Dic3xHe3 - GroupNames

G = Dic₃xHe₃ order 324 = 2²·3⁴

Direct product of Dic₃ and He₃

direct product, metabelian, supersoluble, monomial

Aliases: Dic₃xHe₃, C₃³:₃C₁₂, C₃:(C₄xHe₃), C₆.(C₂xHe₃), C₂.(S₃xHe₃), (C₃xHe₃):₄C₄, (C₆xHe₃).₂C₂, C₆.₉(S₃xC₃²), (C₃²xC₆).₅C₆, (C₃²xDic₃):C₃, (C₂xHe₃).₁₄S₃, C₃².₇(C₃xC₁₂), C₃²:₅(C₃xDic₃), C₃.₅(C₃²xDic₃), (C₃xDic₃).₂C₃², (C₃xC₆).₁₆(C₃xC₆), (C₃xC₆).₂₂(C₃xS₃), SmallGroup(324,93)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₃² — Dic₃xHe₃

Chief series

C₁ — C₃ — C₃² — C₃xC₆ — C₃²xC₆ — C₆xHe₃ — Dic₃xHe₃

Lower central

C₃ — C₃² — Dic₃xHe₃

Upper central

C₁ — C₆ — C₂xHe₃

Generators and relations for Dic₃xHe₃
G = < a,b,c,d,e | a⁶=c³=d³=e³=1, b²=a³, bab^-1=a^-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ece^-1=cd^-1, de=ed >

Subgroups: 284 in 100 conjugacy classes, 35 normal (15 characteristic)
C₁, C₂, C₃, C₃, C₄, C₆, C₆, C₃², C₃², C₃², Dic₃, C₁₂, C₃xC₆, C₃xC₆, C₃xC₆, He₃, He₃, C₃³, C₃xDic₃, C₃xDic₃, C₃xC₁₂, C₂xHe₃, C₂xHe₃, C₃²xC₆, C₃xHe₃, C₄xHe₃, C₃²xDic₃, C₆xHe₃, Dic₃xHe₃
Quotients: C₁, C₂, C₃, C₄, S₃, C₆, C₃², Dic₃, C₁₂, C₃xS₃, C₃xC₆, He₃, C₃xDic₃, C₃xC₁₂, C₂xHe₃, S₃xC₃², C₄xHe₃, C₃²xDic₃, S₃xHe₃, Dic₃xHe₃

Smallest permutation representation of Dic₃xHe₃
►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 22 4 19)(2 21 5 24)(3 20 6 23)(7 26 10 29)(8 25 11 28)(9 30 12 27)(13 32 16 35)(14 31 17 34)(15 36 18 33)

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

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

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

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

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

66 conjugacy classes

class	1	2	3A	3B	3C	3D	3E	3F	···	3M	3N	···	3U	4A	4B	6A	6B	6C	6D	6E	6F	···	6M	6N	···	6U	12A	12B	12C	12D	12E	···	12T
order	1	2	3	3	3	3	3	3	···	3	3	···	3	4	4	6	6	6	6	6	6	···	6	6	···	6	12	12	12	12	12	···	12
size	1	1	1	1	2	2	2	3	···	3	6	···	6	3	3	1	1	2	2	2	3	···	3	6	···	6	3	3	3	3	9	···	9

66 irreducible representations

dim	1	1	1	1	1	1	2	2	2	2	3	3	3	6	6
type	+	+					+	-
image	C₁	C₂	C₃	C₄	C₆	C₁₂	S₃	Dic₃	C₃xS₃	C₃xDic₃	He₃	C₂xHe₃	C₄xHe₃	S₃xHe₃	Dic₃xHe₃
kernel	Dic₃xHe₃	C₆xHe₃	C₃²xDic₃	C₃xHe₃	C₃²xC₆	C₃³	C₂xHe₃	He₃	C₃xC₆	C₃²	Dic₃	C₆	C₃	C₂	C₁
# reps	1	1	8	2	8	16	1	1	8	8	2	2	4	2	2

Matrix representation of Dic₃xHe₃ ►in GL₅(F₁₃)

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

5	0	0	0	0
8	8	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	9	0	0
0	0	0	9	0
0	0	0	0	9

9	0	0	0	0
0	9	0	0	0
0	0	0	0	9
0	0	1	0	0
0	0	0	3	0

G:=sub<GL(5,GF(13))| [1,12,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[5,8,0,0,0,0,8,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[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],[1,0,0,0,0,0,1,0,0,0,0,0,9,0,0,0,0,0,9,0,0,0,0,0,9],[9,0,0,0,0,0,9,0,0,0,0,0,0,1,0,0,0,0,0,3,0,0,9,0,0] >;

Dic₃xHe₃ in GAP, Magma, Sage, TeX

{\rm Dic}_3\times {\rm He}_3

% in TeX

G:=Group("Dic3xHe3");

// GroupNames label

G:=SmallGroup(324,93);

// by ID

G=gap.SmallGroup(324,93);

# by ID

G:=PCGroup([6,-2,-3,-3,-2,-3,-3,108,386,7781]);

// Polycyclic

G:=Group<a,b,c,d,e|a^6=c^3=d^3=e^3=1,b^2=a^3,b*a*b^-1=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

G = Dic3xHe3 order 324 = 22·34

Direct product of Dic3 and He3

G = Dic₃xHe₃ order 324 = 2²·3⁴

Direct product of Dic₃ and He₃