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G = C422A4order 192 = 26·3

The semidirect product of C42 and A4 acting via A4/C22=C3

metabelian, soluble, monomial, A-group

Aliases: C422A4, C24.10A4, C22⋊(C42⋊C3), (C22×C42)⋊3C3, C22.1(C22⋊A4), SmallGroup(192,1020)

Series: Derived Chief Lower central Upper central

C1C22×C42 — C422A4
C1C22C24C22×C42 — C422A4
C22×C42 — C422A4
C1

Generators and relations for C422A4
 G = < a,b,c,d,e | a4=b4=c2=d2=e3=1, ab=ba, ac=ca, ad=da, eae-1=ab-1, bc=cb, bd=db, ebe-1=a-1b2, ece-1=cd=dc, ede-1=c >

Subgroups: 414 in 103 conjugacy classes, 13 normal (5 characteristic)
C1, C2 [×5], C3, C4 [×8], C22, C22 [×4], C22 [×10], C2×C4 [×28], C23 [×5], A4 [×5], C42 [×4], C42 [×4], C22×C4 [×14], C24, C2×C42 [×4], C23×C4, C42⋊C3 [×4], C22⋊A4, C22×C42, C422A4
Quotients: C1, C3, A4 [×5], C42⋊C3 [×4], C22⋊A4, C422A4

Character table of C422A4

 class 12A2B2C2D2E3A3B4A4B4C4D4E4F4G4H4I4J4K4L4M4N4O4P
 size 13333364643333333333333333
ρ1111111111111111111111111    trivial
ρ2111111ζ32ζ31111111111111111    linear of order 3
ρ3111111ζ3ζ321111111111111111    linear of order 3
ρ43-1-1-1-1300-1-13333-1-1-1-1-1-1-1-1-1-1    orthogonal lifted from A4
ρ53-1-1-1-1300-1-1-1-1-1-1-1-1-1-13333-1-1    orthogonal lifted from A4
ρ63-1-1-1-1300-1-1-1-1-1-13333-1-1-1-1-1-1    orthogonal lifted from A4
ρ733333300-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1    orthogonal lifted from A4
ρ83-1-1-1-130033-1-1-1-1-1-1-1-1-1-1-1-133    orthogonal lifted from A4
ρ933-1-1-1-100-1+2i11-1+2i-1-2i1-1-2i11-1+2i-1+2i11-1-2i1-1-2i    complex lifted from C42⋊C3
ρ103-13-1-1-1001-1-2i-1+2i11-1-2i-1+2i11-1-2i-1+2i11-1-2i-1+2i1    complex lifted from C42⋊C3
ρ113-1-1-13-100-1-2i1-1-2i11-1+2i1-1-2i-1+2i1-1+2i11-1-2i1-1+2i    complex lifted from C42⋊C3
ρ1233-1-1-1-1001-1-2i-1-2i11-1+2i1-1+2i-1-2i11-1-2i-1+2i1-1+2i1    complex lifted from C42⋊C3
ρ133-13-1-1-100-1+2i11-1-2i-1+2i11-1-2i-1+2i11-1-2i-1+2i11-1-2i    complex lifted from C42⋊C3
ρ143-1-13-1-1001-1-2i1-1+2i-1-2i11-1-2i-1+2i1-1-2i11-1+2i-1+2i1    complex lifted from C42⋊C3
ρ1533-1-1-1-1001-1+2i-1+2i11-1-2i1-1-2i-1+2i11-1+2i-1-2i1-1-2i1    complex lifted from C42⋊C3
ρ163-1-1-13-100-1+2i1-1+2i11-1-2i1-1+2i-1-2i1-1-2i11-1+2i1-1-2i    complex lifted from C42⋊C3
ρ1733-1-1-1-100-1-2i11-1-2i-1+2i1-1+2i11-1-2i-1-2i11-1+2i1-1+2i    complex lifted from C42⋊C3
ρ183-1-13-1-100-1-2i1-1+2i11-1-2i-1-2i11-1+2i1-1-2i-1+2i11-1+2i    complex lifted from C42⋊C3
ρ193-1-13-1-100-1+2i1-1-2i11-1+2i-1+2i11-1-2i1-1+2i-1-2i11-1-2i    complex lifted from C42⋊C3
ρ203-1-1-13-1001-1+2i1-1+2i-1-2i1-1+2i11-1-2i1-1-2i-1+2i1-1-2i1    complex lifted from C42⋊C3
ρ213-13-1-1-100-1-2i11-1+2i-1-2i11-1+2i-1-2i11-1+2i-1-2i11-1+2i    complex lifted from C42⋊C3
ρ223-1-13-1-1001-1+2i1-1-2i-1+2i11-1+2i-1-2i1-1+2i11-1-2i-1-2i1    complex lifted from C42⋊C3
ρ233-13-1-1-1001-1+2i-1-2i11-1+2i-1-2i11-1+2i-1-2i11-1+2i-1-2i1    complex lifted from C42⋊C3
ρ243-1-1-13-1001-1-2i1-1-2i-1+2i1-1-2i11-1+2i1-1+2i-1-2i1-1+2i1    complex lifted from C42⋊C3

Permutation representations of C422A4
On 24 points - transitive group 24T388
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 5 2 6)(3 7 4 8)(9 12 11 10)(13 16 15 14)
(1 4)(2 3)(5 8)(6 7)(9 14)(10 15)(11 16)(12 13)(17 19)(18 20)(21 23)(22 24)
(1 3)(2 4)(5 7)(6 8)(9 11)(10 12)(13 15)(14 16)(17 22)(18 23)(19 24)(20 21)
(1 20 9)(2 18 11)(3 23 14)(4 21 16)(5 17 10)(6 19 12)(7 24 15)(8 22 13)

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

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

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

G:=TransitiveGroup(24,388);

Matrix representation of C422A4 in GL6(𝔽13)

100000
0120000
0012000
000100
000050
000008
,
1200000
0120000
001000
000500
000050
0000012
,
1200000
0120000
001000
0001200
000010
0000012
,
1200000
010000
0012000
000100
0000120
0000012
,
010000
001000
100000
000010
000001
000100

G:=sub<GL(6,GF(13))| [1,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,5,0,0,0,0,0,0,8],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,5,0,0,0,0,0,0,5,0,0,0,0,0,0,12],[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],[12,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[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] >;

C422A4 in GAP, Magma, Sage, TeX

C_4^2\rtimes_2A_4
% in TeX

G:=Group("C4^2:2A4");
// GroupNames label

G:=SmallGroup(192,1020);
// by ID

G=gap.SmallGroup(192,1020);
# by ID

G:=PCGroup([7,-3,-2,2,-2,2,-2,2,85,680,2207,184,675,1264,4037,7062]);
// Polycyclic

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

Export

Character table of C422A4 in TeX

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