C3xDic3^2 - GroupNames

G = C₃×Dic₃² order 432 = 2⁴·3³

Direct product of C₃, Dic₃ and Dic₃

direct product, metabelian, supersoluble, monomial, A-group

Aliases: C₃×Dic₃², C₃³⋊₄C₄², C₆².₁₀₂D₆, C₆.₃₀(S₃×C₁₂), C₃⋊Dic₃⋊₃C₁₂, C₃²⋊₃(C₄×C₁₂), C₃⋊₁(Dic₃×C₁₂), C₆.₃(C₆×Dic₃), (C₃×Dic₃)⋊₂C₁₂, C₆².₁₇(C₂×C₆), (C₆×Dic₃).₉C₆, C₆.₃₄(S₃×Dic₃), C₃²⋊₈(C₄×Dic₃), (C₆×Dic₃).₂₀S₃, (C₃²×Dic₃)⋊₅C₄, (C₃×C₆²).₁C₂², C₆.₂₅(C₆.D₆), (C₂×C₆).₆₆S₃², C₂².₃(C₃×S₃²), C₂.₂(C₃×S₃×Dic₃), (C₃×C₆).₆₀(C₄×S₃), (C₂×C₆).₂₁(S₃×C₆), (C₃×C₃⋊Dic₃)⋊₁C₄, (C₃×C₆).₂₁(C₂×C₁₂), (C₂×C₃⋊Dic₃).₅C₆, (C₆×C₃⋊Dic₃).₁C₂, (Dic₃×C₃×C₆).₁₀C₂, C₂.₂(C₃×C₆.D₆), (C₃²×C₆).₂₆(C₂×C₄), (C₂×Dic₃).₄(C₃×S₃), (C₃×C₆).₄₇(C₂×Dic₃), SmallGroup(432,425)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₃² — C₃×Dic₃²

Chief series

C₁ — C₃ — C₃² — C₃×C₆ — C₆² — C₃×C₆² — Dic₃×C₃×C₆ — C₃×Dic₃²

Lower central

C₃² — C₃×Dic₃²

Upper central

C₁ — C₂×C₆

Generators and relations for C₃×Dic₃²
G = < a,b,c,d,e | a³=b⁶=d⁶=1, c²=b³, e²=d³, ab=ba, ac=ca, ad=da, ae=ea, cbc^-1=b^-1, bd=db, be=eb, cd=dc, ce=ec, ede^-1=d^-1 >

Subgroups: 448 in 178 conjugacy classes, 72 normal (16 characteristic)
C₁, C₂, C₂, C₃, C₃, C₃, C₄, C₂², C₆, C₆, C₆, C₂×C₄, C₃², C₃², C₃², Dic₃, Dic₃, C₁₂, C₂×C₆, C₂×C₆, C₂×C₆, C₄², C₃×C₆, C₃×C₆, C₃×C₆, C₂×Dic₃, C₂×Dic₃, C₂×C₁₂, C₃³, C₃×Dic₃, C₃×Dic₃, C₃⋊Dic₃, C₃×C₁₂, C₆², C₆², C₆², C₄×Dic₃, C₄×C₁₂, C₃²×C₆, C₃²×C₆, C₆×Dic₃, C₆×Dic₃, C₂×C₃⋊Dic₃, C₆×C₁₂, C₃²×Dic₃, C₃×C₃⋊Dic₃, C₃×C₆², Dic₃², Dic₃×C₁₂, Dic₃×C₃×C₆, C₆×C₃⋊Dic₃, C₃×Dic₃²
Quotients: C₁, C₂, C₃, C₄, C₂², S₃, C₆, C₂×C₄, Dic₃, C₁₂, D₆, C₂×C₆, C₄², C₃×S₃, C₄×S₃, C₂×Dic₃, C₂×C₁₂, C₃×Dic₃, S₃², S₃×C₆, C₄×Dic₃, C₄×C₁₂, S₃×Dic₃, C₆.D₆, S₃×C₁₂, C₆×Dic₃, C₃×S₃², Dic₃², Dic₃×C₁₂, C₃×S₃×Dic₃, C₃×C₆.D₆, C₃×Dic₃²

Smallest permutation representation of C₃×Dic₃²
►On 48 points

Generators in S₄₈

(1 5 3)(2 6 4)(7 9 11)(8 10 12)(13 17 15)(14 18 16)(19 21 23)(20 22 24)(25 27 29)(26 28 30)(31 35 33)(32 36 34)(37 41 39)(38 42 40)(43 45 47)(44 46 48)

(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)(37 38 39 40 41 42)(43 44 45 46 47 48)

(1 30 4 27)(2 29 5 26)(3 28 6 25)(7 36 10 33)(8 35 11 32)(9 34 12 31)(13 20 16 23)(14 19 17 22)(15 24 18 21)(37 44 40 47)(38 43 41 46)(39 48 42 45)

(1 14 5 18 3 16)(2 15 6 13 4 17)(7 44 11 48 9 46)(8 45 12 43 10 47)(19 26 21 28 23 30)(20 27 22 29 24 25)(31 41 33 37 35 39)(32 42 34 38 36 40)

(1 42 18 36)(2 37 13 31)(3 38 14 32)(4 39 15 33)(5 40 16 34)(6 41 17 35)(7 27 48 24)(8 28 43 19)(9 29 44 20)(10 30 45 21)(11 25 46 22)(12 26 47 23)

G:=sub<Sym(48)| (1,5,3)(2,6,4)(7,9,11)(8,10,12)(13,17,15)(14,18,16)(19,21,23)(20,22,24)(25,27,29)(26,28,30)(31,35,33)(32,36,34)(37,41,39)(38,42,40)(43,45,47)(44,46,48), (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)(37,38,39,40,41,42)(43,44,45,46,47,48), (1,30,4,27)(2,29,5,26)(3,28,6,25)(7,36,10,33)(8,35,11,32)(9,34,12,31)(13,20,16,23)(14,19,17,22)(15,24,18,21)(37,44,40,47)(38,43,41,46)(39,48,42,45), (1,14,5,18,3,16)(2,15,6,13,4,17)(7,44,11,48,9,46)(8,45,12,43,10,47)(19,26,21,28,23,30)(20,27,22,29,24,25)(31,41,33,37,35,39)(32,42,34,38,36,40), (1,42,18,36)(2,37,13,31)(3,38,14,32)(4,39,15,33)(5,40,16,34)(6,41,17,35)(7,27,48,24)(8,28,43,19)(9,29,44,20)(10,30,45,21)(11,25,46,22)(12,26,47,23)>;

G:=Group( (1,5,3)(2,6,4)(7,9,11)(8,10,12)(13,17,15)(14,18,16)(19,21,23)(20,22,24)(25,27,29)(26,28,30)(31,35,33)(32,36,34)(37,41,39)(38,42,40)(43,45,47)(44,46,48), (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)(37,38,39,40,41,42)(43,44,45,46,47,48), (1,30,4,27)(2,29,5,26)(3,28,6,25)(7,36,10,33)(8,35,11,32)(9,34,12,31)(13,20,16,23)(14,19,17,22)(15,24,18,21)(37,44,40,47)(38,43,41,46)(39,48,42,45), (1,14,5,18,3,16)(2,15,6,13,4,17)(7,44,11,48,9,46)(8,45,12,43,10,47)(19,26,21,28,23,30)(20,27,22,29,24,25)(31,41,33,37,35,39)(32,42,34,38,36,40), (1,42,18,36)(2,37,13,31)(3,38,14,32)(4,39,15,33)(5,40,16,34)(6,41,17,35)(7,27,48,24)(8,28,43,19)(9,29,44,20)(10,30,45,21)(11,25,46,22)(12,26,47,23) );

G=PermutationGroup([[(1,5,3),(2,6,4),(7,9,11),(8,10,12),(13,17,15),(14,18,16),(19,21,23),(20,22,24),(25,27,29),(26,28,30),(31,35,33),(32,36,34),(37,41,39),(38,42,40),(43,45,47),(44,46,48)], [(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),(37,38,39,40,41,42),(43,44,45,46,47,48)], [(1,30,4,27),(2,29,5,26),(3,28,6,25),(7,36,10,33),(8,35,11,32),(9,34,12,31),(13,20,16,23),(14,19,17,22),(15,24,18,21),(37,44,40,47),(38,43,41,46),(39,48,42,45)], [(1,14,5,18,3,16),(2,15,6,13,4,17),(7,44,11,48,9,46),(8,45,12,43,10,47),(19,26,21,28,23,30),(20,27,22,29,24,25),(31,41,33,37,35,39),(32,42,34,38,36,40)], [(1,42,18,36),(2,37,13,31),(3,38,14,32),(4,39,15,33),(5,40,16,34),(6,41,17,35),(7,27,48,24),(8,28,43,19),(9,29,44,20),(10,30,45,21),(11,25,46,22),(12,26,47,23)]])

108 conjugacy classes

class	1	2A	2B	2C	3A	3B	3C	···	3H	3I	3J	3K	4A	···	4H	4I	4J	4K	4L	6A	···	6F	6G	···	6X	6Y	···	6AG	12A	···	12P	12Q	···	12AN	12AO	···	12AV
order	1	2	2	2	3	3	3	···	3	3	3	3	4	···	4	4	4	4	4	6	···	6	6	···	6	6	···	6	12	···	12	12	···	12	12	···	12
size	1	1	1	1	1	1	2	···	2	4	4	4	3	···	3	9	9	9	9	1	···	1	2	···	2	4	···	4	3	···	3	6	···	6	9	···	9

108 irreducible representations

dim	1	1	1	1	1	1	1	1	1	1	2	2	2	2	2	2	2	2	4	4	4	4	4	4
type	+	+	+								+	-	+						+	-	+
image	C₁	C₂	C₂	C₃	C₄	C₄	C₆	C₆	C₁₂	C₁₂	S₃	Dic₃	D₆	C₃×S₃	C₄×S₃	C₃×Dic₃	S₃×C₆	S₃×C₁₂	S₃²	S₃×Dic₃	C₆.D₆	C₃×S₃²	C₃×S₃×Dic₃	C₃×C₆.D₆
kernel	C₃×Dic₃²	Dic₃×C₃×C₆	C₆×C₃⋊Dic₃	Dic₃²	C₃²×Dic₃	C₃×C₃⋊Dic₃	C₆×Dic₃	C₂×C₃⋊Dic₃	C₃×Dic₃	C₃⋊Dic₃	C₆×Dic₃	C₃×Dic₃	C₆²	C₂×Dic₃	C₃×C₆	Dic₃	C₂×C₆	C₆	C₂×C₆	C₆	C₆	C₂²	C₂	C₂
# reps	1	2	1	2	8	4	4	2	16	8	2	4	2	4	8	8	4	16	1	2	1	2	4	2

Matrix representation of C₃×Dic₃² ►in GL₆(𝔽₁₃)

1	0	0	0	0	0
0	1	0	0	0	0
0	0	9	0	0	0
0	0	0	9	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
0	0	1	0	0	0
0	0	0	1	0	0
0	0	0	0	0	1
0	0	0	0	12	1

12	0	0	0	0	0
0	12	0	0	0	0
0	0	12	0	0	0
0	0	0	12	0	0
0	0	0	0	0	5
0	0	0	0	5	0

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

5	0	0	0	0	0
8	8	0	0	0	0
0	0	1	0	0	0
0	0	12	12	0	0
0	0	0	0	12	0
0	0	0	0	0	12

G:=sub<GL(6,GF(13))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,9,0,0,0,0,0,0,9,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,12,0,0,0,0,1,1],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,5,0,0,0,0,5,0],[1,12,0,0,0,0,1,0,0,0,0,0,0,0,12,1,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[5,8,0,0,0,0,0,8,0,0,0,0,0,0,1,12,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12] >;

C₃×Dic₃² in GAP, Magma, Sage, TeX

C_3\times {\rm Dic}_3^2

% in TeX

G:=Group("C3xDic3^2");

// GroupNames label

G:=SmallGroup(432,425);

// by ID

G=gap.SmallGroup(432,425);

# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,-3,-3,168,92,2028,14118]);

// Polycyclic

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

// generators/relations

G = C3×Dic32 order 432 = 24·33

Direct product of C3, Dic3 and Dic3

G = C₃×Dic₃² order 432 = 2⁴·3³

Direct product of C₃, Dic₃ and Dic₃