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G = C3×C4.A4order 144 = 24·32

Direct product of C3 and C4.A4

direct product, non-abelian, soluble

Aliases: C3×C4.A4, C12.3A4, SL2(𝔽3)⋊2C6, C4.(C3×A4), C4○D4⋊C32, Q8.(C3×C6), C2.3(C6×A4), C6.10(C2×A4), (C3×Q8).4C6, (C3×SL2(𝔽3))⋊5C2, (C3×C4○D4)⋊C3, SmallGroup(144,157)

Series: Derived Chief Lower central Upper central

C1C2Q8 — C3×C4.A4
C1C2Q8C3×Q8C3×SL2(𝔽3) — C3×C4.A4
Q8 — C3×C4.A4
C1C12

Generators and relations for C3×C4.A4
 G = < a,b,c,d,e | a3=b4=e3=1, c2=d2=b2, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, dcd-1=b2c, ece-1=b2cd, ede-1=c >

6C2
4C3
4C3
4C3
3C22
3C4
4C6
4C6
4C6
6C6
4C32
3C2×C4
3D4
3C2×C6
3C12
4C12
4C12
4C12
4C3×C6
3C2×C12
3C3×D4
4C3×C12

Smallest permutation representation of C3×C4.A4
On 48 points
Generators in S48
(1 31 35)(2 32 36)(3 29 33)(4 30 34)(5 14 10)(6 15 11)(7 16 12)(8 13 9)(17 27 21)(18 28 22)(19 25 23)(20 26 24)(37 45 41)(38 46 42)(39 47 43)(40 48 44)
(1 2 3 4)(5 6 7 8)(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 17 3 19)(2 18 4 20)(5 48 7 46)(6 45 8 47)(9 39 11 37)(10 40 12 38)(13 43 15 41)(14 44 16 42)(21 33 23 35)(22 34 24 36)(25 31 27 29)(26 32 28 30)
(1 37 3 39)(2 38 4 40)(5 28 7 26)(6 25 8 27)(9 17 11 19)(10 18 12 20)(13 21 15 23)(14 22 16 24)(29 47 31 45)(30 48 32 46)(33 43 35 41)(34 44 36 42)
(5 28 46)(6 25 47)(7 26 48)(8 27 45)(9 17 37)(10 18 38)(11 19 39)(12 20 40)(13 21 41)(14 22 42)(15 23 43)(16 24 44)

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

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

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

C3×C4.A4 is a maximal subgroup of
C3⋊U2(𝔽3)  SL2(𝔽3).Dic3  C12.6S4  C12.14S4  C12.7S4  Dic6.A4  D12.A4  C36.A4  C4○D4⋊He3
C3×C4.A4 is a maximal quotient of
C12×SL2(𝔽3)  C36.A4  Q8⋊C94C6  C4○D4⋊He3

42 conjugacy classes

class 1 2A2B3A3B3C···3H4A4B4C6A6B6C···6H6I6J12A12B12C12D12E···12P12Q12R
order122333···3444666···6661212121212···121212
size116114···4116114···46611114···466

42 irreducible representations

dim111111223333
type++++
imageC1C2C3C3C6C6C4.A4C3×C4.A4A4C2×A4C3×A4C6×A4
kernelC3×C4.A4C3×SL2(𝔽3)C4.A4C3×C4○D4SL2(𝔽3)C3×Q8C3C1C12C6C4C2
# reps1162626121122

Matrix representation of C3×C4.A4 in GL2(𝔽13) generated by

90
09
,
80
08
,
04
30
,
50
08
,
127
105
G:=sub<GL(2,GF(13))| [9,0,0,9],[8,0,0,8],[0,3,4,0],[5,0,0,8],[12,10,7,5] >;

C3×C4.A4 in GAP, Magma, Sage, TeX

C_3\times C_4.A_4
% in TeX

G:=Group("C3xC4.A4");
// GroupNames label

G:=SmallGroup(144,157);
// by ID

G=gap.SmallGroup(144,157);
# by ID

G:=PCGroup([6,-2,-3,-3,-2,2,-2,432,441,117,820,202,88]);
// Polycyclic

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

Export

Subgroup lattice of C3×C4.A4 in TeX

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