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## G = C8×C3.A4order 288 = 25·32

### Direct product of C8 and C3.A4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — C8×C3.A4
 Chief series C1 — C22 — C23 — C22×C6 — C22×C12 — C4×C3.A4 — C8×C3.A4
 Lower central C22 — C8×C3.A4
 Upper central C1 — C24

Generators and relations for C8×C3.A4
G = < a,b,c,d,e | a8=b3=c2=d2=1, e3=b, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, ece-1=cd=dc, ede-1=c >

Smallest permutation representation of C8×C3.A4
On 72 points
Generators in S72
(1 28 59 25 53 66 41 13)(2 29 60 26 54 67 42 14)(3 30 61 27 46 68 43 15)(4 31 62 19 47 69 44 16)(5 32 63 20 48 70 45 17)(6 33 55 21 49 71 37 18)(7 34 56 22 50 72 38 10)(8 35 57 23 51 64 39 11)(9 36 58 24 52 65 40 12)
(1 4 7)(2 5 8)(3 6 9)(10 13 16)(11 14 17)(12 15 18)(19 22 25)(20 23 26)(21 24 27)(28 31 34)(29 32 35)(30 33 36)(37 40 43)(38 41 44)(39 42 45)(46 49 52)(47 50 53)(48 51 54)(55 58 61)(56 59 62)(57 60 63)(64 67 70)(65 68 71)(66 69 72)
(2 54)(3 46)(5 48)(6 49)(8 51)(9 52)(11 23)(12 24)(14 26)(15 27)(17 20)(18 21)(29 67)(30 68)(32 70)(33 71)(35 64)(36 65)(37 55)(39 57)(40 58)(42 60)(43 61)(45 63)
(1 53)(3 46)(4 47)(6 49)(7 50)(9 52)(10 22)(12 24)(13 25)(15 27)(16 19)(18 21)(28 66)(30 68)(31 69)(33 71)(34 72)(36 65)(37 55)(38 56)(40 58)(41 59)(43 61)(44 62)
(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)

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

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

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

96 conjugacy classes

 class 1 2A 2B 2C 3A 3B 4A 4B 4C 4D 6A 6B 6C 6D 6E 6F 8A 8B 8C 8D 8E 8F 8G 8H 9A ··· 9F 12A 12B 12C 12D 12E 12F 12G 12H 18A ··· 18F 24A ··· 24H 24I ··· 24P 36A ··· 36L 72A ··· 72X order 1 2 2 2 3 3 4 4 4 4 6 6 6 6 6 6 8 8 8 8 8 8 8 8 9 ··· 9 12 12 12 12 12 12 12 12 18 ··· 18 24 ··· 24 24 ··· 24 36 ··· 36 72 ··· 72 size 1 1 3 3 1 1 1 1 3 3 1 1 3 3 3 3 1 1 1 1 3 3 3 3 4 ··· 4 1 1 1 1 3 3 3 3 4 ··· 4 1 ··· 1 3 ··· 3 4 ··· 4 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 C4 C6 C8 C9 C12 C18 C24 C36 C72 A4 C2×A4 C3.A4 C4×A4 C2×C3.A4 C8×A4 C4×C3.A4 C8×C3.A4 kernel C8×C3.A4 C4×C3.A4 C22×C24 C2×C3.A4 C22×C12 C3.A4 C22×C8 C22×C6 C22×C4 C2×C6 C23 C22 C24 C12 C8 C6 C4 C3 C2 C1 # reps 1 1 2 2 2 4 6 4 6 8 12 24 1 1 2 2 2 4 4 8

Matrix representation of C8×C3.A4 in GL3(𝔽73) generated by

 63 0 0 0 63 0 0 0 63
,
 8 0 0 0 8 0 0 0 8
,
 1 0 0 0 72 0 0 0 72
,
 72 0 0 0 72 0 0 0 1
,
 0 1 0 0 0 1 8 0 0
G:=sub<GL(3,GF(73))| [63,0,0,0,63,0,0,0,63],[8,0,0,0,8,0,0,0,8],[1,0,0,0,72,0,0,0,72],[72,0,0,0,72,0,0,0,1],[0,0,8,1,0,0,0,1,0] >;

C8×C3.A4 in GAP, Magma, Sage, TeX

C_8\times C_3.A_4
% in TeX

G:=Group("C8xC3.A4");
// GroupNames label

G:=SmallGroup(288,76);
// by ID

G=gap.SmallGroup(288,76);
# by ID

G:=PCGroup([7,-2,-3,-2,-3,-2,-2,2,42,92,142,3036,5305]);
// Polycyclic

G:=Group<a,b,c,d,e|a^8=b^3=c^2=d^2=1,e^3=b,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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