D4xES-(3,1) - GroupNames

G = D₄×3_-¹⁺² order 216 = 2³·3³

Direct product of D₄ and 3_-¹⁺²

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

Aliases: D₄×3_-¹⁺², C₃₆⋊₃C₆, C₆².₁C₆, C₆.₁₅C₆², (D₄×C₉)⋊C₃, C₉⋊₃(C₃×D₄), (C₂×C₁₈)⋊₅C₆, C₃².(C₃×D₄), (C₃×C₁₂).₃C₆, C₁₈.₇(C₂×C₆), C₁₂.₅(C₃×C₆), (D₄×C₃²).C₃, C₃.₃(D₄×C₃²), C₄⋊(C₂×3_-¹⁺²), (C₃×D₄).₃C₃², (C₄×3_-¹⁺²)⋊₃C₂, C₂²⋊₃(C₂×3_-¹⁺²), (C₂²×3_-¹⁺²)⋊₃C₂, C₂.₂(C₂²×3_-¹⁺²), (C₂×3_-¹⁺²).₇C₂², (C₂×C₆).₈(C₃×C₆), (C₃×C₆).₁₄(C₂×C₆), SmallGroup(216,78)

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 Dic₁₈⋊₂C₆

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

G = D4×3- 1+2 order 216 = 23·33

Direct product of D4 and 3- 1+2

G = D₄×3_-¹⁺² order 216 = 2³·3³

Direct product of D₄ and 3_-¹⁺²