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## G = D4×He3⋊C2order 432 = 24·33

### Direct product of D4 and He3⋊C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3 — C2×He3 — D4×He3⋊C2
 Chief series C1 — C3 — C32 — He3 — C2×He3 — C2×He3⋊C2 — C22×He3⋊C2 — D4×He3⋊C2
 Lower central He3 — C2×He3 — D4×He3⋊C2
 Upper central C1 — C6 — C3×D4

Generators and relations for D4×He3⋊C2
G = < a,b,c,d,e,f | a4=b2=c3=d3=e3=f2=1, bab=a-1, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, cd=dc, ece-1=cd-1, fcf=c-1, de=ed, df=fd, fef=e-1 >

Subgroups: 1265 in 297 conjugacy classes, 55 normal (17 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, C22, S3, C6, C6, C2×C4, D4, D4, C23, C32, Dic3, C12, C12, D6, C2×C6, C2×C6, C2×D4, C3×S3, C3×C6, C3×C6, C4×S3, D12, C3⋊D4, C2×C12, C3×D4, C3×D4, C22×S3, C22×C6, He3, C3×Dic3, C3×C12, S3×C6, C62, S3×D4, C6×D4, He3⋊C2, He3⋊C2, C2×He3, C2×He3, S3×C12, C3×D12, C3×C3⋊D4, D4×C32, S3×C2×C6, He33C4, C4×He3, C2×He3⋊C2, C2×He3⋊C2, C2×He3⋊C2, C22×He3, C3×S3×D4, C4×He3⋊C2, He35D4, He37D4, D4×He3, C22×He3⋊C2, D4×He3⋊C2
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C3⋊S3, C22×S3, C2×C3⋊S3, S3×D4, He3⋊C2, C22×C3⋊S3, C2×He3⋊C2, D4×C3⋊S3, C22×He3⋊C2, D4×He3⋊C2

Smallest permutation representation of D4×He3⋊C2
On 36 points
Generators in S36
(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)
(2 4)(6 8)(10 12)(13 15)(17 19)(22 24)(26 28)(30 32)(34 36)
(1 7 35)(2 8 36)(3 5 33)(4 6 34)(9 29 16)(10 30 13)(11 31 14)(12 32 15)(17 26 22)(18 27 23)(19 28 24)(20 25 21)
(1 21 31)(2 22 32)(3 23 29)(4 24 30)(5 18 16)(6 19 13)(7 20 14)(8 17 15)(9 33 27)(10 34 28)(11 35 25)(12 36 26)
(1 25 20)(2 26 17)(3 27 18)(4 28 19)(5 29 33)(6 30 34)(7 31 35)(8 32 36)(9 16 23)(10 13 24)(11 14 21)(12 15 22)
(1 3)(2 4)(5 35)(6 36)(7 33)(8 34)(9 14)(10 15)(11 16)(12 13)(17 28)(18 25)(19 26)(20 27)(21 23)(22 24)(29 31)(30 32)

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

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), (2,4)(6,8)(10,12)(13,15)(17,19)(22,24)(26,28)(30,32)(34,36), (1,7,35)(2,8,36)(3,5,33)(4,6,34)(9,29,16)(10,30,13)(11,31,14)(12,32,15)(17,26,22)(18,27,23)(19,28,24)(20,25,21), (1,21,31)(2,22,32)(3,23,29)(4,24,30)(5,18,16)(6,19,13)(7,20,14)(8,17,15)(9,33,27)(10,34,28)(11,35,25)(12,36,26), (1,25,20)(2,26,17)(3,27,18)(4,28,19)(5,29,33)(6,30,34)(7,31,35)(8,32,36)(9,16,23)(10,13,24)(11,14,21)(12,15,22), (1,3)(2,4)(5,35)(6,36)(7,33)(8,34)(9,14)(10,15)(11,16)(12,13)(17,28)(18,25)(19,26)(20,27)(21,23)(22,24)(29,31)(30,32) );

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)], [(2,4),(6,8),(10,12),(13,15),(17,19),(22,24),(26,28),(30,32),(34,36)], [(1,7,35),(2,8,36),(3,5,33),(4,6,34),(9,29,16),(10,30,13),(11,31,14),(12,32,15),(17,26,22),(18,27,23),(19,28,24),(20,25,21)], [(1,21,31),(2,22,32),(3,23,29),(4,24,30),(5,18,16),(6,19,13),(7,20,14),(8,17,15),(9,33,27),(10,34,28),(11,35,25),(12,36,26)], [(1,25,20),(2,26,17),(3,27,18),(4,28,19),(5,29,33),(6,30,34),(7,31,35),(8,32,36),(9,16,23),(10,13,24),(11,14,21),(12,15,22)], [(1,3),(2,4),(5,35),(6,36),(7,33),(8,34),(9,14),(10,15),(11,16),(12,13),(17,28),(18,25),(19,26),(20,27),(21,23),(22,24),(29,31),(30,32)]])

50 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3A 3B 3C 3D 3E 3F 4A 4B 6A 6B 6C 6D 6E 6F 6G 6H 6I 6J 6K 6L 6M 6N 6O ··· 6V 6W 6X 6Y 6Z 12A 12B 12C 12D 12E 12F 12G 12H order 1 2 2 2 2 2 2 2 3 3 3 3 3 3 4 4 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 ··· 6 6 6 6 6 12 12 12 12 12 12 12 12 size 1 1 2 2 9 9 18 18 1 1 6 6 6 6 2 18 1 1 2 2 2 2 6 6 6 6 9 9 9 9 12 ··· 12 18 18 18 18 2 2 12 12 12 12 18 18

50 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 3 3 3 4 6 type + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 S3 D4 D6 D6 He3⋊C2 C2×He3⋊C2 C2×He3⋊C2 S3×D4 D4×He3⋊C2 kernel D4×He3⋊C2 C4×He3⋊C2 He3⋊5D4 He3⋊7D4 D4×He3 C22×He3⋊C2 D4×C32 He3⋊C2 C3×C12 C62 D4 C4 C22 C32 C1 # reps 1 1 1 2 1 2 4 2 4 8 4 4 8 4 4

Matrix representation of D4×He3⋊C2 in GL5(𝔽13)

 1 10 0 0 0 5 12 0 0 0 0 0 12 0 0 0 0 0 12 0 0 0 0 0 12
,
 1 0 0 0 0 5 12 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1
,
 1 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 1 0 0
,
 1 0 0 0 0 0 1 0 0 0 0 0 3 0 0 0 0 0 3 0 0 0 0 0 3
,
 1 0 0 0 0 0 1 0 0 0 0 0 0 0 9 0 0 3 0 0 0 0 0 1 0
,
 12 0 0 0 0 0 12 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 1 0

G:=sub<GL(5,GF(13))| [1,5,0,0,0,10,12,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,12],[1,5,0,0,0,0,12,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,1,0],[1,0,0,0,0,0,1,0,0,0,0,0,3,0,0,0,0,0,3,0,0,0,0,0,3],[1,0,0,0,0,0,1,0,0,0,0,0,0,3,0,0,0,0,0,1,0,0,9,0,0],[12,0,0,0,0,0,12,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,1,0] >;

D4×He3⋊C2 in GAP, Magma, Sage, TeX

D_4\times {\rm He}_3\rtimes C_2
% in TeX

G:=Group("D4xHe3:C2");
// GroupNames label

G:=SmallGroup(432,390);
// by ID

G=gap.SmallGroup(432,390);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,135,1124,4037,537]);
// Polycyclic

G:=Group<a,b,c,d,e,f|a^4=b^2=c^3=d^3=e^3=f^2=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,b*c=c*b,b*d=d*b,b*e=e*b,b*f=f*b,c*d=d*c,e*c*e^-1=c*d^-1,f*c*f=c^-1,d*e=e*d,d*f=f*d,f*e*f=e^-1>;
// generators/relations

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