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## G = C3×S3×M4(2)  order 288 = 25·32

### Direct product of C3, S3 and M4(2)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C6 — C3×S3×M4(2)
 Chief series C1 — C3 — C6 — C12 — C3×C12 — S3×C12 — S3×C2×C12 — C3×S3×M4(2)
 Lower central C3 — C6 — C3×S3×M4(2)
 Upper central C1 — C12 — C3×M4(2)

Generators and relations for C3×S3×M4(2)
G = < a,b,c,d,e | a3=b3=c2=d8=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede=d5 >

Subgroups: 282 in 146 conjugacy classes, 78 normal (54 characteristic)
C1, C2, C2 [×4], C3 [×2], C3, C4 [×2], C4 [×2], C22, C22 [×4], S3 [×2], S3, C6 [×2], C6 [×7], C8 [×2], C8 [×2], C2×C4, C2×C4 [×5], C23, C32, Dic3 [×2], C12 [×4], C12 [×4], D6 [×2], D6 [×2], C2×C6 [×2], C2×C6 [×5], C2×C8 [×2], M4(2), M4(2) [×3], C22×C4, C3×S3 [×2], C3×S3, C3×C6, C3×C6, C3⋊C8 [×2], C24 [×4], C24 [×4], C4×S3 [×4], C2×Dic3, C2×C12 [×2], C2×C12 [×6], C22×S3, C22×C6, C2×M4(2), C3×Dic3 [×2], C3×C12 [×2], S3×C6 [×2], S3×C6 [×2], C62, S3×C8 [×2], C8⋊S3 [×2], C4.Dic3, C2×C24 [×2], C3×M4(2) [×2], C3×M4(2) [×4], S3×C2×C4, C22×C12, C3×C3⋊C8 [×2], C3×C24 [×2], S3×C12 [×4], C6×Dic3, C6×C12, S3×C2×C6, S3×M4(2), C6×M4(2), S3×C24 [×2], C3×C8⋊S3 [×2], C3×C4.Dic3, C32×M4(2), S3×C2×C12, C3×S3×M4(2)
Quotients: C1, C2 [×7], C3, C4 [×4], C22 [×7], S3, C6 [×7], C2×C4 [×6], C23, C12 [×4], D6 [×3], C2×C6 [×7], M4(2) [×2], C22×C4, C3×S3, C4×S3 [×2], C2×C12 [×6], C22×S3, C22×C6, C2×M4(2), S3×C6 [×3], C3×M4(2) [×2], S3×C2×C4, C22×C12, S3×C12 [×2], S3×C2×C6, S3×M4(2), C6×M4(2), S3×C2×C12, C3×S3×M4(2)

Smallest permutation representation of C3×S3×M4(2)
On 48 points
Generators in S48
(1 30 39)(2 31 40)(3 32 33)(4 25 34)(5 26 35)(6 27 36)(7 28 37)(8 29 38)(9 42 18)(10 43 19)(11 44 20)(12 45 21)(13 46 22)(14 47 23)(15 48 24)(16 41 17)
(1 30 39)(2 31 40)(3 32 33)(4 25 34)(5 26 35)(6 27 36)(7 28 37)(8 29 38)(9 18 42)(10 19 43)(11 20 44)(12 21 45)(13 22 46)(14 23 47)(15 24 48)(16 17 41)
(1 13)(2 14)(3 15)(4 16)(5 9)(6 10)(7 11)(8 12)(17 34)(18 35)(19 36)(20 37)(21 38)(22 39)(23 40)(24 33)(25 41)(26 42)(27 43)(28 44)(29 45)(30 46)(31 47)(32 48)
(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)
(2 6)(4 8)(10 14)(12 16)(17 21)(19 23)(25 29)(27 31)(34 38)(36 40)(41 45)(43 47)

G:=sub<Sym(48)| (1,30,39)(2,31,40)(3,32,33)(4,25,34)(5,26,35)(6,27,36)(7,28,37)(8,29,38)(9,42,18)(10,43,19)(11,44,20)(12,45,21)(13,46,22)(14,47,23)(15,48,24)(16,41,17), (1,30,39)(2,31,40)(3,32,33)(4,25,34)(5,26,35)(6,27,36)(7,28,37)(8,29,38)(9,18,42)(10,19,43)(11,20,44)(12,21,45)(13,22,46)(14,23,47)(15,24,48)(16,17,41), (1,13)(2,14)(3,15)(4,16)(5,9)(6,10)(7,11)(8,12)(17,34)(18,35)(19,36)(20,37)(21,38)(22,39)(23,40)(24,33)(25,41)(26,42)(27,43)(28,44)(29,45)(30,46)(31,47)(32,48), (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), (2,6)(4,8)(10,14)(12,16)(17,21)(19,23)(25,29)(27,31)(34,38)(36,40)(41,45)(43,47)>;

G:=Group( (1,30,39)(2,31,40)(3,32,33)(4,25,34)(5,26,35)(6,27,36)(7,28,37)(8,29,38)(9,42,18)(10,43,19)(11,44,20)(12,45,21)(13,46,22)(14,47,23)(15,48,24)(16,41,17), (1,30,39)(2,31,40)(3,32,33)(4,25,34)(5,26,35)(6,27,36)(7,28,37)(8,29,38)(9,18,42)(10,19,43)(11,20,44)(12,21,45)(13,22,46)(14,23,47)(15,24,48)(16,17,41), (1,13)(2,14)(3,15)(4,16)(5,9)(6,10)(7,11)(8,12)(17,34)(18,35)(19,36)(20,37)(21,38)(22,39)(23,40)(24,33)(25,41)(26,42)(27,43)(28,44)(29,45)(30,46)(31,47)(32,48), (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), (2,6)(4,8)(10,14)(12,16)(17,21)(19,23)(25,29)(27,31)(34,38)(36,40)(41,45)(43,47) );

G=PermutationGroup([(1,30,39),(2,31,40),(3,32,33),(4,25,34),(5,26,35),(6,27,36),(7,28,37),(8,29,38),(9,42,18),(10,43,19),(11,44,20),(12,45,21),(13,46,22),(14,47,23),(15,48,24),(16,41,17)], [(1,30,39),(2,31,40),(3,32,33),(4,25,34),(5,26,35),(6,27,36),(7,28,37),(8,29,38),(9,18,42),(10,19,43),(11,20,44),(12,21,45),(13,22,46),(14,23,47),(15,24,48),(16,17,41)], [(1,13),(2,14),(3,15),(4,16),(5,9),(6,10),(7,11),(8,12),(17,34),(18,35),(19,36),(20,37),(21,38),(22,39),(23,40),(24,33),(25,41),(26,42),(27,43),(28,44),(29,45),(30,46),(31,47),(32,48)], [(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)], [(2,6),(4,8),(10,14),(12,16),(17,21),(19,23),(25,29),(27,31),(34,38),(36,40),(41,45),(43,47)])

90 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3A 3B 3C 3D 3E 4A 4B 4C 4D 4E 4F 6A 6B 6C ··· 6G 6H 6I 6J 6K 6L 6M 6N 6O 6P 8A 8B 8C 8D 8E 8F 8G 8H 12A 12B 12C 12D 12E ··· 12L 12M 12N 12O 12P 12Q 12R 12S 12T 12U 24A ··· 24H 24I ··· 24T 24U ··· 24AB order 1 2 2 2 2 2 3 3 3 3 3 4 4 4 4 4 4 6 6 6 ··· 6 6 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 12 12 12 12 12 24 ··· 24 24 ··· 24 24 ··· 24 size 1 1 2 3 3 6 1 1 2 2 2 1 1 2 3 3 6 1 1 2 ··· 2 3 3 3 3 4 4 4 6 6 2 2 2 2 6 6 6 6 1 1 1 1 2 ··· 2 3 3 3 3 4 4 4 6 6 2 ··· 2 4 ··· 4 6 ··· 6

90 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 4 4 type + + + + + + + + + image C1 C2 C2 C2 C2 C2 C3 C4 C4 C4 C6 C6 C6 C6 C6 C12 C12 C12 S3 D6 D6 M4(2) C3×S3 C4×S3 C4×S3 S3×C6 S3×C6 C3×M4(2) S3×C12 S3×C12 S3×M4(2) C3×S3×M4(2) kernel C3×S3×M4(2) S3×C24 C3×C8⋊S3 C3×C4.Dic3 C32×M4(2) S3×C2×C12 S3×M4(2) S3×C12 C6×Dic3 S3×C2×C6 S3×C8 C8⋊S3 C4.Dic3 C3×M4(2) S3×C2×C4 C4×S3 C2×Dic3 C22×S3 C3×M4(2) C24 C2×C12 C3×S3 M4(2) C12 C2×C6 C8 C2×C4 S3 C4 C22 C3 C1 # reps 1 2 2 1 1 1 2 4 2 2 4 4 2 2 2 8 4 4 1 2 1 4 2 2 2 4 2 8 4 4 2 4

Matrix representation of C3×S3×M4(2) in GL4(𝔽73) generated by

 1 0 0 0 0 1 0 0 0 0 8 0 0 0 0 8
,
 1 0 0 0 0 1 0 0 0 0 8 0 0 0 0 64
,
 72 0 0 0 0 72 0 0 0 0 0 1 0 0 1 0
,
 0 1 0 0 46 0 0 0 0 0 46 0 0 0 0 46
,
 1 0 0 0 0 72 0 0 0 0 1 0 0 0 0 1
G:=sub<GL(4,GF(73))| [1,0,0,0,0,1,0,0,0,0,8,0,0,0,0,8],[1,0,0,0,0,1,0,0,0,0,8,0,0,0,0,64],[72,0,0,0,0,72,0,0,0,0,0,1,0,0,1,0],[0,46,0,0,1,0,0,0,0,0,46,0,0,0,0,46],[1,0,0,0,0,72,0,0,0,0,1,0,0,0,0,1] >;

C3×S3×M4(2) in GAP, Magma, Sage, TeX

C_3\times S_3\times M_4(2)
% in TeX

G:=Group("C3xS3xM4(2)");
// GroupNames label

G:=SmallGroup(288,677);
// by ID

G=gap.SmallGroup(288,677);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-3,555,142,102,9414]);
// Polycyclic

G:=Group<a,b,c,d,e|a^3=b^3=c^2=d^8=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=b^-1,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e=d^5>;
// generators/relations

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