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## G = C4×C22⋊A4order 192 = 26·3

### Direct product of C4 and C22⋊A4

Aliases: C4×C22⋊A4, C246C12, C25.4C6, C22⋊(C4×A4), (C24×C4)⋊2C3, (C22×C4)⋊4A4, C23.21(C2×A4), C2.1(C2×C22⋊A4), (C2×C22⋊A4).3C2, SmallGroup(192,1505)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C24 — C4×C22⋊A4
 Chief series C1 — C22 — C24 — C25 — C2×C22⋊A4 — C4×C22⋊A4
 Lower central C24 — C4×C22⋊A4
 Upper central C1 — C4

Generators and relations for C4×C22⋊A4
G = < a,b,c,d,e,f | a4=b2=c2=d2=e2=f3=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, fbf-1=bc=cb, bd=db, be=eb, cd=dc, ce=ec, fcf-1=b, fdf-1=de=ed, fef-1=d >

Subgroups: 792 in 262 conjugacy classes, 24 normal (9 characteristic)
C1, C2, C2 [×10], C3, C4, C4 [×5], C22 [×5], C22 [×50], C6, C2×C4 [×40], C23 [×5], C23 [×50], C12, A4 [×5], C22×C4 [×5], C22×C4 [×45], C24, C24 [×10], C2×A4 [×5], C23×C4 [×10], C25, C4×A4 [×5], C22⋊A4, C24×C4, C2×C22⋊A4, C4×C22⋊A4
Quotients: C1, C2, C3, C4, C6, C12, A4 [×5], C2×A4 [×5], C4×A4 [×5], C22⋊A4, C2×C22⋊A4, C4×C22⋊A4

Permutation representations of C4×C22⋊A4
On 24 points - transitive group 24T385
Generators in S24
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)(17 18 19 20)(21 22 23 24)
(1 9)(2 10)(3 11)(4 12)(5 22)(6 23)(7 24)(8 21)(13 15)(14 16)(17 19)(18 20)
(1 11)(2 12)(3 9)(4 10)(5 7)(6 8)(13 17)(14 18)(15 19)(16 20)(21 23)(22 24)
(1 11)(2 12)(3 9)(4 10)(13 19)(14 20)(15 17)(16 18)
(5 22)(6 23)(7 24)(8 21)(13 19)(14 20)(15 17)(16 18)
(1 19 6)(2 20 7)(3 17 8)(4 18 5)(9 15 21)(10 16 22)(11 13 23)(12 14 24)

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

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

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

G:=TransitiveGroup(24,385);

32 conjugacy classes

 class 1 2A 2B ··· 2K 3A 3B 4A 4B 4C ··· 4L 6A 6B 12A 12B 12C 12D order 1 2 2 ··· 2 3 3 4 4 4 ··· 4 6 6 12 12 12 12 size 1 1 3 ··· 3 16 16 1 1 3 ··· 3 16 16 16 16 16 16

32 irreducible representations

 dim 1 1 1 1 1 1 3 3 3 type + + + + image C1 C2 C3 C4 C6 C12 A4 C2×A4 C4×A4 kernel C4×C22⋊A4 C2×C22⋊A4 C24×C4 C22⋊A4 C25 C24 C22×C4 C23 C22 # reps 1 1 2 2 2 4 5 5 10

Matrix representation of C4×C22⋊A4 in GL6(𝔽13)

 12 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 0 8 0 0 0 0 0 0 8 0 0 0 0 0 0 8
,
 12 0 0 0 0 0 0 12 0 0 0 0 0 0 1 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 0 1
,
 12 0 0 0 0 0 0 1 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 0 1 0 0 0 0 0 0 12
,
 1 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 0 1
,
 12 0 0 0 0 0 0 12 0 0 0 0 0 0 1 0 0 0 0 0 0 12 0 0 0 0 0 0 1 0 0 0 0 0 0 12
,
 0 1 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 1 0 0

G:=sub<GL(6,GF(13))| [12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,8,0,0,0,0,0,0,8,0,0,0,0,0,0,8],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,1],[12,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,12],[1,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,1],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,12],[0,0,1,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,1,0] >;

C4×C22⋊A4 in GAP, Magma, Sage, TeX

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

G:=Group("C4xC2^2:A4");
// GroupNames label

G:=SmallGroup(192,1505);
// by ID

G=gap.SmallGroup(192,1505);
# by ID

G:=PCGroup([7,-2,-3,-2,-2,2,-2,2,42,346,641,2028,3541]);
// Polycyclic

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

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