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## G = M4(2)×C3⋊S3order 288 = 25·32

### Direct product of M4(2) and C3⋊S3

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C6 — M4(2)×C3⋊S3
 Chief series C1 — C3 — C32 — C3×C6 — C3×C12 — C4×C3⋊S3 — C2×C4×C3⋊S3 — M4(2)×C3⋊S3
 Lower central C32 — C3×C6 — M4(2)×C3⋊S3
 Upper central C1 — C4 — M4(2)

Generators and relations for M4(2)×C3⋊S3
G = < a,b,c,d,e | a8=b2=c3=d3=e2=1, bab=a5, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ece=c-1, ede=d-1 >

Subgroups: 660 in 204 conjugacy classes, 75 normal (27 characteristic)
C1, C2, C2 [×4], C3 [×4], C4 [×2], C4 [×2], C22, C22 [×4], S3 [×12], C6 [×4], C6 [×4], C8 [×2], C8 [×2], C2×C4, C2×C4 [×5], C23, C32, Dic3 [×8], C12 [×8], D6 [×16], C2×C6 [×4], C2×C8 [×2], M4(2), M4(2) [×3], C22×C4, C3⋊S3 [×2], C3⋊S3, C3×C6, C3×C6, C3⋊C8 [×8], C24 [×8], C4×S3 [×16], C2×Dic3 [×4], C2×C12 [×4], C22×S3 [×4], C2×M4(2), C3⋊Dic3 [×2], C3×C12 [×2], C2×C3⋊S3 [×2], C2×C3⋊S3 [×2], C62, S3×C8 [×8], C8⋊S3 [×8], C4.Dic3 [×4], C3×M4(2) [×4], S3×C2×C4 [×4], C324C8 [×2], C3×C24 [×2], C4×C3⋊S3 [×4], C2×C3⋊Dic3, C6×C12, C22×C3⋊S3, S3×M4(2) [×4], C8×C3⋊S3 [×2], C24⋊S3 [×2], C12.58D6, C32×M4(2), C2×C4×C3⋊S3, M4(2)×C3⋊S3
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3 [×4], C2×C4 [×6], C23, D6 [×12], M4(2) [×2], C22×C4, C3⋊S3, C4×S3 [×8], C22×S3 [×4], C2×M4(2), C2×C3⋊S3 [×3], S3×C2×C4 [×4], C4×C3⋊S3 [×2], C22×C3⋊S3, S3×M4(2) [×4], C2×C4×C3⋊S3, M4(2)×C3⋊S3

Smallest permutation representation of M4(2)×C3⋊S3
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)
(2 6)(4 8)(9 13)(11 15)(17 21)(19 23)(26 30)(28 32)(33 37)(35 39)(42 46)(44 48)(49 53)(51 55)(57 61)(59 63)(66 70)(68 72)
(1 40 18)(2 33 19)(3 34 20)(4 35 21)(5 36 22)(6 37 23)(7 38 24)(8 39 17)(9 59 48)(10 60 41)(11 61 42)(12 62 43)(13 63 44)(14 64 45)(15 57 46)(16 58 47)(25 50 69)(26 51 70)(27 52 71)(28 53 72)(29 54 65)(30 55 66)(31 56 67)(32 49 68)
(1 47 25)(2 48 26)(3 41 27)(4 42 28)(5 43 29)(6 44 30)(7 45 31)(8 46 32)(9 51 33)(10 52 34)(11 53 35)(12 54 36)(13 55 37)(14 56 38)(15 49 39)(16 50 40)(17 57 68)(18 58 69)(19 59 70)(20 60 71)(21 61 72)(22 62 65)(23 63 66)(24 64 67)
(1 5)(2 6)(3 7)(4 8)(9 66)(10 67)(11 68)(12 69)(13 70)(14 71)(15 72)(16 65)(17 35)(18 36)(19 37)(20 38)(21 39)(22 40)(23 33)(24 34)(25 43)(26 44)(27 45)(28 46)(29 47)(30 48)(31 41)(32 42)(49 61)(50 62)(51 63)(52 64)(53 57)(54 58)(55 59)(56 60)

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

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

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

60 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3A 3B 3C 3D 4A 4B 4C 4D 4E 4F 6A 6B 6C 6D 6E 6F 6G 6H 8A 8B 8C 8D 8E 8F 8G 8H 12A ··· 12H 12I 12J 12K 12L 24A ··· 24P order 1 2 2 2 2 2 3 3 3 3 4 4 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 24 ··· 24 size 1 1 2 9 9 18 2 2 2 2 1 1 2 9 9 18 2 2 2 2 4 4 4 4 2 2 2 2 18 18 18 18 2 ··· 2 4 4 4 4 4 ··· 4

60 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 4 type + + + + + + + + + image C1 C2 C2 C2 C2 C2 C4 C4 C4 S3 D6 D6 M4(2) C4×S3 C4×S3 S3×M4(2) kernel M4(2)×C3⋊S3 C8×C3⋊S3 C24⋊S3 C12.58D6 C32×M4(2) C2×C4×C3⋊S3 C4×C3⋊S3 C2×C3⋊Dic3 C22×C3⋊S3 C3×M4(2) C24 C2×C12 C3⋊S3 C12 C2×C6 C3 # reps 1 2 2 1 1 1 4 2 2 4 8 4 4 8 8 8

Matrix representation of M4(2)×C3⋊S3 in GL6(𝔽73)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 72 0 0 0 0 0 0 72 0 0 0 0 0 0 0 1 0 0 0 0 27 0
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 72
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 72 1 0 0 0 0 72 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 72 72 0 0 0 0 1 0 0 0 0 0 0 0 0 72 0 0 0 0 1 72 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 72 72 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 72 0 0 0 0 0 0 72

G:=sub<GL(6,GF(73))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,0,27,0,0,0,0,1,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,72],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,72,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[72,1,0,0,0,0,72,0,0,0,0,0,0,0,0,1,0,0,0,0,72,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,72,0,0,0,0,0,72,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,72,0,0,0,0,0,0,72] >;

M4(2)×C3⋊S3 in GAP, Magma, Sage, TeX

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

G:=Group("M4(2)xC3:S3");
// GroupNames label

G:=SmallGroup(288,763);
// by ID

G=gap.SmallGroup(288,763);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,219,58,80,2693,9414]);
// Polycyclic

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

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