direct product, non-abelian, soluble, monomial
Aliases: C2×A4⋊C4, C24.S3, C23⋊Dic3, C22.6S4, C23.4D6, (C2×A4)⋊C4, A4⋊2(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
| A4 — C2×A4⋊C4 |
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, C2, C3, C4, C22, C22, C6, C2×C4, C23, C23, C23, Dic3, A4, C2×C6, C22⋊C4, C22×C4, C24, C2×Dic3, C2×A4, C2×A4, C2×C22⋊C4, A4⋊C4, C22×A4, C2×A4⋊C4
Quotients: C1, C2, C4, C22, S3, C2×C4, Dic3, D6, C2×Dic3, S4, A4⋊C4, 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 |
►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 16)(2 13)(3 14)(4 15)(5 10)(6 11)(7 12)(8 9)(17 22)(18 23)(19 24)(20 21)
(2 13)(4 15)(5 10)(6 11)(7 12)(8 9)(18 23)(20 21)
(1 16)(2 13)(3 14)(4 15)(17 22)(18 23)(19 24)(20 21)
(1 9 23)(2 24 10)(3 11 21)(4 22 12)(5 13 19)(6 20 14)(7 15 17)(8 18 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)| (1,16)(2,13)(3,14)(4,15)(5,10)(6,11)(7,12)(8,9)(17,22)(18,23)(19,24)(20,21), (2,13)(4,15)(5,10)(6,11)(7,12)(8,9)(18,23)(20,21), (1,16)(2,13)(3,14)(4,15)(17,22)(18,23)(19,24)(20,21), (1,9,23)(2,24,10)(3,11,21)(4,22,12)(5,13,19)(6,20,14)(7,15,17)(8,18,16), (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( (1,16)(2,13)(3,14)(4,15)(5,10)(6,11)(7,12)(8,9)(17,22)(18,23)(19,24)(20,21), (2,13)(4,15)(5,10)(6,11)(7,12)(8,9)(18,23)(20,21), (1,16)(2,13)(3,14)(4,15)(17,22)(18,23)(19,24)(20,21), (1,9,23)(2,24,10)(3,11,21)(4,22,12)(5,13,19)(6,20,14)(7,15,17)(8,18,16), (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([[(1,16),(2,13),(3,14),(4,15),(5,10),(6,11),(7,12),(8,9),(17,22),(18,23),(19,24),(20,21)], [(2,13),(4,15),(5,10),(6,11),(7,12),(8,9),(18,23),(20,21)], [(1,16),(2,13),(3,14),(4,15),(17,22),(18,23),(19,24),(20,21)], [(1,9,23),(2,24,10),(3,11,21),(4,22,12),(5,13,19),(6,20,14),(7,15,17),(8,18,16)], [(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(24,124);
►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 D4⋊2S4
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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