C3xD6.Dic3 - GroupNames

G = C₃×D₆.Dic₃ order 432 = 2⁴·3³

Direct product of C₃ and D₆.Dic₃

direct product, metabelian, supersoluble, monomial

Aliases: C₃×D₆.Dic₃, C₃³⋊₅M₄(2), C₁₂.₁₀₃S₃², D₆.(C₃×Dic₃), (S₃×C₆).₄C₁₂, (S₃×C₁₂).₁C₆, C₆.₂₉(S₃×C₁₂), C₁₂.₄₆(S₃×C₆), C₆.₂(C₆×Dic₃), (S₃×C₁₂).₁₃S₃, C₃²⋊₄C₈⋊₁₄C₆, (C₃×C₁₂).₁₈₀D₆, Dic₃.(C₃×Dic₃), (S₃×C₆).₅Dic₃, C₆.₃₃(S₃×Dic₃), (C₃×Dic₃).₁C₁₂, C₃²⋊₄(C₃×M₄(2)), C₃²⋊₁₃(C₈⋊S₃), (C₃×Dic₃).₇Dic₃, C₃²⋊₈(C₄.Dic₃), (C₃²×Dic₃).₄C₄, (C₃²×C₁₂).₆₂C₂², (C₃×C₃⋊C₈)⋊₇C₆, C₃⋊C₈⋊₄(C₃×S₃), (C₃×C₃⋊C₈)⋊₁₁S₃, C₄.₁₅(C₃×S₃²), (S₃×C₃×C₆).₄C₄, C₃⋊₃(C₃×C₈⋊S₃), (S₃×C₃×C₁₂).₂C₂, C₂.₃(C₃×S₃×Dic₃), (C₄×S₃).₂(C₃×S₃), (C₃²×C₃⋊C₈)⋊₁₄C₂, (C₃×C₆).₈₆(C₄×S₃), C₃⋊₁(C₃×C₄.Dic₃), (C₃×C₁₂).₆₃(C₂×C₆), (C₃×C₆).₁₉(C₂×C₁₂), (C₃×C₃²⋊₄C₈)⋊₁₈C₂, (C₃²×C₆).₂₄(C₂×C₄), (C₃×C₆).₄₆(C₂×Dic₃), SmallGroup(432,416)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₃×C₆ — C₃×D₆.Dic₃

Chief series

C₁ — C₃ — C₃² — C₃×C₆ — C₃×C₁₂ — C₃²×C₁₂ — S₃×C₃×C₁₂ — C₃×D₆.Dic₃

Lower central

C₃² — C₃×C₆ — C₃×D₆.Dic₃

Upper central

C₁ — C₁₂

Generators and relations for C₃×D₆.Dic₃
G = < a,b,c,d,e | a³=b⁶=c²=1, d⁶=b³, e²=b³d³, ab=ba, ac=ca, ad=da, ae=ea, cbc=b^-1, bd=db, be=eb, cd=dc, ece^-1=b³c, ede^-1=d⁵ >

Subgroups: 304 in 118 conjugacy classes, 44 normal (all characteristic)
C₁, C₂, C₂, C₃, C₃, C₄, C₄, C₂², S₃, C₆, C₆, C₈, C₂×C₄, C₃², C₃², Dic₃, C₁₂, C₁₂, D₆, C₂×C₆, M₄(2), C₃×S₃, C₃×C₆, C₃×C₆, C₃⋊C₈, C₃⋊C₈, C₂₄, C₄×S₃, C₂×C₁₂, C₃³, C₃×Dic₃, C₃×Dic₃, C₃×C₁₂, C₃×C₁₂, S₃×C₆, S₃×C₆, C₆², C₈⋊S₃, C₄.Dic₃, C₃×M₄(2), S₃×C₃², C₃²×C₆, C₃×C₃⋊C₈, C₃×C₃⋊C₈, C₃²⋊₄C₈, C₃×C₂₄, S₃×C₁₂, S₃×C₁₂, C₆×C₁₂, C₃²×Dic₃, C₃²×C₁₂, S₃×C₃×C₆, D₆.Dic₃, C₃×C₈⋊S₃, C₃×C₄.Dic₃, C₃²×C₃⋊C₈, C₃×C₃²⋊₄C₈, S₃×C₃×C₁₂, C₃×D₆.Dic₃
Quotients: C₁, C₂, C₃, C₄, C₂², S₃, C₆, C₂×C₄, Dic₃, C₁₂, D₆, C₂×C₆, M₄(2), C₃×S₃, C₄×S₃, C₂×Dic₃, C₂×C₁₂, C₃×Dic₃, S₃², S₃×C₆, C₈⋊S₃, C₄.Dic₃, C₃×M₄(2), S₃×Dic₃, S₃×C₁₂, C₆×Dic₃, C₃×S₃², D₆.Dic₃, C₃×C₈⋊S₃, C₃×C₄.Dic₃, C₃×S₃×Dic₃, C₃×D₆.Dic₃

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

Generators in S₄₈

(1 9 5)(2 10 6)(3 11 7)(4 12 8)(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 41 45)(38 42 46)(39 43 47)(40 44 48)

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

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

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

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

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

90 conjugacy classes

class	1	2A	2B	3A	3B	3C	···	3H	3I	3J	3K	4A	4B	4C	6A	6B	6C	···	6H	6I	6J	6K	6L	···	6S	8A	8B	8C	8D	12A	12B	12C	12D	12E	···	12P	12Q	···	12V	12W	···	12AD	24A	···	24P	24Q	24R	24S	24T
order	1	2	2	3	3	3	···	3	3	3	3	4	4	4	6	6	6	···	6	6	6	6	6	···	6	8	8	8	8	12	12	12	12	12	···	12	12	···	12	12	···	12	24	···	24	24	24	24	24
size	1	1	6	1	1	2	···	2	4	4	4	1	1	6	1	1	2	···	2	4	4	4	6	···	6	6	6	18	18	1	1	1	1	2	···	2	4	···	4	6	···	6	6	···	6	18	18	18	18

90 irreducible representations

dim	1	1	1	1	1	1	1	1	1	1	1	1	2	2	2	2	2	2	2	2	2	2	2	2	2	2	2	2	2	2	4	4	4	4	4	4
type	+	+	+	+									+	+	-	+	-														+	-
image	C₁	C₂	C₂	C₂	C₃	C₄	C₄	C₆	C₆	C₆	C₁₂	C₁₂	S₃	S₃	Dic₃	D₆	Dic₃	M₄(2)	C₃×S₃	C₃×S₃	C₄×S₃	C₃×Dic₃	S₃×C₆	C₃×Dic₃	C₈⋊S₃	C₄.Dic₃	C₃×M₄(2)	S₃×C₁₂	C₃×C₈⋊S₃	C₃×C₄.Dic₃	S₃²	S₃×Dic₃	C₃×S₃²	D₆.Dic₃	C₃×S₃×Dic₃	C₃×D₆.Dic₃
kernel	C₃×D₆.Dic₃	C₃²×C₃⋊C₈	C₃×C₃²⋊₄C₈	S₃×C₃×C₁₂	D₆.Dic₃	C₃²×Dic₃	S₃×C₃×C₆	C₃×C₃⋊C₈	C₃²⋊₄C₈	S₃×C₁₂	C₃×Dic₃	S₃×C₆	C₃×C₃⋊C₈	S₃×C₁₂	C₃×Dic₃	C₃×C₁₂	S₃×C₆	C₃³	C₃⋊C₈	C₄×S₃	C₃×C₆	Dic₃	C₁₂	D₆	C₃²	C₃²	C₃²	C₆	C₃	C₃	C₁₂	C₆	C₄	C₃	C₂	C₁
# reps	1	1	1	1	2	2	2	2	2	2	4	4	1	1	1	2	1	2	2	2	2	2	4	2	4	4	4	4	8	8	1	1	2	2	2	4

Matrix representation of C₃×D₆.Dic₃ ►in GL₆(𝔽₇₃)

64	0	0	0	0	0
0	64	0	0	0	0
0	0	8	0	0	0
0	0	0	8	0	0
0	0	0	0	1	0
0	0	0	0	0	1

72	0	0	0	0	0
0	72	0	0	0	0
0	0	0	72	0	0
0	0	1	72	0	0
0	0	0	0	1	0
0	0	0	0	0	1

57	60	0	0	0	0
14	16	0	0	0	0
0	0	72	0	0	0
0	0	72	1	0	0
0	0	0	0	1	0
0	0	0	0	0	1

27	0	0	0	0	0
0	27	0	0	0	0
0	0	1	0	0	0
0	0	0	1	0	0
0	0	0	0	72	1
0	0	0	0	72	0

0	27	0	0	0	0
1	0	0	0	0	0
0	0	72	0	0	0
0	0	0	72	0	0
0	0	0	0	0	1
0	0	0	0	1	0

G:=sub<GL(6,GF(73))| [64,0,0,0,0,0,0,64,0,0,0,0,0,0,8,0,0,0,0,0,0,8,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,0,1,0,0,0,0,72,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[57,14,0,0,0,0,60,16,0,0,0,0,0,0,72,72,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[27,0,0,0,0,0,0,27,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,72,0,0,0,0,1,0],[0,1,0,0,0,0,27,0,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;

C₃×D₆.Dic₃ in GAP, Magma, Sage, TeX

C_3\times D_6.{\rm Dic}_3

% in TeX

G:=Group("C3xD6.Dic3");

// GroupNames label

G:=SmallGroup(432,416);

// by ID

G=gap.SmallGroup(432,416);

# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,-3,-3,365,92,80,2028,14118]);

// Polycyclic

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

// generators/relations

G = C3×D6.Dic3 order 432 = 24·33

Direct product of C3 and D6.Dic3

G = C₃×D₆.Dic₃ order 432 = 2⁴·3³

Direct product of C₃ and D₆.Dic₃