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G = A4×C22⋊C4order 192 = 26·3

Direct product of A4 and C22⋊C4

direct product, metabelian, soluble, monomial

Aliases: A4×C22⋊C4, C242C12, C25.2C6, C2.1(D4×A4), C222(C4×A4), (C23×C4)⋊1C6, (C22×A4)⋊1C4, (C2×A4).12D4, (C23×A4).1C2, C24.24(C2×C6), C23.24(C2×A4), C23.21(C3×D4), C23.15(C2×C12), C22.11(C22×A4), (C22×A4).14C22, (C2×C4×A4)⋊2C2, C2.3(C2×C4×A4), (C2×C4)⋊1(C2×A4), C22⋊(C3×C22⋊C4), (C22×C22⋊C4)⋊C3, (C2×A4).11(C2×C4), SmallGroup(192,994)

Series: Derived Chief Lower central Upper central

C1C23 — A4×C22⋊C4
C1C22C23C24C22×A4C23×A4 — A4×C22⋊C4
C22C23 — A4×C22⋊C4
C1C22C22⋊C4

Generators and relations for A4×C22⋊C4
 G = < a,b,c,d,e,f | a2=b2=c3=d2=e2=f4=1, cac-1=ab=ba, ad=da, ae=ea, af=fa, cbc-1=a, bd=db, be=eb, bf=fb, cd=dc, ce=ec, cf=fc, fdf-1=de=ed, ef=fe >

Subgroups: 620 in 169 conjugacy classes, 33 normal (15 characteristic)
C1, C2, C2 [×2], C2 [×8], C3, C4 [×4], C22 [×2], C22 [×2], C22 [×33], C6 [×5], C2×C4 [×2], C2×C4 [×10], C23 [×2], C23 [×2], C23 [×33], C12 [×2], A4, C2×C6 [×5], C22⋊C4, C22⋊C4 [×5], C22×C4 [×8], C24, C24 [×2], C24 [×8], C2×C12 [×2], C2×A4, C2×A4 [×2], C2×A4 [×2], C22×C6, C2×C22⋊C4 [×4], C23×C4 [×2], C25, C3×C22⋊C4, C4×A4 [×2], C22×A4, C22×A4 [×2], C22×A4 [×2], C22×C22⋊C4, C2×C4×A4 [×2], C23×A4, A4×C22⋊C4
Quotients: C1, C2 [×3], C3, C4 [×2], C22, C6 [×3], C2×C4, D4 [×2], C12 [×2], A4, C2×C6, C22⋊C4, C2×C12, C3×D4 [×2], C2×A4 [×3], C3×C22⋊C4, C4×A4 [×2], C22×A4, C2×C4×A4, D4×A4 [×2], A4×C22⋊C4

Permutation representations of A4×C22⋊C4
On 24 points - transitive group 24T408
Generators in S24
(1 5)(2 6)(3 7)(4 8)(9 11)(10 12)(13 15)(14 16)(17 21)(18 22)(19 23)(20 24)
(1 7)(2 8)(3 5)(4 6)(9 15)(10 16)(11 13)(12 14)(17 19)(18 20)(21 23)(22 24)
(1 15 23)(2 16 24)(3 13 21)(4 14 22)(5 9 17)(6 10 18)(7 11 19)(8 12 20)
(1 3)(2 8)(4 6)(5 7)(9 11)(10 14)(12 16)(13 15)(17 19)(18 22)(20 24)(21 23)
(1 5)(2 6)(3 7)(4 8)(9 15)(10 16)(11 13)(12 14)(17 23)(18 24)(19 21)(20 22)
(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)| (1,5)(2,6)(3,7)(4,8)(9,11)(10,12)(13,15)(14,16)(17,21)(18,22)(19,23)(20,24), (1,7)(2,8)(3,5)(4,6)(9,15)(10,16)(11,13)(12,14)(17,19)(18,20)(21,23)(22,24), (1,15,23)(2,16,24)(3,13,21)(4,14,22)(5,9,17)(6,10,18)(7,11,19)(8,12,20), (1,3)(2,8)(4,6)(5,7)(9,11)(10,14)(12,16)(13,15)(17,19)(18,22)(20,24)(21,23), (1,5)(2,6)(3,7)(4,8)(9,15)(10,16)(11,13)(12,14)(17,23)(18,24)(19,21)(20,22), (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( (1,5)(2,6)(3,7)(4,8)(9,11)(10,12)(13,15)(14,16)(17,21)(18,22)(19,23)(20,24), (1,7)(2,8)(3,5)(4,6)(9,15)(10,16)(11,13)(12,14)(17,19)(18,20)(21,23)(22,24), (1,15,23)(2,16,24)(3,13,21)(4,14,22)(5,9,17)(6,10,18)(7,11,19)(8,12,20), (1,3)(2,8)(4,6)(5,7)(9,11)(10,14)(12,16)(13,15)(17,19)(18,22)(20,24)(21,23), (1,5)(2,6)(3,7)(4,8)(9,15)(10,16)(11,13)(12,14)(17,23)(18,24)(19,21)(20,22), (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([(1,5),(2,6),(3,7),(4,8),(9,11),(10,12),(13,15),(14,16),(17,21),(18,22),(19,23),(20,24)], [(1,7),(2,8),(3,5),(4,6),(9,15),(10,16),(11,13),(12,14),(17,19),(18,20),(21,23),(22,24)], [(1,15,23),(2,16,24),(3,13,21),(4,14,22),(5,9,17),(6,10,18),(7,11,19),(8,12,20)], [(1,3),(2,8),(4,6),(5,7),(9,11),(10,14),(12,16),(13,15),(17,19),(18,22),(20,24),(21,23)], [(1,5),(2,6),(3,7),(4,8),(9,15),(10,16),(11,13),(12,14),(17,23),(18,24),(19,21),(20,22)], [(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(24,408);

On 24 points - transitive group 24T409
Generators in S24
(5 7)(6 8)(13 15)(14 16)(17 19)(18 20)(21 23)(22 24)
(1 3)(2 4)(5 7)(6 8)(9 11)(10 12)(21 23)(22 24)
(1 19 6)(2 20 7)(3 17 8)(4 18 5)(9 15 21)(10 16 22)(11 13 23)(12 14 24)
(1 3)(2 10)(4 12)(5 24)(6 8)(7 22)(9 11)(13 15)(14 18)(16 20)(17 19)(21 23)
(1 11)(2 12)(3 9)(4 10)(5 22)(6 23)(7 24)(8 21)(13 19)(14 20)(15 17)(16 18)
(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)| (5,7)(6,8)(13,15)(14,16)(17,19)(18,20)(21,23)(22,24), (1,3)(2,4)(5,7)(6,8)(9,11)(10,12)(21,23)(22,24), (1,19,6)(2,20,7)(3,17,8)(4,18,5)(9,15,21)(10,16,22)(11,13,23)(12,14,24), (1,3)(2,10)(4,12)(5,24)(6,8)(7,22)(9,11)(13,15)(14,18)(16,20)(17,19)(21,23), (1,11)(2,12)(3,9)(4,10)(5,22)(6,23)(7,24)(8,21)(13,19)(14,20)(15,17)(16,18), (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( (5,7)(6,8)(13,15)(14,16)(17,19)(18,20)(21,23)(22,24), (1,3)(2,4)(5,7)(6,8)(9,11)(10,12)(21,23)(22,24), (1,19,6)(2,20,7)(3,17,8)(4,18,5)(9,15,21)(10,16,22)(11,13,23)(12,14,24), (1,3)(2,10)(4,12)(5,24)(6,8)(7,22)(9,11)(13,15)(14,18)(16,20)(17,19)(21,23), (1,11)(2,12)(3,9)(4,10)(5,22)(6,23)(7,24)(8,21)(13,19)(14,20)(15,17)(16,18), (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([(5,7),(6,8),(13,15),(14,16),(17,19),(18,20),(21,23),(22,24)], [(1,3),(2,4),(5,7),(6,8),(9,11),(10,12),(21,23),(22,24)], [(1,19,6),(2,20,7),(3,17,8),(4,18,5),(9,15,21),(10,16,22),(11,13,23),(12,14,24)], [(1,3),(2,10),(4,12),(5,24),(6,8),(7,22),(9,11),(13,15),(14,18),(16,20),(17,19),(21,23)], [(1,11),(2,12),(3,9),(4,10),(5,22),(6,23),(7,24),(8,21),(13,19),(14,20),(15,17),(16,18)], [(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(24,409);

On 24 points - transitive group 24T410
Generators in S24
(5 16)(6 13)(7 14)(8 15)(17 21)(18 22)(19 23)(20 24)
(1 11)(2 12)(3 9)(4 10)(17 21)(18 22)(19 23)(20 24)
(1 7 19)(2 8 20)(3 5 17)(4 6 18)(9 16 21)(10 13 22)(11 14 23)(12 15 24)
(2 12)(4 10)(6 13)(8 15)(18 22)(20 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 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)| (5,16)(6,13)(7,14)(8,15)(17,21)(18,22)(19,23)(20,24), (1,11)(2,12)(3,9)(4,10)(17,21)(18,22)(19,23)(20,24), (1,7,19)(2,8,20)(3,5,17)(4,6,18)(9,16,21)(10,13,22)(11,14,23)(12,15,24), (2,12)(4,10)(6,13)(8,15)(18,22)(20,24), (1,11)(2,12)(3,9)(4,10)(5,16)(6,13)(7,14)(8,15)(17,21)(18,22)(19,23)(20,24), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24)>;

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

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

G:=TransitiveGroup(24,410);

40 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I2J2K3A3B4A4B4C4D4E4F4G4H6A···6F6G6H6I6J12A···12H
order12222222222233444444446···6666612···12
size11112233336644222266664···488888···8

40 irreducible representations

dim111111112233336
type++++++++
imageC1C2C2C3C4C6C6C12D4C3×D4A4C2×A4C2×A4C4×A4D4×A4
kernelA4×C22⋊C4C2×C4×A4C23×A4C22×C22⋊C4C22×A4C23×C4C25C24C2×A4C23C22⋊C4C2×C4C23C22C2
# reps121244282412142

Matrix representation of A4×C22⋊C4 in GL7(𝔽13)

1000000
0100000
0010000
0001000
00001200
00000120
0000001
,
1000000
0100000
0010000
0001000
0000100
00000120
00000012
,
1000000
0100000
0030000
0003000
0000001
00001200
00000120
,
12000000
0100000
0010000
00012000
0000100
0000010
0000001
,
12000000
01200000
00120000
00012000
0000100
0000010
0000001
,
01200000
1000000
0005000
0080000
00001200
00000120
00000012

G:=sub<GL(7,GF(13))| [1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,12],[1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,12,0,0,0,0,1,0,0],[12,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[12,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[0,1,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,8,0,0,0,0,0,5,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,12] >;

A4×C22⋊C4 in GAP, Magma, Sage, TeX

A_4\times C_2^2\rtimes C_4
% in TeX

G:=Group("A4xC2^2:C4");
// GroupNames label

G:=SmallGroup(192,994);
// by ID

G=gap.SmallGroup(192,994);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,-2,2,365,92,1027,1784]);
// Polycyclic

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

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