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## G = A4×C24order 288 = 25·32

### Direct product of C24 and A4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — A4×C24
 Chief series C1 — C22 — C23 — C22×C4 — C22×C12 — C12×A4 — A4×C24
 Lower central C22 — A4×C24
 Upper central C1 — C24

Generators and relations for A4×C24
G = < a,b,c,d | a24=b2=c2=d3=1, ab=ba, ac=ca, ad=da, dbd-1=bc=cb, dcd-1=b >

Subgroups: 156 in 68 conjugacy classes, 32 normal (20 characteristic)
C1, C2, C2 [×2], C3, C3 [×3], C4, C4, C22, C22 [×2], C6, C6 [×5], C8, C8, C2×C4 [×2], C23, C32, C12, C12 [×4], A4 [×3], C2×C6, C2×C6 [×2], C2×C8 [×2], C22×C4, C3×C6, C24, C24 [×4], C2×C12 [×2], C2×A4 [×3], C22×C6, C22×C8, C3×C12, C3×A4, C2×C24 [×2], C4×A4 [×3], C22×C12, C3×C24, C6×A4, C8×A4 [×3], C22×C24, C12×A4, A4×C24
Quotients: C1, C2, C3 [×4], C4, C6 [×4], C8, C32, C12 [×4], A4, C3×C6, C24 [×4], C2×A4, C3×C12, C3×A4, C4×A4, C3×C24, C6×A4, C8×A4, C12×A4, A4×C24

Smallest permutation representation of A4×C24
On 72 points
Generators in S72
(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)
(25 37)(26 38)(27 39)(28 40)(29 41)(30 42)(31 43)(32 44)(33 45)(34 46)(35 47)(36 48)(49 61)(50 62)(51 63)(52 64)(53 65)(54 66)(55 67)(56 68)(57 69)(58 70)(59 71)(60 72)
(1 13)(2 14)(3 15)(4 16)(5 17)(6 18)(7 19)(8 20)(9 21)(10 22)(11 23)(12 24)(25 37)(26 38)(27 39)(28 40)(29 41)(30 42)(31 43)(32 44)(33 45)(34 46)(35 47)(36 48)
(1 53 45)(2 54 46)(3 55 47)(4 56 48)(5 57 25)(6 58 26)(7 59 27)(8 60 28)(9 61 29)(10 62 30)(11 63 31)(12 64 32)(13 65 33)(14 66 34)(15 67 35)(16 68 36)(17 69 37)(18 70 38)(19 71 39)(20 72 40)(21 49 41)(22 50 42)(23 51 43)(24 52 44)

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

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

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

96 conjugacy classes

 class 1 2A 2B 2C 3A 3B 3C ··· 3H 4A 4B 4C 4D 6A 6B 6C 6D 6E 6F 6G ··· 6L 8A 8B 8C 8D 8E 8F 8G 8H 12A 12B 12C 12D 12E 12F 12G 12H 12I ··· 12T 24A ··· 24H 24I ··· 24P 24Q ··· 24AN order 1 2 2 2 3 3 3 ··· 3 4 4 4 4 6 6 6 6 6 6 6 ··· 6 8 8 8 8 8 8 8 8 12 12 12 12 12 12 12 12 12 ··· 12 24 ··· 24 24 ··· 24 24 ··· 24 size 1 1 3 3 1 1 4 ··· 4 1 1 3 3 1 1 3 3 3 3 4 ··· 4 1 1 1 1 3 3 3 3 1 1 1 1 3 3 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 C3 C3 C4 C6 C6 C8 C12 C12 C24 C24 A4 C2×A4 C3×A4 C4×A4 C6×A4 C8×A4 C12×A4 A4×C24 kernel A4×C24 C12×A4 C8×A4 C22×C24 C6×A4 C4×A4 C22×C12 C3×A4 C2×A4 C22×C6 A4 C2×C6 C24 C12 C8 C6 C4 C3 C2 C1 # reps 1 1 6 2 2 6 2 4 12 4 24 8 1 1 2 2 2 4 4 8

Matrix representation of A4×C24 in GL4(𝔽73) generated by

 17 0 0 0 0 63 0 0 0 0 63 0 0 0 0 63
,
 1 0 0 0 0 1 64 68 0 0 72 0 0 0 0 72
,
 1 0 0 0 0 72 0 5 0 0 72 0 0 0 0 1
,
 8 0 0 0 0 68 22 17 0 0 0 1 0 71 9 5
G:=sub<GL(4,GF(73))| [17,0,0,0,0,63,0,0,0,0,63,0,0,0,0,63],[1,0,0,0,0,1,0,0,0,64,72,0,0,68,0,72],[1,0,0,0,0,72,0,0,0,0,72,0,0,5,0,1],[8,0,0,0,0,68,0,71,0,22,0,9,0,17,1,5] >;

A4×C24 in GAP, Magma, Sage, TeX

A_4\times C_{24}
% in TeX

G:=Group("A4xC24");
// GroupNames label

G:=SmallGroup(288,637);
// by ID

G=gap.SmallGroup(288,637);
# by ID

G:=PCGroup([7,-2,-3,-3,-2,-2,-2,2,126,80,3036,5305]);
// Polycyclic

G:=Group<a,b,c,d|a^24=b^2=c^2=d^3=1,a*b=b*a,a*c=c*a,a*d=d*a,d*b*d^-1=b*c=c*b,d*c*d^-1=b>;
// generators/relations

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