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## G = D4×3- 1+2order 216 = 23·33

### Direct product of D4 and 3- 1+2

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C6 — D4×3- 1+2
 Chief series C1 — C3 — C6 — C3×C6 — C2×3- 1+2 — C22×3- 1+2 — D4×3- 1+2
 Lower central C1 — C6 — D4×3- 1+2
 Upper central C1 — C6 — D4×3- 1+2

Generators and relations for D4×3- 1+2
G = < a,b,c,d | a4=b2=c9=d3=1, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c4 >

Subgroups: 100 in 64 conjugacy classes, 42 normal (16 characteristic)
C1, C2, C2, C3, C3, C4, C22, C6, C6, D4, C9, C32, C12, C12, C2×C6, C2×C6, C18, C18, C3×C6, C3×C6, C3×D4, C3×D4, 3- 1+2, C36, C2×C18, C3×C12, C62, C2×3- 1+2, C2×3- 1+2, D4×C9, D4×C32, C4×3- 1+2, C22×3- 1+2, D4×3- 1+2
Quotients: C1, C2, C3, C22, C6, D4, C32, C2×C6, C3×C6, C3×D4, 3- 1+2, C62, C2×3- 1+2, D4×C32, C22×3- 1+2, D4×3- 1+2

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

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

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

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

D4×3- 1+2 is a maximal subgroup of   Dic18⋊C6  D36⋊C6  Dic182C6

55 conjugacy classes

 class 1 2A 2B 2C 3A 3B 3C 3D 4 6A 6B 6C 6D 6E 6F 6G 6H 6I 6J 6K 6L 9A ··· 9F 12A 12B 12C 12D 18A ··· 18F 18G ··· 18R 36A ··· 36F order 1 2 2 2 3 3 3 3 4 6 6 6 6 6 6 6 6 6 6 6 6 9 ··· 9 12 12 12 12 18 ··· 18 18 ··· 18 36 ··· 36 size 1 1 2 2 1 1 3 3 2 1 1 2 2 2 2 3 3 6 6 6 6 3 ··· 3 2 2 6 6 3 ··· 3 6 ··· 6 6 ··· 6

55 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 2 2 3 3 3 6 type + + + + image C1 C2 C2 C3 C3 C6 C6 C6 C6 D4 C3×D4 C3×D4 3- 1+2 C2×3- 1+2 C2×3- 1+2 D4×3- 1+2 kernel D4×3- 1+2 C4×3- 1+2 C22×3- 1+2 D4×C9 D4×C32 C36 C2×C18 C3×C12 C62 3- 1+2 C9 C32 D4 C4 C22 C1 # reps 1 1 2 6 2 6 12 2 4 1 6 2 2 2 4 2

Matrix representation of D4×3- 1+2 in GL5(𝔽37)

 36 2 0 0 0 36 1 0 0 0 0 0 36 0 0 0 0 0 36 0 0 0 0 0 36
,
 1 0 0 0 0 1 36 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1
,
 10 0 0 0 0 0 10 0 0 0 0 0 0 1 0 0 0 0 0 26 0 0 1 0 0
,
 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 26 0 0 0 0 0 10

G:=sub<GL(5,GF(37))| [36,36,0,0,0,2,1,0,0,0,0,0,36,0,0,0,0,0,36,0,0,0,0,0,36],[1,1,0,0,0,0,36,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[10,0,0,0,0,0,10,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,26,0],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,26,0,0,0,0,0,10] >;

D4×3- 1+2 in GAP, Magma, Sage, TeX

D_4\times 3_-^{1+2}
% in TeX

G:=Group("D4xES-(3,1)");
// GroupNames label

G:=SmallGroup(216,78);
// by ID

G=gap.SmallGroup(216,78);
# by ID

G:=PCGroup([6,-2,-2,-3,-3,-2,-3,457,338,519]);
// Polycyclic

G:=Group<a,b,c,d|a^4=b^2=c^9=d^3=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^4>;
// generators/relations

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