C3xDic6:S3 - GroupNames

G = C₃×Dic₆⋊S₃ order 432 = 2⁴·3³

Direct product of C₃ and Dic₆⋊S₃

direct product, metabelian, supersoluble, monomial

Aliases: C₃×Dic₆⋊S₃, C₃³⋊₉SD₁₆, C₁₂.₉₇S₃², C₁₂.₂₇(S₃×C₆), C₃²⋊₄C₈⋊₉C₆, (C₃×Dic₆)⋊₁S₃, (C₃×Dic₆)⋊₁C₆, Dic₆⋊₂(C₃×S₃), D₁₂.₂(C₃×S₃), (C₃×D₁₂).₅C₆, (C₃×D₁₂).₉S₃, (C₃×C₁₂).₁₁₀D₆, C₃²⋊₆(C₃×SD₁₆), (C₃²×C₆).₂₀D₄, (C₃²×Dic₆)⋊₁C₂, (C₃²×D₁₂).₁C₂, C₆.₂₈(D₆⋊S₃), C₃²⋊₁₂(D₄.S₃), (C₃²×C₁₂).₃C₂², C₃²⋊₁₂(Q₈⋊₂S₃), C₄.₉(C₃×S₃²), C₃⋊₂(C₃×D₄.S₃), C₆.₈(C₃×C₃⋊D₄), C₃⋊₂(C₃×Q₈⋊₂S₃), (C₃×C₆).₁₉(C₃×D₄), (C₃×C₁₂).₃₇(C₂×C₆), C₂.₄(C₃×D₆⋊S₃), (C₃×C₃²⋊₄C₈)⋊₁₅C₂, (C₃×C₆).₈₃(C₃⋊D₄), SmallGroup(432,420)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₃×C₁₂ — C₃×Dic₆⋊S₃

Chief series

C₁ — C₃ — C₃² — C₃×C₆ — C₃×C₁₂ — C₃²×C₁₂ — C₃²×D₁₂ — C₃×Dic₆⋊S₃

Lower central

C₃² — C₃×C₆ — C₃×C₁₂ — C₃×Dic₆⋊S₃

Upper central

C₁ — C₆ — C₁₂

Generators and relations for C₃×Dic₆⋊S₃
G = < a,b,c,d,e | a³=b¹²=d³=e²=1, c²=b⁶, ab=ba, ac=ca, ad=da, ae=ea, cbc^-1=b^-1, bd=db, ebe=b⁷, cd=dc, ece=b³c, ede=d^-1 >

Subgroups: 400 in 122 conjugacy classes, 36 normal (all characteristic)
C₁, C₂, C₂, C₃, C₃, C₄, C₄, C₂², S₃, C₆, C₆, C₈, D₄, Q₈, C₃², C₃², Dic₃, C₁₂, C₁₂, D₆, C₂×C₆, SD₁₆, C₃×S₃, C₃×C₆, C₃×C₆, C₃⋊C₈, C₂₄, Dic₆, D₁₂, C₃×D₄, C₃×Q₈, C₃³, C₃×Dic₃, C₃×C₁₂, C₃×C₁₂, S₃×C₆, C₆², D₄.S₃, Q₈⋊₂S₃, C₃×SD₁₆, S₃×C₃², C₃²×C₆, C₃×C₃⋊C₈, C₃²⋊₄C₈, C₃×Dic₆, C₃×Dic₆, C₃×D₁₂, C₃×D₁₂, D₄×C₃², Q₈×C₃², C₃²×Dic₃, C₃²×C₁₂, S₃×C₃×C₆, Dic₆⋊S₃, C₃×D₄.S₃, C₃×Q₈⋊₂S₃, C₃×C₃²⋊₄C₈, C₃²×Dic₆, C₃²×D₁₂, C₃×Dic₆⋊S₃
Quotients: C₁, C₂, C₃, C₂², S₃, C₆, D₄, D₆, C₂×C₆, SD₁₆, C₃×S₃, C₃⋊D₄, C₃×D₄, S₃², S₃×C₆, D₄.S₃, Q₈⋊₂S₃, C₃×SD₁₆, D₆⋊S₃, C₃×C₃⋊D₄, C₃×S₃², Dic₆⋊S₃, C₃×D₄.S₃, C₃×Q₈⋊₂S₃, C₃×D₆⋊S₃, C₃×Dic₆⋊S₃

Smallest permutation representation of C₃×Dic₆⋊S₃
►On 48 points

Generators in S₄₈

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

(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 29 33)(26 30 34)(27 31 35)(28 32 36)(37 45 41)(38 46 42)(39 47 43)(40 48 44)

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

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

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

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

63 conjugacy classes

class	1	2A	2B	3A	3B	3C	···	3H	3I	3J	3K	4A	4B	6A	6B	6C	···	6H	6I	6J	6K	6L	···	6S	8A	8B	12A	12B	12C	···	12N	12O	···	12V	24A	24B	24C	24D
order	1	2	2	3	3	3	···	3	3	3	3	4	4	6	6	6	···	6	6	6	6	6	···	6	8	8	12	12	12	···	12	12	···	12	24	24	24	24
size	1	1	12	1	1	2	···	2	4	4	4	2	12	1	1	2	···	2	4	4	4	12	···	12	18	18	2	2	4	···	4	12	···	12	18	18	18	18

63 irreducible representations

dim	1	1	1	1	1	1	1	1	2	2	2	2	2	2	2	2	2	2	2	2	4	4	4	4	4	4	4	4	4	4
type	+	+	+	+					+	+	+	+									+	-	+	-
image	C₁	C₂	C₂	C₂	C₃	C₆	C₆	C₆	S₃	S₃	D₄	D₆	SD₁₆	C₃×S₃	C₃×S₃	C₃⋊D₄	C₃×D₄	S₃×C₆	C₃×SD₁₆	C₃×C₃⋊D₄	S₃²	D₄.S₃	Q₈⋊₂S₃	D₆⋊S₃	C₃×S₃²	Dic₆⋊S₃	C₃×D₄.S₃	C₃×Q₈⋊₂S₃	C₃×D₆⋊S₃	C₃×Dic₆⋊S₃
kernel	C₃×Dic₆⋊S₃	C₃×C₃²⋊₄C₈	C₃²×Dic₆	C₃²×D₁₂	Dic₆⋊S₃	C₃²⋊₄C₈	C₃×Dic₆	C₃×D₁₂	C₃×Dic₆	C₃×D₁₂	C₃²×C₆	C₃×C₁₂	C₃³	Dic₆	D₁₂	C₃×C₆	C₃×C₆	C₁₂	C₃²	C₆	C₁₂	C₃²	C₃²	C₆	C₄	C₃	C₃	C₃	C₂	C₁
# reps	1	1	1	1	2	2	2	2	1	1	1	2	2	2	2	4	2	4	4	8	1	1	1	1	2	2	2	2	2	4

Matrix representation of C₃×Dic₆⋊S₃ ►in GL₆(𝔽₇₃)

1	0	0	0	0	0
0	1	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

1	71	0	0	0	0
1	72	0	0	0	0
0	0	1	0	0	0
0	0	0	1	0	0
0	0	0	0	0	72
0	0	0	0	1	72

19	22	0	0	0	0
30	54	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

1	0	0	0	0	0
0	1	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

59	19	0	0	0	0
32	14	0	0	0	0
0	0	0	1	0	0
0	0	1	0	0	0
0	0	0	0	1	0
0	0	0	0	0	1

G:=sub<GL(6,GF(73))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,8,0,0,0,0,0,0,8,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,1,0,0,0,0,71,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,72,72],[19,30,0,0,0,0,22,54,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,72,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[59,32,0,0,0,0,19,14,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;

C₃×Dic₆⋊S₃ in GAP, Magma, Sage, TeX

C_3\times {\rm Dic}_6\rtimes S_3

% in TeX

G:=Group("C3xDic6:S3");

// GroupNames label

G:=SmallGroup(432,420);

// by ID

G=gap.SmallGroup(432,420);

# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,-3,-3,197,176,1011,514,80,2028,14118]);

// Polycyclic

G:=Group<a,b,c,d,e|a^3=b^12=d^3=e^2=1,c^2=b^6,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,e*b*e=b^7,c*d=d*c,e*c*e=b^3*c,e*d*e=d^-1>;

// generators/relations

G = C3×Dic6⋊S3 order 432 = 24·33

Direct product of C3 and Dic6⋊S3

G = C₃×Dic₆⋊S₃ order 432 = 2⁴·3³

Direct product of C₃ and Dic₆⋊S₃