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## G = C3×C4.A4order 144 = 24·32

### Direct product of C3 and C4.A4

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

 Derived series C1 — C2 — Q8 — C3×C4.A4
 Chief series C1 — C2 — Q8 — C3×Q8 — C3×SL2(𝔽3) — C3×C4.A4
 Lower central Q8 — C3×C4.A4
 Upper central C1 — C12

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 >

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 2A 2B 3A 3B 3C ··· 3H 4A 4B 4C 6A 6B 6C ··· 6H 6I 6J 12A 12B 12C 12D 12E ··· 12P 12Q 12R order 1 2 2 3 3 3 ··· 3 4 4 4 6 6 6 ··· 6 6 6 12 12 12 12 12 ··· 12 12 12 size 1 1 6 1 1 4 ··· 4 1 1 6 1 1 4 ··· 4 6 6 1 1 1 1 4 ··· 4 6 6

42 irreducible representations

 dim 1 1 1 1 1 1 2 2 3 3 3 3 type + + + + image C1 C2 C3 C3 C6 C6 C4.A4 C3×C4.A4 A4 C2×A4 C3×A4 C6×A4 kernel C3×C4.A4 C3×SL2(𝔽3) C4.A4 C3×C4○D4 SL2(𝔽3) C3×Q8 C3 C1 C12 C6 C4 C2 # reps 1 1 6 2 6 2 6 12 1 1 2 2

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

 9 0 0 9
,
 8 0 0 8
,
 0 4 3 0
,
 5 0 0 8
,
 12 7 10 5
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

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