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G = C42.A4  order 192 = 26·3

The non-split extension by C42 of A4 acting faithfully

non-abelian, soluble, monomial

Aliases: C42.A4, C22.58C24⋊C3, C22.3(C22⋊A4), SmallGroup(192,1025)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C22 — C22.58C24 — C42.A4
Chief series
C1 — C22 — C42 — C22.58C24 — C42.A4

Subgroups: 202 in 42 conjugacy classes, 9 normal (4 characteristic)
C1, C2, C3, C4 [×5], C22, C2×C4 [×5], A4, C42 [×5], C4⋊C4 [×10], C42.C2 [×5], C42⋊C3 [×5], C22.58C24, C42.A4

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

Generators and relations
 G = < a,b,c,d,e | a4=b4=e3=1, c2=b2, d2=a2b2, ab=ba, cac-1=ebe-1=a-1, dad-1=a-1b2, eae-1=a-1b, cbc-1=a2b-1, dbd-1=b-1, cd=dc, ece-1=a2b2cd, ede-1=a2c >

Smallest permutation representation
►On 48 points
Generators in S48
(9 10)(11 12)(13 14)(15 16)(17 18 19 20)(21 22 23 24)(25 26 27 28)(29 30 31 32)(33 34 35 36)(37 38 39 40)(41 42 43 44)(45 46 47 48)

(1 4 2 7)(3 6 5 8)(9 13 10 14)(11 15 12 16)(17 19)(18 20)(21 24 23 22)(25 26 27 28)(29 32 31 30)(33 34 35 36)(37 39)(38 40)

(1 5 2 3)(4 6 7 8)(9 11 10 12)(13 16 14 15)(17 48)(18 47)(19 46)(20 45)(21 27 23 25)(22 26 24 28)(29 35 31 33)(30 34 32 36)(37 41)(38 44)(39 43)(40 42)

(1 11 2 12)(3 9 5 10)(4 16 7 15)(6 14 8 13)(17 39 19 37)(18 38 20 40)(21 36)(22 33)(23 34)(24 35)(25 32)(26 29)(27 30)(28 31)(41 48 43 46)(42 47 44 45)

(1 43 33)(2 41 35)(3 48 28)(4 44 36)(5 46 26)(6 45 27)(7 42 34)(8 47 25)(9 39 24)(10 37 22)(11 19 31)(12 17 29)(13 38 23)(14 40 21)(15 18 30)(16 20 32)
 
G:=sub<Sym(48)| (9,10)(11,12)(13,14)(15,16)(17,18,19,20)(21,22,23,24)(25,26,27,28)(29,30,31,32)(33,34,35,36)(37,38,39,40)(41,42,43,44)(45,46,47,48), (1,4,2,7)(3,6,5,8)(9,13,10,14)(11,15,12,16)(17,19)(18,20)(21,24,23,22)(25,26,27,28)(29,32,31,30)(33,34,35,36)(37,39)(38,40), (1,5,2,3)(4,6,7,8)(9,11,10,12)(13,16,14,15)(17,48)(18,47)(19,46)(20,45)(21,27,23,25)(22,26,24,28)(29,35,31,33)(30,34,32,36)(37,41)(38,44)(39,43)(40,42), (1,11,2,12)(3,9,5,10)(4,16,7,15)(6,14,8,13)(17,39,19,37)(18,38,20,40)(21,36)(22,33)(23,34)(24,35)(25,32)(26,29)(27,30)(28,31)(41,48,43,46)(42,47,44,45), (1,43,33)(2,41,35)(3,48,28)(4,44,36)(5,46,26)(6,45,27)(7,42,34)(8,47,25)(9,39,24)(10,37,22)(11,19,31)(12,17,29)(13,38,23)(14,40,21)(15,18,30)(16,20,32)>;
 
G:=Group( (9,10)(11,12)(13,14)(15,16)(17,18,19,20)(21,22,23,24)(25,26,27,28)(29,30,31,32)(33,34,35,36)(37,38,39,40)(41,42,43,44)(45,46,47,48), (1,4,2,7)(3,6,5,8)(9,13,10,14)(11,15,12,16)(17,19)(18,20)(21,24,23,22)(25,26,27,28)(29,32,31,30)(33,34,35,36)(37,39)(38,40), (1,5,2,3)(4,6,7,8)(9,11,10,12)(13,16,14,15)(17,48)(18,47)(19,46)(20,45)(21,27,23,25)(22,26,24,28)(29,35,31,33)(30,34,32,36)(37,41)(38,44)(39,43)(40,42), (1,11,2,12)(3,9,5,10)(4,16,7,15)(6,14,8,13)(17,39,19,37)(18,38,20,40)(21,36)(22,33)(23,34)(24,35)(25,32)(26,29)(27,30)(28,31)(41,48,43,46)(42,47,44,45), (1,43,33)(2,41,35)(3,48,28)(4,44,36)(5,46,26)(6,45,27)(7,42,34)(8,47,25)(9,39,24)(10,37,22)(11,19,31)(12,17,29)(13,38,23)(14,40,21)(15,18,30)(16,20,32) );
 
G=PermutationGroup([(9,10),(11,12),(13,14),(15,16),(17,18,19,20),(21,22,23,24),(25,26,27,28),(29,30,31,32),(33,34,35,36),(37,38,39,40),(41,42,43,44),(45,46,47,48)], [(1,4,2,7),(3,6,5,8),(9,13,10,14),(11,15,12,16),(17,19),(18,20),(21,24,23,22),(25,26,27,28),(29,32,31,30),(33,34,35,36),(37,39),(38,40)], [(1,5,2,3),(4,6,7,8),(9,11,10,12),(13,16,14,15),(17,48),(18,47),(19,46),(20,45),(21,27,23,25),(22,26,24,28),(29,35,31,33),(30,34,32,36),(37,41),(38,44),(39,43),(40,42)], [(1,11,2,12),(3,9,5,10),(4,16,7,15),(6,14,8,13),(17,39,19,37),(18,38,20,40),(21,36),(22,33),(23,34),(24,35),(25,32),(26,29),(27,30),(28,31),(41,48,43,46),(42,47,44,45)], [(1,43,33),(2,41,35),(3,48,28),(4,44,36),(5,46,26),(6,45,27),(7,42,34),(8,47,25),(9,39,24),(10,37,22),(11,19,31),(12,17,29),(13,38,23),(14,40,21),(15,18,30),(16,20,32)])
 

Matrix representation ►G ⊆ GL12(𝔽13)

100000000000
0120000000000
0012000000000
000100000000
000000080000
000000800000
000008000000
000080000000
000000000005
000000000080
000000000800
000000005000
,
000500000000
005000000000
050000000000
500000000000
000000080000
000000500000
000005000000
000080000000
000000001000
0000000001200
0000000000120
000000000001
,
0001200000000
001000000000
0120000000000
100000000000
000001000000
0000120000000
000000010000
0000001200000
000000000010
000000000001
000000001000
000000000100
,
0120000000000
100000000000
0001200000000
001000000000
0000001200000
0000000120000
0000120000000
0000012000000
0000000000012
000000000010
0000000001200
000000001000
,
000010000000
000001000000
000000100000
000000010000
000000001000
000000000100
000000000010
000000000001
100000000000
010000000000
001000000000
000100000000

G:=sub<GL(12,GF(13))| [1,0,0,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,0,0,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,0,0,5,0,0,0],[0,0,0,5,0,0,0,0,0,0,0,0,0,0,5,0,0,0,0,0,0,0,0,0,0,5,0,0,0,0,0,0,0,0,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,0,0,5,0,0,0,0,0,0,0,0,0,0,5,0,0,0,0,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,1],[0,0,0,1,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0],[0,1,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,12,0,0,0],[0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0] >;
 

Character table of C42.A4

 class 123A3B4A4B4C4D4E
 size 1364641212121212
ρ1111111111    trivial
ρ211ζ3ζ3211111    linear of order 3
ρ311ζ32ζ311111    linear of order 3
ρ43300-1-1-1-13    orthogonal lifted from A4
ρ53300-13-1-1-1    orthogonal lifted from A4
ρ633003-1-1-1-1    orthogonal lifted from A4
ρ73300-1-13-1-1    orthogonal lifted from A4
ρ83300-1-1-13-1    orthogonal lifted from A4
ρ912-40000000    symplectic faithful, Schur index 2

In GAP, Magma, Sage, TeX 

C_4^2.A_4
 
% in TeX
 
G:=Group("C4^2.A4");
 
// GroupNames label
 
G:=SmallGroup(192,1025);
 
// by ID
 
G=gap.SmallGroup(192,1025);
 
# by ID
 
G:=PCGroup([7,-3,-2,2,-2,2,-2,2,85,680,2207,184,675,570,745,360,4624,1971,718,102,4037,7062]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^4=b^4=e^3=1,c^2=b^2,d^2=a^2*b^2,a*b=b*a,c*a*c^-1=e*b*e^-1=a^-1,d*a*d^-1=a^-1*b^2,e*a*e^-1=a^-1*b,c*b*c^-1=a^2*b^-1,d*b*d^-1=b^-1,c*d=d*c,e*c*e^-1=a^2*b^2*c*d,e*d*e^-1=a^2*c>;
 
// generators/relations
 

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