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## G = C3×C32⋊M4(2)  order 432 = 24·33

### Direct product of C3 and C32⋊M4(2)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C6 — C3×C32⋊M4(2)
 Chief series C1 — C32 — C3×C6 — C3⋊Dic3 — C3×C3⋊Dic3 — C3×C32⋊2C8 — C3×C32⋊M4(2)
 Lower central C32 — C3×C6 — C3×C32⋊M4(2)
 Upper central C1 — C6 — C12

Generators and relations for C3×C32⋊M4(2)
G = < a,b,c,d,e | a3=b3=c3=d8=e2=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, dbd-1=bc-1, ebe=b-1, dcd-1=b-1c-1, ece=c-1, ede=d5 >

Subgroups: 364 in 84 conjugacy classes, 24 normal (20 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, S3, C6, C6, C8, C2×C4, C32, C32, Dic3, C12, C12, D6, C2×C6, M4(2), C3×S3, C3⋊S3, C3×C6, C3×C6, C24, C4×S3, C2×C12, C33, C3×Dic3, C3⋊Dic3, C3×C12, C3×C12, S3×C6, C2×C3⋊S3, C3×M4(2), C3×C3⋊S3, C32×C6, C322C8, S3×C12, C4×C3⋊S3, C3×C3⋊Dic3, C32×C12, C6×C3⋊S3, C32⋊M4(2), C3×C322C8, C12×C3⋊S3, C3×C32⋊M4(2)
Quotients: C1, C2, C3, C4, C22, C6, C2×C4, C12, C2×C6, M4(2), C2×C12, C32⋊C4, C3×M4(2), C2×C32⋊C4, C3×C32⋊C4, C32⋊M4(2), C6×C32⋊C4, C3×C32⋊M4(2)

Smallest permutation representation of C3×C32⋊M4(2)
On 48 points
Generators in S48
(1 15 37)(2 16 38)(3 9 39)(4 10 40)(5 11 33)(6 12 34)(7 13 35)(8 14 36)(17 47 30)(18 48 31)(19 41 32)(20 42 25)(21 43 26)(22 44 27)(23 45 28)(24 46 29)
(2 38 16)(4 10 40)(6 34 12)(8 14 36)(18 31 48)(20 42 25)(22 27 44)(24 46 29)
(1 15 37)(2 38 16)(3 39 9)(4 10 40)(5 11 33)(6 34 12)(7 35 13)(8 14 36)(17 47 30)(18 31 48)(19 32 41)(20 42 25)(21 43 26)(22 27 44)(23 28 45)(24 46 29)
(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)
(1 23)(2 20)(3 17)(4 22)(5 19)(6 24)(7 21)(8 18)(9 47)(10 44)(11 41)(12 46)(13 43)(14 48)(15 45)(16 42)(25 38)(26 35)(27 40)(28 37)(29 34)(30 39)(31 36)(32 33)

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

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

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

54 conjugacy classes

 class 1 2A 2B 3A 3B 3C ··· 3H 4A 4B 4C 6A 6B 6C ··· 6H 6I 6J 8A 8B 8C 8D 12A 12B 12C ··· 12N 12O 12P 12Q 12R 24A ··· 24H order 1 2 2 3 3 3 ··· 3 4 4 4 6 6 6 ··· 6 6 6 8 8 8 8 12 12 12 ··· 12 12 12 12 12 24 ··· 24 size 1 1 18 1 1 4 ··· 4 2 9 9 1 1 4 ··· 4 18 18 18 18 18 18 2 2 4 ··· 4 9 9 9 9 18 ··· 18

54 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 4 4 4 4 4 4 type + + + + + image C1 C2 C2 C3 C4 C4 C6 C6 C12 C12 M4(2) C3×M4(2) C32⋊C4 C2×C32⋊C4 C3×C32⋊C4 C32⋊M4(2) C6×C32⋊C4 C3×C32⋊M4(2) kernel C3×C32⋊M4(2) C3×C32⋊2C8 C12×C3⋊S3 C32⋊M4(2) C32×C12 C6×C3⋊S3 C32⋊2C8 C4×C3⋊S3 C3×C12 C2×C3⋊S3 C33 C32 C12 C6 C4 C3 C2 C1 # reps 1 2 1 2 2 2 4 2 4 4 2 4 2 2 4 4 4 8

Matrix representation of C3×C32⋊M4(2) in GL4(𝔽73) generated by

 64 0 0 0 0 64 0 0 0 0 64 0 0 0 0 64
,
 1 0 0 0 0 1 0 0 48 13 8 0 18 20 0 64
,
 64 0 0 0 0 8 0 0 54 0 8 0 0 34 0 64
,
 54 60 0 43 18 14 30 0 59 11 59 18 31 46 60 19
,
 0 27 0 0 46 0 0 0 32 67 0 46 6 32 27 0
G:=sub<GL(4,GF(73))| [64,0,0,0,0,64,0,0,0,0,64,0,0,0,0,64],[1,0,48,18,0,1,13,20,0,0,8,0,0,0,0,64],[64,0,54,0,0,8,0,34,0,0,8,0,0,0,0,64],[54,18,59,31,60,14,11,46,0,30,59,60,43,0,18,19],[0,46,32,6,27,0,67,32,0,0,0,27,0,0,46,0] >;

C3×C32⋊M4(2) in GAP, Magma, Sage, TeX

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

G:=Group("C3xC3^2:M4(2)");
// GroupNames label

G:=SmallGroup(432,629);
// by ID

G=gap.SmallGroup(432,629);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,-3,3,84,365,176,80,14117,362,18822,1203]);
// Polycyclic

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

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