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## G = D4×He3order 216 = 23·33

### Direct product of D4 and He3

direct product, metabelian, nilpotent (class 2), monomial

Series: Derived Chief Lower central Upper central

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

Generators and relations for D4×He3
G = < a,b,c,d,e | a4=b2=c3=d3=e3=1, bab=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ece-1=cd-1, de=ed >

Subgroups: 190 in 88 conjugacy classes, 42 normal (12 characteristic)
C1, C2, C2, C3, C3, C4, C22, C6, C6, D4, C32, C12, C12, C2×C6, C2×C6, C3×C6, C3×C6, C3×D4, C3×D4, He3, C3×C12, C62, C2×He3, C2×He3, D4×C32, C4×He3, C22×He3, D4×He3
Quotients: C1, C2, C3, C22, C6, D4, C32, C2×C6, C3×C6, C3×D4, He3, C62, C2×He3, D4×C32, C22×He3, D4×He3

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

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

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

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

D4×He3 is a maximal subgroup of   He38SD16  He36D8  He37D8  He39SD16  C62.13D6  C62.16D6

55 conjugacy classes

 class 1 2A 2B 2C 3A 3B 3C ··· 3J 4 6A 6B 6C 6D 6E 6F 6G ··· 6N 6O ··· 6AD 12A 12B 12C ··· 12J order 1 2 2 2 3 3 3 ··· 3 4 6 6 6 6 6 6 6 ··· 6 6 ··· 6 12 12 12 ··· 12 size 1 1 2 2 1 1 3 ··· 3 2 1 1 2 2 2 2 3 ··· 3 6 ··· 6 2 2 6 ··· 6

55 irreducible representations

 dim 1 1 1 1 1 1 2 2 3 3 3 6 type + + + + image C1 C2 C2 C3 C6 C6 D4 C3×D4 He3 C2×He3 C2×He3 D4×He3 kernel D4×He3 C4×He3 C22×He3 D4×C32 C3×C12 C62 He3 C32 D4 C4 C22 C1 # reps 1 1 2 8 8 16 1 8 2 2 4 2

Matrix representation of D4×He3 in GL5(𝔽13)

 0 1 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 12 0 0 0 0 0 12
,
 0 1 0 0 0 1 0 0 0 0 0 0 12 0 0 0 0 0 12 0 0 0 0 0 12
,
 9 0 0 0 0 0 9 0 0 0 0 0 0 1 0 0 0 4 10 2 0 0 0 0 3
,
 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
,
 9 0 0 0 0 0 9 0 0 0 0 0 0 9 0 0 0 12 4 6 0 0 12 1 9

G:=sub<GL(5,GF(13))| [0,12,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,12],[0,1,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,12],[9,0,0,0,0,0,9,0,0,0,0,0,0,4,0,0,0,1,10,0,0,0,0,2,3],[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],[9,0,0,0,0,0,9,0,0,0,0,0,0,12,12,0,0,9,4,1,0,0,0,6,9] >;

D4×He3 in GAP, Magma, Sage, TeX

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

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

G:=SmallGroup(216,77);
// by ID

G=gap.SmallGroup(216,77);
# by ID

G:=PCGroup([6,-2,-2,-3,-3,-2,-3,457,519]);
// Polycyclic

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

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