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G = A4⋊M4(2)  order 192 = 26·3

The semidirect product of A4 and M4(2) acting via M4(2)/C2×C4=C2

non-abelian, soluble, monomial

Aliases: A42M4(2), C24.2Dic3, A4⋊C85C2, C4.31(C2×S4), (C4×A4).1C4, (C2×C4).12S4, C4.1(A4⋊C4), (C23×C4).4S3, (C22×A4).3C4, (C22×C4).11D6, C22⋊(C4.Dic3), C22.4(A4⋊C4), (C4×A4).15C22, C23.2(C2×Dic3), (C22×C4).3Dic3, (C2×C4×A4).4C2, C2.3(C2×A4⋊C4), (C2×A4).7(C2×C4), SmallGroup(192,968)

Series: Derived Chief Lower central Upper central

C1C22C2×A4 — A4⋊M4(2)
C1C22A4C2×A4C4×A4A4⋊C8 — A4⋊M4(2)
A4C2×A4 — A4⋊M4(2)
C1C4C2×C4

Generators and relations for A4⋊M4(2)
 G = < a,b,c,d,e | a2=b2=c3=d8=e2=1, cac-1=dad-1=ab=ba, ae=ea, cbc-1=a, bd=db, be=eb, dcd-1=c-1, ce=ec, ede=d5 >

Subgroups: 254 in 81 conjugacy classes, 21 normal (19 characteristic)
C1, C2, C2 [×4], C3, C4 [×2], C4 [×2], C22 [×2], C22 [×8], C6 [×2], C8 [×4], C2×C4, C2×C4 [×8], C23, C23 [×4], C12 [×2], A4, C2×C6, C2×C8 [×4], M4(2) [×4], C22×C4 [×2], C22×C4 [×4], C24, C3⋊C8 [×2], C2×C12, C2×A4, C2×A4, C22⋊C8 [×4], C2×M4(2) [×2], C23×C4, C4.Dic3, C4×A4 [×2], C22×A4, C24.4C4, A4⋊C8 [×2], C2×C4×A4, A4⋊M4(2)
Quotients: C1, C2 [×3], C4 [×2], C22, S3, C2×C4, Dic3 [×2], D6, M4(2), C2×Dic3, S4, C4.Dic3, A4⋊C4 [×2], C2×S4, C2×A4⋊C4, A4⋊M4(2)

Character table of A4⋊M4(2)

 class 12A2B2C2D2E34A4B4C4D4E4F6A6B6C8A8B8C8D8E8F8G8H12A12B12C12D
 size 112336811233688812121212121212128888
ρ11111111111111111111111111111    trivial
ρ21111111111111111-1-1-1-1-1-1-1-11111    linear of order 2
ρ311-111-1111-111-1-11-1-11111-1-1-1-1-111    linear of order 2
ρ411-111-1111-111-1-11-11-1-1-1-1111-1-111    linear of order 2
ρ511-111-11-1-11-1-11-11-1iii-i-i-i-ii11-1-1    linear of order 4
ρ61111111-1-1-1-1-1-1111-iii-i-iii-i-1-1-1-1    linear of order 4
ρ711-111-11-1-11-1-11-11-1-i-i-iiiii-i11-1-1    linear of order 4
ρ81111111-1-1-1-1-1-1111i-i-iii-i-ii-1-1-1-1    linear of order 4
ρ9222222-1222222-1-1-100000000-1-1-1-1    orthogonal lifted from S3
ρ1022-222-2-122-222-21-110000000011-1-1    orthogonal lifted from D6
ρ11222222-1-2-2-2-2-2-2-1-1-1000000001111    symplectic lifted from Dic3, Schur index 2
ρ1222-222-2-1-2-22-2-221-1100000000-1-111    symplectic lifted from Dic3, Schur index 2
ρ132-20-22022i-2i02i-2i00-200000000000-2i2i    complex lifted from M4(2)
ρ142-20-2202-2i2i0-2i2i00-2000000000002i-2i    complex lifted from M4(2)
ρ152-20-220-1-2i2i0-2i2i0--31-300000000-33-ii    complex lifted from C4.Dic3
ρ162-20-220-12i-2i02i-2i0--31-3000000003-3i-i    complex lifted from C4.Dic3
ρ172-20-220-1-2i2i0-2i2i0-31--3000000003-3-ii    complex lifted from C4.Dic3
ρ182-20-220-12i-2i02i-2i0-31--300000000-33i-i    complex lifted from C4.Dic3
ρ1933-3-1-11033-3-1-11000-1-11-111-110000    orthogonal lifted from C2×S4
ρ2033-3-1-11033-3-1-1100011-11-1-11-10000    orthogonal lifted from C2×S4
ρ21333-1-1-10333-1-1-10001-11-11-11-10000    orthogonal lifted from S4
ρ22333-1-1-10333-1-1-1000-11-11-11-110000    orthogonal lifted from S4
ρ2333-3-1-110-3-3311-1000i-iii-ii-i-i0000    complex lifted from A4⋊C4
ρ24333-1-1-10-3-3-3111000ii-i-iii-i-i0000    complex lifted from A4⋊C4
ρ2533-3-1-110-3-3311-1000-ii-i-ii-iii0000    complex lifted from A4⋊C4
ρ26333-1-1-10-3-3-3111000-i-iii-i-iii0000    complex lifted from A4⋊C4
ρ276-602-2006i-6i0-2i2i0000000000000000    complex faithful
ρ286-602-200-6i6i02i-2i0000000000000000    complex faithful

Permutation representations of A4⋊M4(2)
On 24 points - transitive group 24T294
Generators in S24
(1 5)(3 7)(9 13)(10 14)(11 15)(12 16)(17 21)(19 23)
(1 5)(2 6)(3 7)(4 8)(17 21)(18 22)(19 23)(20 24)
(1 22 11)(2 12 23)(3 24 13)(4 14 17)(5 18 15)(6 16 19)(7 20 9)(8 10 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)
(2 6)(4 8)(10 14)(12 16)(17 21)(19 23)

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

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

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

G:=TransitiveGroup(24,294);

Matrix representation of A4⋊M4(2) in GL5(𝔽73)

10000
01000
00001
00727272
00100
,
10000
01000
00727272
00001
00010
,
6471000
08000
00100
00727272
00010
,
851000
565000
00100
00001
00010
,
7247000
01000
00100
00010
00001

G:=sub<GL(5,GF(73))| [1,0,0,0,0,0,1,0,0,0,0,0,0,72,1,0,0,0,72,0,0,0,1,72,0],[1,0,0,0,0,0,1,0,0,0,0,0,72,0,0,0,0,72,0,1,0,0,72,1,0],[64,0,0,0,0,71,8,0,0,0,0,0,1,72,0,0,0,0,72,1,0,0,0,72,0],[8,5,0,0,0,51,65,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,1,0],[72,0,0,0,0,47,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1] >;

A4⋊M4(2) in GAP, Magma, Sage, TeX

A_4\rtimes M_4(2)
% in TeX

G:=Group("A4:M4(2)");
// GroupNames label

G:=SmallGroup(192,968);
// by ID

G=gap.SmallGroup(192,968);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,-2,2,28,141,58,1124,4037,285,2358,475]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^2=c^3=d^8=e^2=1,c*a*c^-1=d*a*d^-1=a*b=b*a,a*e=e*a,c*b*c^-1=a,b*d=d*b,b*e=e*b,d*c*d^-1=c^-1,c*e=e*c,e*d*e=d^5>;
// generators/relations

Export

Character table of A4⋊M4(2) in TeX

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