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## G = C4×C42⋊C3order 192 = 26·3

### Direct product of C4 and C42⋊C3

Aliases: C4×C42⋊C3, C4C3, C43⋊C3, C426C12, C23.8(C2×A4), (C22×C4).7A4, (C2×C42).3C6, C22.1(C4×A4), C2.1(C2×C42⋊C3), (C2×C42⋊C3).4C2, SmallGroup(192,188)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C42 — C4×C42⋊C3
 Chief series C1 — C22 — C42 — C2×C42 — C2×C42⋊C3 — C4×C42⋊C3
 Lower central C42 — C4×C42⋊C3
 Upper central C1 — C4

Generators and relations for C4×C42⋊C3
G = < a,b,c,d | a4=b4=c4=d3=1, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=bc-1, dcd-1=b-1c2 >

Subgroups: 192 in 58 conjugacy classes, 12 normal (all characteristic)
C1, C2, C2 [×2], C3, C4, C4 [×9], C22, C22 [×2], C6, C2×C4 [×14], C23, C12, A4, C42, C42 [×9], C22×C4, C22×C4 [×2], C2×A4, C2×C42, C2×C42 [×2], C42⋊C3, C4×A4, C43, C2×C42⋊C3, C4×C42⋊C3
Quotients: C1, C2, C3, C4, C6, C12, A4, C2×A4, C42⋊C3, C4×A4, C2×C42⋊C3, C4×C42⋊C3

Permutation representations of C4×C42⋊C3
On 12 points - transitive group 12T94
Generators in S12
(1 2 3 4)(5 6 7 8)(9 10 11 12)
(1 4 3 2)(5 8 7 6)(9 11)(10 12)
(1 2 3 4)(9 12 11 10)
(1 11 7)(2 12 8)(3 9 5)(4 10 6)

G:=sub<Sym(12)| (1,2,3,4)(5,6,7,8)(9,10,11,12), (1,4,3,2)(5,8,7,6)(9,11)(10,12), (1,2,3,4)(9,12,11,10), (1,11,7)(2,12,8)(3,9,5)(4,10,6)>;

G:=Group( (1,2,3,4)(5,6,7,8)(9,10,11,12), (1,4,3,2)(5,8,7,6)(9,11)(10,12), (1,2,3,4)(9,12,11,10), (1,11,7)(2,12,8)(3,9,5)(4,10,6) );

G=PermutationGroup([(1,2,3,4),(5,6,7,8),(9,10,11,12)], [(1,4,3,2),(5,8,7,6),(9,11),(10,12)], [(1,2,3,4),(9,12,11,10)], [(1,11,7),(2,12,8),(3,9,5),(4,10,6)])

G:=TransitiveGroup(12,94);

On 24 points - transitive group 24T469
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 6 7 8)(13 17 15 19)(14 18 16 20)(21 22 23 24)
(1 12 3 10)(2 9 4 11)(5 24)(6 21)(7 22)(8 23)(13 16 15 14)(17 20 19 18)
(1 24 16)(2 21 13)(3 22 14)(4 23 15)(5 19 11)(6 20 12)(7 17 9)(8 18 10)

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

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

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

G:=TransitiveGroup(24,469);

On 24 points - transitive group 24T470
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 11 3 9)(2 12 4 10)(17 23 19 21)(18 24 20 22)
(1 11 3 9)(2 12 4 10)(5 16 7 14)(6 13 8 15)(17 19)(18 20)(21 23)(22 24)
(1 6 19)(2 7 20)(3 8 17)(4 5 18)(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), (1,11,3,9)(2,12,4,10)(17,23,19,21)(18,24,20,22), (1,11,3,9)(2,12,4,10)(5,16,7,14)(6,13,8,15)(17,19)(18,20)(21,23)(22,24), (1,6,19)(2,7,20)(3,8,17)(4,5,18)(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), (1,11,3,9)(2,12,4,10)(17,23,19,21)(18,24,20,22), (1,11,3,9)(2,12,4,10)(5,16,7,14)(6,13,8,15)(17,19)(18,20)(21,23)(22,24), (1,6,19)(2,7,20)(3,8,17)(4,5,18)(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)], [(1,11,3,9),(2,12,4,10),(17,23,19,21),(18,24,20,22)], [(1,11,3,9),(2,12,4,10),(5,16,7,14),(6,13,8,15),(17,19),(18,20),(21,23),(22,24)], [(1,6,19),(2,7,20),(3,8,17),(4,5,18),(9,15,21),(10,16,22),(11,13,23),(12,14,24)])

G:=TransitiveGroup(24,470);

Polynomial with Galois group C4×C42⋊C3 over ℚ
actionf(x)Disc(f)
12T94x12-18x10+119x8-383x6+644x4-543x2+181212·78·1813

32 conjugacy classes

 class 1 2A 2B 2C 3A 3B 4A 4B 4C ··· 4T 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 3 3 3 type + + + + image C1 C2 C3 C4 C6 C12 A4 C2×A4 C42⋊C3 C4×A4 C2×C42⋊C3 C4×C42⋊C3 kernel C4×C42⋊C3 C2×C42⋊C3 C43 C42⋊C3 C2×C42 C42 C22×C4 C23 C4 C22 C2 C1 # reps 1 1 2 2 2 4 1 1 4 2 4 8

Matrix representation of C4×C42⋊C3 in GL3(𝔽5) generated by

 3 0 0 0 3 0 0 0 3
,
 4 0 1 0 0 3 1 2 2
,
 4 3 3 2 4 1 3 4 2
,
 1 1 1 0 4 2 0 2 0
G:=sub<GL(3,GF(5))| [3,0,0,0,3,0,0,0,3],[4,0,1,0,0,2,1,3,2],[4,2,3,3,4,4,3,1,2],[1,0,0,1,4,2,1,2,0] >;

C4×C42⋊C3 in GAP, Magma, Sage, TeX

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

G:=Group("C4xC4^2:C3");
// GroupNames label

G:=SmallGroup(192,188);
// by ID

G=gap.SmallGroup(192,188);
# by ID

G:=PCGroup([7,-2,-3,-2,-2,2,-2,2,42,346,360,2321,102,2028,3541]);
// Polycyclic

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

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