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G = C42⋊A4  order 192 = 26·3

The semidirect product of C42 and A4 acting faithfully

non-abelian, soluble, monomial

Aliases: C42⋊A4, C24⋊4A4, C24⋊C22⋊2C3, C22.2(C22⋊A4), SmallGroup(192,1023)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C22 — C24⋊C22 — C42⋊A4
Chief series
C1 — C22 — C24 — C24⋊C22 — C42⋊A4
Lower central
C24⋊C22 — C42⋊A4
Upper central
C1

Generators and relations for C42⋊A4
 G = < a,b,c,d,e | a4=b4=c2=d2=e3=1, ab=ba, cac=ab2, dad=a-1, eae-1=a-1b-1, cbc=a2b, dbd=a2b-1, ebe-1=a, ece-1=cd=dc, ede-1=c >

Subgroups: 482 in 70 conjugacy classes, 9 normal (4 characteristic)
C1, C2, C3, C4, C22, C22, C2×C4, D4, Q8, C23, A4, C42, C22⋊C4, C2×D4, C2×Q8, C24, C22≀C2, C4.4D4, C42⋊C3, C22⋊A4, C24⋊C22, C42⋊A4
Quotients: C1, C3, A4, C22⋊A4, C42⋊A4

Character table of C42⋊A4

 class 12A2B2C3A3B4A4B4C
 size 1312126464121212
ρ1111111111    trivial
ρ21111ζ32ζ3111    linear of order 3
ρ31111ζ3ζ32111    linear of order 3
ρ4333-100-1-1-1    orthogonal lifted from A4
ρ533-1-100-1-13    orthogonal lifted from A4
ρ633-1300-1-1-1    orthogonal lifted from A4
ρ733-1-1003-1-1    orthogonal lifted from A4
ρ833-1-100-13-1    orthogonal lifted from A4
ρ912-40000000    orthogonal faithful

Permutation representations of C42⋊A4
►On 16 points - transitive group 16T440
Generators in S16
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)

(1 9 15 6)(2 10 16 7)(3 11 13 8)(4 12 14 5)

(1 3)(2 14)(4 16)(5 12)(7 10)(13 15)

(1 15)(2 14)(3 13)(4 16)(6 8)(9 11)

(2 9 5)(3 15 13)(4 6 10)(7 16 8)(11 12 14)
 
G:=sub<Sym(16)| (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16), (1,9,15,6)(2,10,16,7)(3,11,13,8)(4,12,14,5), (1,3)(2,14)(4,16)(5,12)(7,10)(13,15), (1,15)(2,14)(3,13)(4,16)(6,8)(9,11), (2,9,5)(3,15,13)(4,6,10)(7,16,8)(11,12,14)>;
 
G:=Group( (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16), (1,9,15,6)(2,10,16,7)(3,11,13,8)(4,12,14,5), (1,3)(2,14)(4,16)(5,12)(7,10)(13,15), (1,15)(2,14)(3,13)(4,16)(6,8)(9,11), (2,9,5)(3,15,13)(4,6,10)(7,16,8)(11,12,14) );
 
G=PermutationGroup([[(1,2,3,4),(5,6,7,8),(9,10,11,12),(13,14,15,16)], [(1,9,15,6),(2,10,16,7),(3,11,13,8),(4,12,14,5)], [(1,3),(2,14),(4,16),(5,12),(7,10),(13,15)], [(1,15),(2,14),(3,13),(4,16),(6,8),(9,11)], [(2,9,5),(3,15,13),(4,6,10),(7,16,8),(11,12,14)]])
 
G:=TransitiveGroup(16,440);
 

►On 24 points - transitive group 24T372
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 7 6 4)(2 8 5 3)(9 18)(10 19)(11 20)(12 17)(13 21 15 23)(14 22 16 24)

(2 5)(3 8)(9 11)(10 12)(13 15)(22 24)

(1 6)(2 5)(9 11)(14 16)(17 19)(22 24)

(1 12 24)(2 17 15)(3 9 14)(4 18 23)(5 19 13)(6 10 22)(7 20 21)(8 11 16)
 
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), (1,7,6,4)(2,8,5,3)(9,18)(10,19)(11,20)(12,17)(13,21,15,23)(14,22,16,24), (2,5)(3,8)(9,11)(10,12)(13,15)(22,24), (1,6)(2,5)(9,11)(14,16)(17,19)(22,24), (1,12,24)(2,17,15)(3,9,14)(4,18,23)(5,19,13)(6,10,22)(7,20,21)(8,11,16)>;
 
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), (1,7,6,4)(2,8,5,3)(9,18)(10,19)(11,20)(12,17)(13,21,15,23)(14,22,16,24), (2,5)(3,8)(9,11)(10,12)(13,15)(22,24), (1,6)(2,5)(9,11)(14,16)(17,19)(22,24), (1,12,24)(2,17,15)(3,9,14)(4,18,23)(5,19,13)(6,10,22)(7,20,21)(8,11,16) );
 
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)], [(1,7,6,4),(2,8,5,3),(9,18),(10,19),(11,20),(12,17),(13,21,15,23),(14,22,16,24)], [(2,5),(3,8),(9,11),(10,12),(13,15),(22,24)], [(1,6),(2,5),(9,11),(14,16),(17,19),(22,24)], [(1,12,24),(2,17,15),(3,9,14),(4,18,23),(5,19,13),(6,10,22),(7,20,21),(8,11,16)]])
 
G:=TransitiveGroup(24,372);
 

►On 24 points - transitive group 24T391
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 7 4 5)(2 8 3 6)(9 24 11 22)(10 21 12 23)(13 20)(14 17)(15 18)(16 19)

(1 4)(5 7)(9 12)(10 11)(13 19)(14 20)(15 17)(16 18)(21 24)(22 23)

(1 5)(2 6)(3 8)(4 7)(9 10)(11 12)(13 15)(17 19)(21 24)(22 23)

(1 13 9)(2 20 21)(3 18 23)(4 15 11)(5 17 12)(6 14 24)(7 19 10)(8 16 22)
 
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), (1,7,4,5)(2,8,3,6)(9,24,11,22)(10,21,12,23)(13,20)(14,17)(15,18)(16,19), (1,4)(5,7)(9,12)(10,11)(13,19)(14,20)(15,17)(16,18)(21,24)(22,23), (1,5)(2,6)(3,8)(4,7)(9,10)(11,12)(13,15)(17,19)(21,24)(22,23), (1,13,9)(2,20,21)(3,18,23)(4,15,11)(5,17,12)(6,14,24)(7,19,10)(8,16,22)>;
 
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), (1,7,4,5)(2,8,3,6)(9,24,11,22)(10,21,12,23)(13,20)(14,17)(15,18)(16,19), (1,4)(5,7)(9,12)(10,11)(13,19)(14,20)(15,17)(16,18)(21,24)(22,23), (1,5)(2,6)(3,8)(4,7)(9,10)(11,12)(13,15)(17,19)(21,24)(22,23), (1,13,9)(2,20,21)(3,18,23)(4,15,11)(5,17,12)(6,14,24)(7,19,10)(8,16,22) );
 
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)], [(1,7,4,5),(2,8,3,6),(9,24,11,22),(10,21,12,23),(13,20),(14,17),(15,18),(16,19)], [(1,4),(5,7),(9,12),(10,11),(13,19),(14,20),(15,17),(16,18),(21,24),(22,23)], [(1,5),(2,6),(3,8),(4,7),(9,10),(11,12),(13,15),(17,19),(21,24),(22,23)], [(1,13,9),(2,20,21),(3,18,23),(4,15,11),(5,17,12),(6,14,24),(7,19,10),(8,16,22)]])
 
G:=TransitiveGroup(24,391);
 

►On 24 points - transitive group 24T392
Generators in S24
(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 7 6 8)(9 12 11 10)(13 14 15 16)(21 23)(22 24)

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

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

(1 19 9)(2 17 11)(3 18 10)(4 20 12)(5 21 13)(6 23 15)(7 22 14)(8 24 16)
 
G:=sub<Sym(24)| (5,6)(7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24), (1,3,2,4)(5,7,6,8)(9,12,11,10)(13,14,15,16)(21,23)(22,24), (1,5)(2,6)(3,7)(4,8)(9,10)(11,12)(13,14)(15,16)(17,24)(18,21)(19,22)(20,23), (1,3)(2,4)(5,7)(6,8)(9,14)(10,13)(11,16)(12,15)(17,23)(18,22)(19,21)(20,24), (1,19,9)(2,17,11)(3,18,10)(4,20,12)(5,21,13)(6,23,15)(7,22,14)(8,24,16)>;
 
G:=Group( (5,6)(7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24), (1,3,2,4)(5,7,6,8)(9,12,11,10)(13,14,15,16)(21,23)(22,24), (1,5)(2,6)(3,7)(4,8)(9,10)(11,12)(13,14)(15,16)(17,24)(18,21)(19,22)(20,23), (1,3)(2,4)(5,7)(6,8)(9,14)(10,13)(11,16)(12,15)(17,23)(18,22)(19,21)(20,24), (1,19,9)(2,17,11)(3,18,10)(4,20,12)(5,21,13)(6,23,15)(7,22,14)(8,24,16) );
 
G=PermutationGroup([[(5,6),(7,8),(9,10,11,12),(13,14,15,16),(17,18,19,20),(21,22,23,24)], [(1,3,2,4),(5,7,6,8),(9,12,11,10),(13,14,15,16),(21,23),(22,24)], [(1,5),(2,6),(3,7),(4,8),(9,10),(11,12),(13,14),(15,16),(17,24),(18,21),(19,22),(20,23)], [(1,3),(2,4),(5,7),(6,8),(9,14),(10,13),(11,16),(12,15),(17,23),(18,22),(19,21),(20,24)], [(1,19,9),(2,17,11),(3,18,10),(4,20,12),(5,21,13),(6,23,15),(7,22,14),(8,24,16)]])
 
G:=TransitiveGroup(24,392);
 

Matrix representation of C42⋊A4 ►in GL12(ℤ)

000000-1-1-1000
000000001000
000000010000
000000000001
000000000-1-1-1
000000000100
100000000000
010000000000
001000000000
000010000000
000100000000
000-1-1-1000000
,
000001000000
000-1-1-1000000
000100000000
100000000000
010000000000
001000000000
000000000-1-1-1
000000000001
000000000010
000000010000
000000100000
000000-1-1-1000
,
010000000000
100000000000
-1-1-1000000000
000001000000
000-1-1-1000000
000100000000
000000-1-1-1000
000000001000
000000010000
000000000100
000000000010
000000000001
,
-1-1-1000000000
001000000000
010000000000
000001000000
000-1-1-1000000
000100000000
000000100000
000000010000
000000001000
000000000010
000000000100
000000000-1-1-1
,
100000000000
001000000000
-1-1-1000000000
000000100000
000000001000
000000-1-1-1000
000000000100
000000000001
000000000-1-1-1
000100000000
000001000000
000-1-1-1000000

G:=sub<GL(12,Integers())| [0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,1,0,-1,0,0,0,0,0,0,0,0,0,0,0,-1,-1,0,0,0,0,0,0,0,0,0,0,0,-1,0,1,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0],[0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,1,0,-1,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,-1,0,1,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0],[0,1,-1,0,0,0,0,0,0,0,0,0,1,0,-1,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,-1,0,1,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1],[-1,0,0,0,0,0,0,0,0,0,0,0,-1,0,1,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,1,0,-1,0,0,0,0,0,0,0,0,0,0,0,-1],[1,0,-1,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,-1,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,1,-1,0,0,0,1,0,-1,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,-1,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,1,-1,0,0,0] >;
 

C42⋊A4 in GAP, Magma, Sage, TeX 

C_4^2\rtimes A_4
 
% in TeX
 
G:=Group("C4^2:A4");
 
// GroupNames label
 
G:=SmallGroup(192,1023);
 
// by ID
 
G=gap.SmallGroup(192,1023);
 
# by ID
 
G:=PCGroup([7,-3,-2,2,-2,2,-2,2,85,191,675,570,745,1264,1971,718,4037,7062]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^4=b^4=c^2=d^2=e^3=1,a*b=b*a,c*a*c=a*b^2,d*a*d=a^-1,e*a*e^-1=a^-1*b^-1,c*b*c=a^2*b,d*b*d=a^2*b^-1,e*b*e^-1=a,e*c*e^-1=c*d=d*c,e*d*e^-1=c>;
 
// generators/relations
 

Export

Character table of C42⋊A4 in TeX

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