C3xD12:5S3 - GroupNames

G = C₃xD₁₂:₅S₃ order 432 = 2⁴·3³

Direct product of C₃ and D₁₂:₅S₃

direct product, metabelian, supersoluble, monomial

Aliases: C₃xD₁₂:₅S₃, C₁₂.₇₉S₃², (S₃xC₁₂):₂C₆, (S₃xC₁₂):₂S₃, (C₃xD₁₂):₄C₆, D₁₂:₅(C₃xS₃), D₆.₁(S₃xC₆), D₆:S₃:₂C₆, (C₃xD₁₂):₁₁S₃, C₁₂.₃₆(S₃xC₆), (S₃xDic₃):₄C₆, (S₃xC₆).₃₆D₆, C₃³:₈(C₄oD₄), (C₃xC₁₂).₁₃₅D₆, (C₃²xD₁₂):₆C₂, Dic₃.₈(S₃xC₆), C₃²:₄Q₈:₁₁C₆, (C₃xDic₃).₄₉D₆, C₃²:₂₂(C₄oD₁₂), (C₃²xC₆).₂₁C₂³, C₃²:₂₁(D₄:₂S₃), (C₃²xC₁₂).₃₆C₂², (C₃²xDic₃).₂₆C₂², C₂.₅(S₃²xC₆), C₄.₆(C₃xS₃²), C₆.₂(S₃xC₂xC₆), (S₃xC₃xC₁₂):₂C₂, (C₄xS₃):₁(C₃xS₃), C₆.₁₀₅(C₂xS₃²), C₃:₃(C₃xC₄oD₁₂), C₃:₂(C₃xD₄:₂S₃), (S₃xC₆).₁(C₂xC₆), (C₃xS₃xDic₃):₁₁C₂, C₃²:₄(C₃xC₄oD₄), (C₃xD₆:S₃):₉C₂, (C₃xC₁₂).₅₁(C₂xC₆), (S₃xC₃xC₆).₁₀C₂², C₃:Dic₃.₉(C₂xC₆), (C₃xC₃²:₄Q₈):₇C₂, (C₃xC₆).₁₂(C₂²xC₆), (C₃xDic₃).₈(C₂xC₆), (C₃xC₆).₁₂₆(C₂²xS₃), (C₃xC₃:Dic₃).₃₅C₂², SmallGroup(432,643)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₃xC₆ — C₃xD₁₂:₅S₃

Chief series

C₁ — C₃ — C₃² — C₃xC₆ — C₃²xC₆ — S₃xC₃xC₆ — C₃xS₃xDic₃ — C₃xD₁₂:₅S₃

Lower central

C₃² — C₃xC₆ — C₃xD₁₂:₅S₃

Upper central

C₁ — C₆ — C₁₂

Generators and relations for C₃xD₁₂:₅S₃
G = < a,b,c,d,e | a³=b¹²=c²=d³=e²=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=b^-1, bd=db, be=eb, cd=dc, ece=b⁶c, ede=d^-1 >

Subgroups: 664 in 210 conjugacy classes, 64 normal (44 characteristic)
C₁, C₂, C₂, C₃, C₃, C₄, C₄, C₂², S₃, C₆, C₆, C₂xC₄, D₄, Q₈, C₃², C₃², Dic₃, Dic₃, C₁₂, C₁₂, D₆, D₆, C₂xC₆, C₄oD₄, C₃xS₃, C₃xC₆, C₃xC₆, Dic₆, C₄xS₃, C₄xS₃, D₁₂, C₂xDic₃, C₃:D₄, C₂xC₁₂, C₃xD₄, C₃xQ₈, C₃³, C₃xDic₃, C₃xDic₃, C₃:Dic₃, C₃xC₁₂, C₃xC₁₂, S₃xC₆, S₃xC₆, S₃xC₆, C₆², C₄oD₁₂, D₄:₂S₃, C₃xC₄oD₄, S₃xC₃², C₃²xC₆, S₃xDic₃, D₆:S₃, C₃xDic₆, S₃xC₁₂, S₃xC₁₂, C₃xD₁₂, C₃xD₁₂, C₆xDic₃, C₃xC₃:D₄, C₃²:₄Q₈, C₆xC₁₂, D₄xC₃², C₃²xDic₃, C₃xC₃:Dic₃, C₃²xC₁₂, S₃xC₃xC₆, S₃xC₃xC₆, D₁₂:₅S₃, C₃xC₄oD₁₂, C₃xD₄:₂S₃, C₃xS₃xDic₃, C₃xD₆:S₃, S₃xC₃xC₁₂, C₃²xD₁₂, C₃xC₃²:₄Q₈, C₃xD₁₂:₅S₃
Quotients: C₁, C₂, C₃, C₂², S₃, C₆, C₂³, D₆, C₂xC₆, C₄oD₄, C₃xS₃, C₂²xS₃, C₂²xC₆, S₃², S₃xC₆, C₄oD₁₂, D₄:₂S₃, C₃xC₄oD₄, C₂xS₃², S₃xC₂xC₆, C₃xS₃², D₁₂:₅S₃, C₃xC₄oD₁₂, C₃xD₄:₂S₃, S₃²xC₆, C₃xD₁₂:₅S₃

Smallest permutation representation of C₃xD₁₂:₅S₃
►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 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 29)(2 28)(3 27)(4 26)(5 25)(6 36)(7 35)(8 34)(9 33)(10 32)(11 31)(12 30)(13 40)(14 39)(15 38)(16 37)(17 48)(18 47)(19 46)(20 45)(21 44)(22 43)(23 42)(24 41)

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

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

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

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

81 conjugacy classes

class	1	2A	2B	2C	2D	3A	3B	3C	···	3H	3I	3J	3K	4A	4B	4C	4D	4E	6A	6B	6C	···	6H	6I	6J	6K	6L	···	6W	6X	···	6AC	12A	···	12H	12I	12J	12K	12L	12M	···	12U	12V	···	12AA	12AB	12AC	12AD	12AE
order	1	2	2	2	2	3	3	3	···	3	3	3	3	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	12	12	12	12
size	1	1	6	6	6	1	1	2	···	2	4	4	4	2	3	3	18	18	1	1	2	···	2	4	4	4	6	···	6	12	···	12	2	···	2	3	3	3	3	4	···	4	6	···	6	18	18	18	18

81 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	4	4	4	4	4	4	4	4
type	+	+	+	+	+	+							+	+	+	+	+										+	-	+		-
image	C₁	C₂	C₂	C₂	C₂	C₂	C₃	C₆	C₆	C₆	C₆	C₆	S₃	S₃	D₆	D₆	D₆	C₄oD₄	C₃xS₃	C₃xS₃	S₃xC₆	S₃xC₆	S₃xC₆	C₄oD₁₂	C₃xC₄oD₄	C₃xC₄oD₁₂	S₃²	D₄:₂S₃	C₂xS₃²	C₃xS₃²	D₁₂:₅S₃	C₃xD₄:₂S₃	S₃²xC₆	C₃xD₁₂:₅S₃
kernel	C₃xD₁₂:₅S₃	C₃xS₃xDic₃	C₃xD₆:S₃	S₃xC₃xC₁₂	C₃²xD₁₂	C₃xC₃²:₄Q₈	D₁₂:₅S₃	S₃xDic₃	D₆:S₃	S₃xC₁₂	C₃xD₁₂	C₃²:₄Q₈	S₃xC₁₂	C₃xD₁₂	C₃xDic₃	C₃xC₁₂	S₃xC₆	C₃³	C₄xS₃	D₁₂	Dic₃	C₁₂	D₆	C₃²	C₃²	C₃	C₁₂	C₃²	C₆	C₄	C₃	C₃	C₂	C₁
# reps	1	2	2	1	1	1	2	4	4	2	2	2	1	1	1	2	3	2	2	2	2	4	6	4	4	8	1	1	1	2	2	2	2	4

Matrix representation of C₃xD₁₂:₅S₃ ►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

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

0	5	0	0	0	0
8	0	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

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	12

12	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	1	0
0	0	0	0	12	12

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],[8,0,0,0,0,0,0,5,0,0,0,0,0,0,1,1,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,8,0,0,0,0,5,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,12,0,0,0,0,1,12],[12,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,12,0,0,0,0,0,12] >;

C₃xD₁₂:₅S₃ in GAP, Magma, Sage, TeX

C_3\times D_{12}\rtimes_5S_3

% in TeX

G:=Group("C3xD12:5S3");

// GroupNames label

G:=SmallGroup(432,643);

// by ID

G=gap.SmallGroup(432,643);

# by ID

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

// Polycyclic

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

// generators/relations

G = C3xD12:5S3 order 432 = 24·33

Direct product of C3 and D12:5S3

G = C₃xD₁₂:₅S₃ order 432 = 2⁴·3³

Direct product of C₃ and D₁₂:₅S₃