C3xDic3.D6 - GroupNames

G = C₃×Dic₃.D₆ order 432 = 2⁴·3³

Direct product of C₃ and Dic₃.D₆

direct product, metabelian, supersoluble, monomial

Aliases: C₃×Dic₃.D₆, C₁₂.₁₀₀S₃², C₃³⋊₈(C₂×Q₈), C₃²⋊₅(C₆×Q₈), C₁₂.₃₈(S₃×C₆), (C₃×Dic₆)⋊₅S₃, (C₃×Dic₆)⋊₆C₆, Dic₆⋊₄(C₃×S₃), C₃²⋊₂Q₈⋊₃C₆, C₃²⋊₁₂(S₃×Q₈), (C₃×C₁₂).₁₃₇D₆, Dic₃.₂(S₃×C₆), C₆.D₆.₁C₆, (C₃×Dic₃).₂₅D₆, (C₃²×Dic₆)⋊₈C₂, (C₃²×C₆).₂₃C₂³, (C₃²×C₁₂).₃₈C₂², (C₃²×Dic₃).₁₀C₂², C₃⋊₁(C₃×S₃×Q₈), C₂.₇(S₃²×C₆), C₆.₄(S₃×C₂×C₆), C₄.₁₂(C₃×S₃²), (C₃×C₃⋊S₃)⋊₄Q₈, C₃⋊S₃⋊₃(C₃×Q₈), C₆.₁₀₇(C₂×S₃²), (C₄×C₃⋊S₃).₄C₆, (C₁₂×C₃⋊S₃).₅C₂, (C₃×C₁₂).₅₃(C₂×C₆), (C₃×C₃²⋊₂Q₈)⋊₉C₂, (C₆×C₃⋊S₃).₄₅C₂², C₃⋊Dic₃.₁₇(C₂×C₆), (C₃×C₆).₁₄(C₂²×C₆), (C₃×Dic₃).₃(C₂×C₆), (C₃×C₆.D₆).₂C₂, (C₃×C₆).₁₂₈(C₂²×S₃), (C₃×C₃⋊Dic₃).₅₄C₂², (C₂×C₃⋊S₃).₁₇(C₂×C₆), SmallGroup(432,645)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₃×C₆ — C₃×Dic₃.D₆

Chief series

C₁ — C₃ — C₃² — C₃×C₆ — C₃²×C₆ — C₃²×Dic₃ — C₃×C₆.D₆ — C₃×Dic₃.D₆

Lower central

C₃² — C₃×C₆ — C₃×Dic₃.D₆

Upper central

C₁ — C₆ — C₁₂

Generators and relations for C₃×Dic₃.D₆
G = < a,b,c,d,e | a³=b⁶=1, c²=d⁶=e²=b³, ab=ba, ac=ca, ad=da, ae=ea, cbc^-1=b^-1, bd=db, be=eb, dcd^-1=ece^-1=b³c, ede^-1=b³d⁵ >

Subgroups: 608 in 198 conjugacy classes, 68 normal (20 characteristic)
C₁, C₂, C₂, C₃, C₃, C₃, C₄, C₄, C₂², S₃, C₆, C₆, C₆, C₂×C₄, Q₈, C₃², C₃², C₃², Dic₃, Dic₃, C₁₂, C₁₂, C₁₂, D₆, C₂×C₆, C₂×Q₈, C₃×S₃, C₃⋊S₃, C₃×C₆, C₃×C₆, C₃×C₆, Dic₆, Dic₆, C₄×S₃, C₂×C₁₂, C₃×Q₈, C₃³, C₃×Dic₃, C₃×Dic₃, C₃⋊Dic₃, C₃×C₁₂, C₃×C₁₂, C₃×C₁₂, S₃×C₆, C₂×C₃⋊S₃, S₃×Q₈, C₆×Q₈, C₃×C₃⋊S₃, C₃²×C₆, C₆.D₆, C₃²⋊₂Q₈, C₃×Dic₆, C₃×Dic₆, S₃×C₁₂, C₄×C₃⋊S₃, Q₈×C₃², C₃²×Dic₃, C₃×C₃⋊Dic₃, C₃²×C₁₂, C₆×C₃⋊S₃, Dic₃.D₆, C₃×S₃×Q₈, C₃×C₆.D₆, C₃×C₃²⋊₂Q₈, C₃²×Dic₆, C₁₂×C₃⋊S₃, C₃×Dic₃.D₆
Quotients: C₁, C₂, C₃, C₂², S₃, C₆, Q₈, C₂³, D₆, C₂×C₆, C₂×Q₈, C₃×S₃, C₃×Q₈, C₂²×S₃, C₂²×C₆, S₃², S₃×C₆, S₃×Q₈, C₆×Q₈, C₂×S₃², S₃×C₂×C₆, C₃×S₃², Dic₃.D₆, C₃×S₃×Q₈, S₃²×C₆, C₃×Dic₃.D₆

Smallest permutation representation of C₃×Dic₃.D₆
►On 48 points

Generators in S₄₈

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

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

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

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

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

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

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

72 conjugacy classes

class	1	2A	2B	2C	3A	3B	3C	···	3H	3I	3J	3K	4A	4B	4C	4D	4E	4F	6A	6B	6C	···	6H	6I	6J	6K	6L	6M	6N	6O	12A	12B	12C	···	12N	12O	···	12V	12W	···	12AH	12AI	12AJ
order	1	2	2	2	3	3	3	···	3	3	3	3	4	4	4	4	4	4	6	6	6	···	6	6	6	6	6	6	6	6	12	12	12	···	12	12	···	12	12	···	12	12	12
size	1	1	9	9	1	1	2	···	2	4	4	4	2	6	6	6	6	18	1	1	2	···	2	4	4	4	9	9	9	9	2	2	4	···	4	6	···	6	12	···	12	18	18

72 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	4	4
type	+	+	+	+	+						+	-	+	+					+	-	+
image	C₁	C₂	C₂	C₂	C₂	C₃	C₆	C₆	C₆	C₆	S₃	Q₈	D₆	D₆	C₃×S₃	C₃×Q₈	S₃×C₆	S₃×C₆	S₃²	S₃×Q₈	C₂×S₃²	C₃×S₃²	Dic₃.D₆	C₃×S₃×Q₈	S₃²×C₆	C₃×Dic₃.D₆
kernel	C₃×Dic₃.D₆	C₃×C₆.D₆	C₃×C₃²⋊₂Q₈	C₃²×Dic₆	C₁₂×C₃⋊S₃	Dic₃.D₆	C₆.D₆	C₃²⋊₂Q₈	C₃×Dic₆	C₄×C₃⋊S₃	C₃×Dic₆	C₃×C₃⋊S₃	C₃×Dic₃	C₃×C₁₂	Dic₆	C₃⋊S₃	Dic₃	C₁₂	C₁₂	C₃²	C₆	C₄	C₃	C₃	C₂	C₁
# reps	1	2	2	2	1	2	4	4	4	2	2	2	4	2	4	4	8	4	1	2	1	2	2	4	2	4

Matrix representation of C₃×Dic₃.D₆ ►in GL₆(𝔽₁₃)

3	0	0	0	0	0
0	3	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

12	0	0	0	0	0
0	12	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	12

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

8	0	0	0	0	0
0	5	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

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

G:=sub<GL(6,GF(13))| [3,0,0,0,0,0,0,3,0,0,0,0,0,0,9,0,0,0,0,0,0,9,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,12,12],[0,12,0,0,0,0,1,0,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[8,0,0,0,0,0,0,5,0,0,0,0,0,0,12,12,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,8,0,0,0,0,8,0,0,0,0,0,0,0,0,12,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;

C₃×Dic₃.D₆ in GAP, Magma, Sage, TeX

C_3\times {\rm Dic}_3.D_6

% in TeX

G:=Group("C3xDic3.D6");

// GroupNames label

G:=SmallGroup(432,645);

// by ID

G=gap.SmallGroup(432,645);

# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-3,-3,168,176,590,303,142,2028,14118]);

// Polycyclic

G:=Group<a,b,c,d,e|a^3=b^6=1,c^2=d^6=e^2=b^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,d*c*d^-1=e*c*e^-1=b^3*c,e*d*e^-1=b^3*d^5>;

// generators/relations

G = C3×Dic3.D6 order 432 = 24·33

Direct product of C3 and Dic3.D6

G = C₃×Dic₃.D₆ order 432 = 2⁴·3³

Direct product of C₃ and Dic₃.D₆