3)(5 11)(6 8)(7 9)(10 12)(13 14)(15 16)(17 23)(18 20)(19 21)(22 24)
(1 3)(2 4)(5 9)(6 10)(7 11)(8 12)(13 15)(14 16)(17 21)(18 22)(19 23)(20 24)
(1 5 10)(2 11 6)(3 7 12)(4 9 8)(13 21 20)(14 17 22)(15 23 18)(16 19 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)| (1,14)(2,15)(3,16)(4,13)(5,17)(6,18)(7,19)(8,20)(9,21)(10,22)(11,23)(12,24), (1,4)(2,3)(5,11)(6,8)(7,9)(10,12)(13,14)(15,16)(17,23)(18,20)(19,21)(22,24), (1,3)(2,4)(5,9)(6,10)(7,11)(8,12)(13,15)(14,16)(17,21)(18,22)(19,23)(20,24), (1,5,10)(2,11,6)(3,7,12)(4,9,8)(13,21,20)(14,17,22)(15,23,18)(16,19,24), (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( (1,14)(2,15)(3,16)(4,13)(5,17)(6,18)(7,19)(8,20)(9,21)(10,22)(11,23)(12,24), (1,4)(2,3)(5,11)(6,8)(7,9)(10,12)(13,14)(15,16)(17,23)(18,20)(19,21)(22,24), (1,3)(2,4)(5,9)(6,10)(7,11)(8,12)(13,15)(14,16)(17,21)(18,22)(19,23)(20,24), (1,5,10)(2,11,6)(3,7,12)(4,9,8)(13,21,20)(14,17,22)(15,23,18)(16,19,24), (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([(1,14),(2,15),(3,16),(4,13),(5,17),(6,18),(7,19),(8,20),(9,21),(10,22),(11,23),(12,24)], [(1,4),(2,3),(5,11),(6,8),(7,9),(10,12),(13,14),(15,16),(17,23),(18,20),(19,21),(22,24)], [(1,3),(2,4),(5,9),(6,10),(7,11),(8,12),(13,15),(14,16),(17,21),(18,22),(19,23),(20,24)], [(1,5,10),(2,11,6),(3,7,12),(4,9,8),(13,21,20),(14,17,22),(15,23,18),(16,19,24)], [(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(24,132);

On 24 points - transitive group 24T148
Generators in S24
(1 23)(2 24)(3 21)(4 22)(5 10)(6 11)(7 12)(8 9)(13 19)(14 20)(15 17)(16 18)
(2 4)(5 7)(6 8)(9 11)(10 12)(13 15)(17 19)(22 24)
(1 3)(2 4)(13 15)(14 16)(17 19)(18 20)(21 23)(22 24)
(1 11 19)(2 20 12)(3 9 17)(4 18 10)(5 22 16)(6 13 23)(7 24 14)(8 15 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)| (1,23)(2,24)(3,21)(4,22)(5,10)(6,11)(7,12)(8,9)(13,19)(14,20)(15,17)(16,18), (2,4)(5,7)(6,8)(9,11)(10,12)(13,15)(17,19)(22,24), (1,3)(2,4)(13,15)(14,16)(17,19)(18,20)(21,23)(22,24), (1,11,19)(2,20,12)(3,9,17)(4,18,10)(5,22,16)(6,13,23)(7,24,14)(8,15,21), (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( (1,23)(2,24)(3,21)(4,22)(5,10)(6,11)(7,12)(8,9)(13,19)(14,20)(15,17)(16,18), (2,4)(5,7)(6,8)(9,11)(10,12)(13,15)(17,19)(22,24), (1,3)(2,4)(13,15)(14,16)(17,19)(18,20)(21,23)(22,24), (1,11,19)(2,20,12)(3,9,17)(4,18,10)(5,22,16)(6,13,23)(7,24,14)(8,15,21), (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([(1,23),(2,24),(3,21),(4,22),(5,10),(6,11),(7,12),(8,9),(13,19),(14,20),(15,17),(16,18)], [(2,4),(5,7),(6,8),(9,11),(10,12),(13,15),(17,19),(22,24)], [(1,3),(2,4),(13,15),(14,16),(17,19),(18,20),(21,23),(22,24)], [(1,11,19),(2,20,12),(3,9,17),(4,18,10),(5,22,16),(6,13,23),(7,24,14),(8,15,21)], [(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(24,148);

C2×A4⋊C4 is a maximal subgroup of   C24.3D6  C24.4D6  C24.5D6  C25.S3  C2×C4×S4  D42S4
C2×A4⋊C4 is a maximal quotient of   A4⋊M4(2)  C24.4D6  C23.15S4  U2(𝔽3)⋊C2  C4.A4⋊C4  (C2×C4).S4  C25.S3

Matrix representation of C2×A4⋊C4 in GL5(𝔽13)

 1 0 0 0 0 0 1 0 0 0 0 0 12 0 0 0 0 0 12 0 0 0 0 0 12
,
 1 0 0 0 0 0 1 0 0 0 0 0 12 0 12 0 0 0 12 0 0 0 0 0 1
,
 1 0 0 0 0 0 1 0 0 0 0 0 1 1 1 0 0 0 12 0 0 0 0 0 12
,
 0 12 0 0 0 1 12 0 0 0 0 0 1 0 0 0 0 11 12 12 0 0 0 1 0
,
 11 4 0 0 0 2 2 0 0 0 0 0 8 0 0 0 0 0 0 8 0 0 0 8 0

G:=sub<GL(5,GF(13))| [1,0,0,0,0,0,1,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,12],[1,0,0,0,0,0,1,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,12,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,1,12,0,0,0,1,0,12],[0,1,0,0,0,12,12,0,0,0,0,0,1,11,0,0,0,0,12,1,0,0,0,12,0],[11,2,0,0,0,4,2,0,0,0,0,0,8,0,0,0,0,0,0,8,0,0,0,8,0] >;

C2×A4⋊C4 in GAP, Magma, Sage, TeX

C_2\times A_4\rtimes C_4
% in TeX

G:=Group("C2xA4:C4");
// GroupNames label

G:=SmallGroup(96,194);
// by ID

G=gap.SmallGroup(96,194);
# by ID

G:=PCGroup([6,-2,-2,-2,-3,-2,2,24,387,1444,202,869,347]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^2=c^2=d^3=e^4=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,d*b*d^-1=e*b*e^-1=b*c=c*b,d*c*d^-1=b,c*e=e*c,e*d*e^-1=d^-1>;
// generators/relations

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