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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

Subgroups: 482 in 70 conjugacy classes, 9 normal (4 characteristic)
C1, C2 [×3], C3, C4 [×3], C22, C22 [×14], C2×C4 [×3], D4 [×3], Q8, C23 [×4], A4 [×9], C42 [×3], C22⋊C4 [×6], C2×D4 [×3], C2×Q8, C24 [×2], C22≀C2 [×2], C4.4D4 [×3], C42⋊C3 [×3], C22⋊A4 [×2], C24⋊C22, C42⋊A4

Quotients:
C1, C3, A4 [×5], C22⋊A4, C42⋊A4

Generators and relations
 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 >

Permutation representations
►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 5)(2 10 16 6)(3 11 13 7)(4 12 14 8)

(1 3)(2 14)(4 16)(6 10)(8 12)(13 15)

(1 15)(2 14)(3 13)(4 16)(5 7)(9 11)

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

(2 6)(3 8)(9 11)(10 12)(13 15)(18 20)

(1 5)(2 6)(9 11)(14 16)(18 20)(22 24)

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

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

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

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

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

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

Matrix representation ►G ⊆ 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] >;
 

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

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
 

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