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G = C22×C42⋊C3order 192 = 26·3

Direct product of C22 and C42⋊C3

direct product, metabelian, soluble, monomial, A-group

Aliases: C22×C42⋊C3, C24.9A4, (C2×C42)⋊3C6, C4216(C2×C6), (C22×C42)⋊1C3, C23.20(C2×A4), C22.1(C22×A4), SmallGroup(192,992)

Series: Derived Chief Lower central Upper central

C1C42 — C22×C42⋊C3
C1C22C42C42⋊C3C2×C42⋊C3 — C22×C42⋊C3
C42 — C22×C42⋊C3
C1C22

Generators and relations for C22×C42⋊C3
 G = < a,b,c,d,e | a2=b2=c4=d4=e3=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ece-1=cd-1, ede-1=c-1d2 >

Subgroups: 354 in 108 conjugacy classes, 20 normal (8 characteristic)
C1, C2 [×3], C2 [×4], C3, C4 [×8], C22 [×2], C22 [×11], C6 [×3], C2×C4 [×28], C23 [×3], C23 [×4], A4, C2×C6, C42, C42 [×5], C22×C4 [×14], C24, C2×A4 [×3], C2×C42 [×3], C2×C42 [×3], C23×C4, C42⋊C3, C22×A4, C22×C42, C2×C42⋊C3 [×3], C22×C42⋊C3
Quotients: C1, C2 [×3], C3, C22, C6 [×3], A4, C2×C6, C2×A4 [×3], C42⋊C3, C22×A4, C2×C42⋊C3 [×3], C22×C42⋊C3

Permutation representations of C22×C42⋊C3
On 24 points - transitive group 24T420
Generators in S24
(1 2)(3 4)(5 6)(7 8)(9 13)(10 14)(11 15)(12 16)(17 22)(18 23)(19 24)(20 21)
(1 7)(2 8)(3 6)(4 5)(9 11)(10 12)(13 15)(14 16)(17 19)(18 20)(21 23)(22 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 3 7 6)(2 4 8 5)(9 14 11 16)(10 15 12 13)(17 24)(18 21)(19 22)(20 23)
(1 20 9)(2 21 13)(3 24 10)(4 19 14)(5 17 16)(6 22 12)(7 18 11)(8 23 15)

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

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

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

G:=TransitiveGroup(24,420);

On 24 points - transitive group 24T421
Generators in S24
(1 6)(2 5)(3 7)(4 8)(9 22)(10 23)(11 24)(12 21)(13 17)(14 18)(15 19)(16 20)
(1 5)(2 6)(3 8)(4 7)(9 24)(10 21)(11 22)(12 23)(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)
(1 3 2 4)(5 8 6 7)(13 16 15 14)(17 20 19 18)
(1 10 15)(2 12 13)(3 11 16)(4 9 14)(5 21 17)(6 23 19)(7 24 20)(8 22 18)

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

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

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

G:=TransitiveGroup(24,421);

32 conjugacy classes

class 1 2A2B2C2D2E2F2G3A3B4A···4P6A···6F
order12222222334···46···6
size1111333316163···316···16

32 irreducible representations

dim11113333
type++++
imageC1C2C3C6A4C2×A4C42⋊C3C2×C42⋊C3
kernelC22×C42⋊C3C2×C42⋊C3C22×C42C2×C42C24C23C22C2
# reps132613412

Matrix representation of C22×C42⋊C3 in GL4(𝔽13) generated by

1000
01200
00120
00012
,
12000
01200
00120
00012
,
1000
01200
0050
0005
,
1000
0500
0080
0001
,
3000
0010
0001
0100
G:=sub<GL(4,GF(13))| [1,0,0,0,0,12,0,0,0,0,12,0,0,0,0,12],[12,0,0,0,0,12,0,0,0,0,12,0,0,0,0,12],[1,0,0,0,0,12,0,0,0,0,5,0,0,0,0,5],[1,0,0,0,0,5,0,0,0,0,8,0,0,0,0,1],[3,0,0,0,0,0,0,1,0,1,0,0,0,0,1,0] >;

C22×C42⋊C3 in GAP, Magma, Sage, TeX

C_2^2\times C_4^2\rtimes C_3
% in TeX

G:=Group("C2^2xC4^2:C3");
// GroupNames label

G:=SmallGroup(192,992);
// by ID

G=gap.SmallGroup(192,992);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,2,-2,2,185,360,1173,102,1027,1784]);
// Polycyclic

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

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