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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 [×2], C2 [×4], C3, C3 [×3], C4 [×2], C4 [×2], C22 [×2], C22 [×11], C6, C6 [×2], C6 [×13], C2×C4, C2×C4 [×9], C23, C23 [×2], C23 [×4], C32, C12 [×2], C12 [×8], A4 [×3], C2×C6 [×2], C2×C6 [×14], C22×C4 [×2], C22×C4 [×4], C24, C3×C6 [×3], C2×C12, C2×C12 [×12], C2×A4 [×9], C22×C6, C22×C6 [×2], C22×C6 [×4], C23×C4, C3×C12 [×2], C3×A4, C62, C4×A4 [×6], C22×C12 [×2], C22×C12 [×4], C22×A4 [×3], C23×C6, C6×C12, C6×A4, C6×A4 [×2], C2×C4×A4 [×3], C23×C12, C12×A4 [×2], A4×C2×C6, A4×C2×C12
Quotients: C1, C2 [×3], C3 [×4], C4 [×2], C22, C6 [×12], C2×C4, C32, C12 [×8], A4, C2×C6 [×4], C3×C6 [×3], C2×C12 [×4], C2×A4 [×3], C3×C12 [×2], C3×A4, C62, C4×A4 [×2], C22×A4, C6×C12, C6×A4 [×3], C2×C4×A4, C12×A4 [×2], A4×C2×C6, A4×C2×C12

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

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

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

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

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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