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G = C2×A4⋊C4order 96 = 25·3

Direct product of C2 and A4⋊C4

direct product, non-abelian, soluble, monomial

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

C1C22A4 — C2×A4⋊C4
C1C22A4C2×A4A4⋊C4 — C2×A4⋊C4
A4 — C2×A4⋊C4
C1C22

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 12A2B2C2D2E2F2G34A4B4C4D4E4F4G4H6A6B6C
 size 11113333866666666888
ρ111111111111111111111    trivial
ρ211-1-1-1-11111-1-1-1-1111-11-1    linear of order 2
ρ3111111111-1-1-1-1-1-1-1-1111    linear of order 2
ρ411-1-1-1-1111-11111-1-1-1-11-1    linear of order 2
ρ51-11-1-11-111-iii-i-iii-i-1-11    linear of order 4
ρ61-11-1-11-111i-i-iii-i-ii-1-11    linear of order 4
ρ71-1-111-1-111-i-i-iiiii-i1-1-1    linear of order 4
ρ81-1-111-1-111iii-i-i-i-ii1-1-1    linear of order 4
ρ922-2-2-2-222-1000000001-11    orthogonal lifted from D6
ρ1022222222-100000000-1-1-1    orthogonal lifted from S3
ρ112-22-2-22-22-10000000011-1    symplectic lifted from Dic3, Schur index 2
ρ122-2-222-2-22-100000000-111    symplectic lifted from Dic3, Schur index 2
ρ133333-1-1-1-10-11-11-11-11000    orthogonal lifted from S4
ρ1433-3-311-1-10-1-11-111-11000    orthogonal lifted from C2×S4
ρ153333-1-1-1-101-11-11-11-1000    orthogonal lifted from S4
ρ1633-3-311-1-1011-11-1-11-1000    orthogonal lifted from C2×S4
ρ173-33-31-11-10-i-iii-i-iii000    complex lifted from A4⋊C4
ρ183-33-31-11-10ii-i-iii-i-i000    complex lifted from A4⋊C4
ρ193-3-33-111-10i-iii-ii-i-i000    complex lifted from A4⋊C4
ρ203-3-33-111-10-ii-i-ii-iii000    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)

10000
01000
001200
000120
000012
,
10000
01000
0012012
000120
00001
,
10000
01000
00111
000120
000012
,
012000
112000
00100
00111212
00010
,
114000
22000
00800
00008
00080

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

Export

Character table of C2×A4⋊C4 in TeX

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