He3:4D4 - GroupNames

G = He₃:₄D₄ order 216 = 2³·3³

1^st semidirect product of He₃ and D₄ acting via D₄/C₄=C₂

Aliases: He₃:₄D₄, C₃²:₃D₁₂, C₁₂:S₃:C₃, (C₃xC₁₂):₁C₆, (C₃xC₁₂):₁S₃, C₄:(C₃²:C₆), (C₃xC₆).₈D₆, C₆.₁₁(S₃xC₆), C₁₂.₄(C₃xS₃), (C₄xHe₃):₁C₂, C₃.₂(C₃xD₁₂), C₃²:₂(C₃xD₄), (C₂xHe₃).₈C₂², (C₂xC₃:S₃):₁C₆, (C₃xC₆).₃(C₂xC₆), (C₂xC₃²:C₆):₃C₂, C₂.₄(C₂xC₃²:C₆), SmallGroup(216,51)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₃xC₆ — He₃:₄D₄

Chief series

C₁ — C₃ — C₃² — C₃xC₆ — C₂xHe₃ — C₂xC₃²:C₆ — He₃:₄D₄

Lower central

C₃² — C₃xC₆ — He₃:₄D₄

Upper central

C₁ — C₂ — C₄

Generators and relations for He₃:₄D₄
G = < a,b,c,d,e | a³=b³=c³=d⁴=e²=1, ab=ba, cac^-1=ab^-1, ad=da, eae=a^-1, bc=cb, bd=db, ebe=b^-1, cd=dc, ce=ec, ede=d^-1 >

Subgroups: 302 in 62 conjugacy classes, 21 normal (17 characteristic)
C₁, C₂, C₂, C₃, C₃, C₄, C₂², S₃, C₆, C₆, D₄, C₃², C₃², C₁₂, C₁₂, D₆, C₂xC₆, C₃xS₃, C₃:S₃, C₃xC₆, C₃xC₆, D₁₂, C₃xD₄, He₃, C₃xC₁₂, C₃xC₁₂, S₃xC₆, C₂xC₃:S₃, C₃²:C₆, C₂xHe₃, C₃xD₁₂, C₁₂:S₃, C₄xHe₃, C₂xC₃²:C₆, He₃:₄D₄
Quotients: C₁, C₂, C₃, C₂², S₃, C₆, D₄, D₆, C₂xC₆, C₃xS₃, D₁₂, C₃xD₄, S₃xC₆, C₃²:C₆, C₃xD₁₂, C₂xC₃²:C₆, He₃:₄D₄

Smallest permutation representation of He₃:₄D₄
►On 36 points

Generators in S₃₆

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

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

(9 16 27)(10 13 28)(11 14 25)(12 15 26)(17 36 23)(18 33 24)(19 34 21)(20 35 22)

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

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

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

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

He₃:₄D₄ is a maximal subgroup of
He₃:₃SD₁₆ He₃:₂D₈ He₃:₃D₈ He₃:₅SD₁₆ He₃:₆SD₁₆ He₃:₄D₈ He₃:₆D₈ He₃:₁₀SD₁₆ C₁₂:S₃:S₃ C₁₂.₈₄S₃² C₃:S₃:D₁₂ C₁₂.₈₆S₃² C₆².₃₆D₆ D₄xC₃²:C₆ (Q₈xHe₃):C₂
He₃:₄D₄ is a maximal quotient of
He₃:₄Q₁₆ He₃:₆SD₁₆ He₃:₄D₈ C₆².₂₀D₆ C₆².₂₁D₆

31 conjugacy classes

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

31 irreducible representations

dim	1	1	1	1	1	1	2	2	2	2	2	2	2	2	6	6	6
type	+	+	+				+	+	+		+				+	+	+
image	C₁	C₂	C₂	C₃	C₆	C₆	S₃	D₄	D₆	C₃xS₃	D₁₂	C₃xD₄	S₃xC₆	C₃xD₁₂	C₃²:C₆	C₂xC₃²:C₆	He₃:₄D₄
kernel	He₃:₄D₄	C₄xHe₃	C₂xC₃²:C₆	C₁₂:S₃	C₃xC₁₂	C₂xC₃:S₃	C₃xC₁₂	He₃	C₃xC₆	C₁₂	C₃²	C₃²	C₆	C₃	C₄	C₂	C₁
# reps	1	1	2	2	2	4	1	1	1	2	2	2	2	4	1	1	2

Matrix representation of He₃:₄D₄ ►in GL₆(F₁₃)

0	0	1	0	0	0
0	0	0	1	0	0
0	0	0	0	1	0
0	0	0	0	0	1
1	0	0	0	0	0
0	1	0	0	0	0

12	1	0	0	0	0
12	0	0	0	0	0
0	0	12	1	0	0
0	0	12	0	0	0
0	0	0	0	12	1
0	0	0	0	12	0

1	0	0	0	0	0
0	1	0	0	0	0
0	0	0	12	0	0
0	0	1	12	0	0
0	0	0	0	12	1
0	0	0	0	12	0

3	7	0	0	0	0
6	10	0	0	0	0
0	0	3	7	0	0
0	0	6	10	0	0
0	0	0	0	3	7
0	0	0	0	6	10

0	12	0	0	0	0
12	0	0	0	0	0
0	0	0	0	0	12
0	0	0	0	12	0
0	0	0	12	0	0
0	0	12	0	0	0

G:=sub<GL(6,GF(13))| [0,0,0,0,1,0,0,0,0,0,0,1,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0],[12,12,0,0,0,0,1,0,0,0,0,0,0,0,12,12,0,0,0,0,1,0,0,0,0,0,0,0,12,12,0,0,0,0,1,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,12,12,0,0,0,0,0,0,12,12,0,0,0,0,1,0],[3,6,0,0,0,0,7,10,0,0,0,0,0,0,3,6,0,0,0,0,7,10,0,0,0,0,0,0,3,6,0,0,0,0,7,10],[0,12,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,12,0,0,0,0,12,0,0,0,0,12,0,0,0] >;

He₃:₄D₄ in GAP, Magma, Sage, TeX

{\rm He}_3\rtimes_4D_4

% in TeX

G:=Group("He3:4D4");

// GroupNames label

G:=SmallGroup(216,51);

// by ID

G=gap.SmallGroup(216,51);

# by ID

G:=PCGroup([6,-2,-2,-3,-2,-3,-3,169,79,1444,736,5189]);

// Polycyclic

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

// generators/relations

G = He3:4D4 order 216 = 23·33

1st semidirect product of He3 and D4 acting via D4/C4=C2

G = He₃:₄D₄ order 216 = 2³·3³

1^st semidirect product of He₃ and D₄ acting via D₄/C₄=C₂