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## G = C22×C42⋊C3order 192 = 26·3

### Direct product of C22 and C42⋊C3

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

 Derived series C1 — C42 — C22×C42⋊C3
 Chief series C1 — C22 — C42 — C42⋊C3 — C2×C42⋊C3 — C22×C42⋊C3
 Lower central C42 — C22×C42⋊C3
 Upper central C1 — C22

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 2A 2B 2C 2D 2E 2F 2G 3A 3B 4A ··· 4P 6A ··· 6F order 1 2 2 2 2 2 2 2 3 3 4 ··· 4 6 ··· 6 size 1 1 1 1 3 3 3 3 16 16 3 ··· 3 16 ··· 16

32 irreducible representations

 dim 1 1 1 1 3 3 3 3 type + + + + image C1 C2 C3 C6 A4 C2×A4 C42⋊C3 C2×C42⋊C3 kernel C22×C42⋊C3 C2×C42⋊C3 C22×C42 C2×C42 C24 C23 C22 C2 # reps 1 3 2 6 1 3 4 12

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

 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 1 0 0 0 0 1 0 1 0 0
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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