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## G = A4×C2×C12order 288 = 25·32

### Direct product of C2×C12 and A4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — A4×C2×C12
 Chief series C1 — C22 — C23 — C22×C6 — C6×A4 — A4×C2×C6 — A4×C2×C12
 Lower central C22 — A4×C2×C12
 Upper central C1 — C2×C12

Generators and relations for A4×C2×C12
G = < a,b,c,d,e | a2=b12=c2=d2=e3=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, ece-1=cd=dc, ede-1=c >

Subgroups: 396 in 164 conjugacy classes, 64 normal (20 characteristic)
C1, C2, C2, C2, C3, C3, C4, C4, C22, C22, C6, C6, C6, C2×C4, C2×C4, C23, C23, C23, C32, C12, C12, A4, C2×C6, C2×C6, C22×C4, C22×C4, C24, C3×C6, C2×C12, C2×C12, C2×A4, C22×C6, C22×C6, C22×C6, C23×C4, C3×C12, C3×A4, C62, C4×A4, C22×C12, C22×C12, C22×A4, C23×C6, C6×C12, C6×A4, C6×A4, C2×C4×A4, C23×C12, C12×A4, A4×C2×C6, A4×C2×C12
Quotients: C1, C2, C3, C4, C22, C6, C2×C4, C32, C12, A4, C2×C6, C3×C6, C2×C12, C2×A4, C3×C12, C3×A4, C62, C4×A4, C22×A4, C6×C12, C6×A4, C2×C4×A4, C12×A4, A4×C2×C6, A4×C2×C12

Smallest permutation representation of A4×C2×C12
On 72 points
Generators in S72
(1 41)(2 42)(3 43)(4 44)(5 45)(6 46)(7 47)(8 48)(9 37)(10 38)(11 39)(12 40)(13 51)(14 52)(15 53)(16 54)(17 55)(18 56)(19 57)(20 58)(21 59)(22 60)(23 49)(24 50)(25 66)(26 67)(27 68)(28 69)(29 70)(30 71)(31 72)(32 61)(33 62)(34 63)(35 64)(36 65)
(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)(49 50 51 52 53 54 55 56 57 58 59 60)(61 62 63 64 65 66 67 68 69 70 71 72)
(13 57)(14 58)(15 59)(16 60)(17 49)(18 50)(19 51)(20 52)(21 53)(22 54)(23 55)(24 56)(25 72)(26 61)(27 62)(28 63)(29 64)(30 65)(31 66)(32 67)(33 68)(34 69)(35 70)(36 71)
(1 47)(2 48)(3 37)(4 38)(5 39)(6 40)(7 41)(8 42)(9 43)(10 44)(11 45)(12 46)(13 57)(14 58)(15 59)(16 60)(17 49)(18 50)(19 51)(20 52)(21 53)(22 54)(23 55)(24 56)
(1 72 17)(2 61 18)(3 62 19)(4 63 20)(5 64 21)(6 65 22)(7 66 23)(8 67 24)(9 68 13)(10 69 14)(11 70 15)(12 71 16)(25 49 47)(26 50 48)(27 51 37)(28 52 38)(29 53 39)(30 54 40)(31 55 41)(32 56 42)(33 57 43)(34 58 44)(35 59 45)(36 60 46)

G:=sub<Sym(72)| (1,41)(2,42)(3,43)(4,44)(5,45)(6,46)(7,47)(8,48)(9,37)(10,38)(11,39)(12,40)(13,51)(14,52)(15,53)(16,54)(17,55)(18,56)(19,57)(20,58)(21,59)(22,60)(23,49)(24,50)(25,66)(26,67)(27,68)(28,69)(29,70)(30,71)(31,72)(32,61)(33,62)(34,63)(35,64)(36,65), (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)(49,50,51,52,53,54,55,56,57,58,59,60)(61,62,63,64,65,66,67,68,69,70,71,72), (13,57)(14,58)(15,59)(16,60)(17,49)(18,50)(19,51)(20,52)(21,53)(22,54)(23,55)(24,56)(25,72)(26,61)(27,62)(28,63)(29,64)(30,65)(31,66)(32,67)(33,68)(34,69)(35,70)(36,71), (1,47)(2,48)(3,37)(4,38)(5,39)(6,40)(7,41)(8,42)(9,43)(10,44)(11,45)(12,46)(13,57)(14,58)(15,59)(16,60)(17,49)(18,50)(19,51)(20,52)(21,53)(22,54)(23,55)(24,56), (1,72,17)(2,61,18)(3,62,19)(4,63,20)(5,64,21)(6,65,22)(7,66,23)(8,67,24)(9,68,13)(10,69,14)(11,70,15)(12,71,16)(25,49,47)(26,50,48)(27,51,37)(28,52,38)(29,53,39)(30,54,40)(31,55,41)(32,56,42)(33,57,43)(34,58,44)(35,59,45)(36,60,46)>;

G:=Group( (1,41)(2,42)(3,43)(4,44)(5,45)(6,46)(7,47)(8,48)(9,37)(10,38)(11,39)(12,40)(13,51)(14,52)(15,53)(16,54)(17,55)(18,56)(19,57)(20,58)(21,59)(22,60)(23,49)(24,50)(25,66)(26,67)(27,68)(28,69)(29,70)(30,71)(31,72)(32,61)(33,62)(34,63)(35,64)(36,65), (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)(49,50,51,52,53,54,55,56,57,58,59,60)(61,62,63,64,65,66,67,68,69,70,71,72), (13,57)(14,58)(15,59)(16,60)(17,49)(18,50)(19,51)(20,52)(21,53)(22,54)(23,55)(24,56)(25,72)(26,61)(27,62)(28,63)(29,64)(30,65)(31,66)(32,67)(33,68)(34,69)(35,70)(36,71), (1,47)(2,48)(3,37)(4,38)(5,39)(6,40)(7,41)(8,42)(9,43)(10,44)(11,45)(12,46)(13,57)(14,58)(15,59)(16,60)(17,49)(18,50)(19,51)(20,52)(21,53)(22,54)(23,55)(24,56), (1,72,17)(2,61,18)(3,62,19)(4,63,20)(5,64,21)(6,65,22)(7,66,23)(8,67,24)(9,68,13)(10,69,14)(11,70,15)(12,71,16)(25,49,47)(26,50,48)(27,51,37)(28,52,38)(29,53,39)(30,54,40)(31,55,41)(32,56,42)(33,57,43)(34,58,44)(35,59,45)(36,60,46) );

G=PermutationGroup([[(1,41),(2,42),(3,43),(4,44),(5,45),(6,46),(7,47),(8,48),(9,37),(10,38),(11,39),(12,40),(13,51),(14,52),(15,53),(16,54),(17,55),(18,56),(19,57),(20,58),(21,59),(22,60),(23,49),(24,50),(25,66),(26,67),(27,68),(28,69),(29,70),(30,71),(31,72),(32,61),(33,62),(34,63),(35,64),(36,65)], [(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),(49,50,51,52,53,54,55,56,57,58,59,60),(61,62,63,64,65,66,67,68,69,70,71,72)], [(13,57),(14,58),(15,59),(16,60),(17,49),(18,50),(19,51),(20,52),(21,53),(22,54),(23,55),(24,56),(25,72),(26,61),(27,62),(28,63),(29,64),(30,65),(31,66),(32,67),(33,68),(34,69),(35,70),(36,71)], [(1,47),(2,48),(3,37),(4,38),(5,39),(6,40),(7,41),(8,42),(9,43),(10,44),(11,45),(12,46),(13,57),(14,58),(15,59),(16,60),(17,49),(18,50),(19,51),(20,52),(21,53),(22,54),(23,55),(24,56)], [(1,72,17),(2,61,18),(3,62,19),(4,63,20),(5,64,21),(6,65,22),(7,66,23),(8,67,24),(9,68,13),(10,69,14),(11,70,15),(12,71,16),(25,49,47),(26,50,48),(27,51,37),(28,52,38),(29,53,39),(30,54,40),(31,55,41),(32,56,42),(33,57,43),(34,58,44),(35,59,45),(36,60,46)]])

96 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3A 3B 3C ··· 3H 4A 4B 4C 4D 4E 4F 4G 4H 6A ··· 6F 6G ··· 6N 6O ··· 6AF 12A ··· 12H 12I ··· 12P 12Q ··· 12AN order 1 2 2 2 2 2 2 2 3 3 3 ··· 3 4 4 4 4 4 4 4 4 6 ··· 6 6 ··· 6 6 ··· 6 12 ··· 12 12 ··· 12 12 ··· 12 size 1 1 1 1 3 3 3 3 1 1 4 ··· 4 1 1 1 1 3 3 3 3 1 ··· 1 3 ··· 3 4 ··· 4 1 ··· 1 3 ··· 3 4 ··· 4

96 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 3 3 3 3 3 3 3 3 type + + + + + + image C1 C2 C2 C3 C3 C4 C6 C6 C6 C6 C12 C12 A4 C2×A4 C2×A4 C3×A4 C4×A4 C6×A4 C6×A4 C12×A4 kernel A4×C2×C12 C12×A4 A4×C2×C6 C2×C4×A4 C23×C12 C6×A4 C4×A4 C22×C12 C22×A4 C23×C6 C2×A4 C22×C6 C2×C12 C12 C2×C6 C2×C4 C6 C4 C22 C2 # reps 1 2 1 6 2 4 12 4 6 2 24 8 1 2 1 2 4 4 2 8

Matrix representation of A4×C2×C12 in GL4(𝔽13) generated by

 1 0 0 0 0 12 0 0 0 0 12 0 0 0 0 12
,
 11 0 0 0 0 3 0 0 0 0 3 0 0 0 0 3
,
 1 0 0 0 0 1 0 0 0 0 12 0 0 4 0 12
,
 1 0 0 0 0 12 0 0 0 0 12 0 0 9 3 1
,
 1 0 0 0 0 0 3 0 0 12 4 7 0 0 0 9
G:=sub<GL(4,GF(13))| [1,0,0,0,0,12,0,0,0,0,12,0,0,0,0,12],[11,0,0,0,0,3,0,0,0,0,3,0,0,0,0,3],[1,0,0,0,0,1,0,4,0,0,12,0,0,0,0,12],[1,0,0,0,0,12,0,9,0,0,12,3,0,0,0,1],[1,0,0,0,0,0,12,0,0,3,4,0,0,0,7,9] >;

A4×C2×C12 in GAP, Magma, Sage, TeX

A_4\times C_2\times C_{12}
% in TeX

G:=Group("A4xC2xC12");
// GroupNames label

G:=SmallGroup(288,979);
// by ID

G=gap.SmallGroup(288,979);
# by ID

G:=PCGroup([7,-2,-2,-3,-3,-2,-2,2,260,1531,2666]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^12=c^2=d^2=e^3=1,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,e*c*e^-1=c*d=d*c,e*d*e^-1=c>;
// generators/relations

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