C2xC3^3:9D4 - GroupNames

G = C₂xC₃³:₉D₄ order 432 = 2⁴·3³

Direct product of C₂ and C₃³:₉D₄

direct product, metabelian, supersoluble, monomial

Aliases: C₂xC₃³:₉D₄, C₆².₉₇D₆, (C₃xC₆):₆D₁₂, (C₃²xC₆):₉D₄, C₃³:₂₇(C₂xD₄), C₃:Dic₃:₂₂D₆, C₆:₃(C₃:D₁₂), C₆:₁(D₆:S₃), C₃²:₁₁(C₂xD₁₂), (C₃xC₆²).₃₅C₂², (C₃²xC₆).₇₂C₂³, C₂².₆(C₃²:₄D₆), (C₂xC₆).₆₄S₃², C₆.₁₀₁(C₂xS₃²), (C₂xC₃:S₃):₂₂D₆, C₃:₂(C₂xD₆:S₃), C₃:₄(C₂xC₃:D₁₂), (C₃xC₆):₈(C₃:D₄), (C₆xC₃:Dic₃):₉C₂, (C₆xC₃:S₃):₁₉C₂², (C₂²xC₃:S₃):₁₁S₃, (C₂xC₃:Dic₃):₁₄S₃, C₃²:₁₄(C₂xC₃:D₄), C₂.₈(C₂xC₃²:₄D₆), (C₃xC₆).₁₂₂(C₂²xS₃), (C₃xC₃:Dic₃):₁₆C₂², (C₂xC₆xC₃:S₃):₆C₂, SmallGroup(432,694)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₃²xC₆ — C₂xC₃³:₉D₄

Chief series

C₁ — C₃ — C₃² — C₃³ — C₃²xC₆ — C₆xC₃:S₃ — C₃³:₉D₄ — C₂xC₃³:₉D₄

Lower central

C₃³ — C₃²xC₆ — C₂xC₃³:₉D₄

Upper central

C₁ — C₂²

Generators and relations for C₂xC₃³:₉D₄
G = < a,b,c,d,e,f | a²=b³=c³=d³=e⁴=f²=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, fbf=b^-1, cd=dc, ece^-1=c^-1, cf=fc, ede^-1=fdf=d^-1, fef=e^-1 >

Subgroups: 1560 in 306 conjugacy classes, 63 normal (15 characteristic)
C₁, C₂, C₂, C₂, C₃, C₃, C₃, C₄, C₂², C₂², S₃, C₆, C₆, C₆, C₂xC₄, D₄, C₂³, C₃², C₃², C₃², Dic₃, C₁₂, D₆, C₂xC₆, C₂xC₆, C₂xC₆, C₂xD₄, C₃xS₃, C₃:S₃, C₃xC₆, C₃xC₆, C₃xC₆, D₁₂, C₂xDic₃, C₃:D₄, C₂xC₁₂, C₂²xS₃, C₂²xC₆, C₃³, C₃xDic₃, C₃:Dic₃, S₃xC₆, C₂xC₃:S₃, C₂xC₃:S₃, C₆², C₆², C₆², C₂xD₁₂, C₂xC₃:D₄, C₃xC₃:S₃, C₃²xC₆, C₃²xC₆, D₆:S₃, C₃:D₁₂, C₆xDic₃, C₂xC₃:Dic₃, S₃xC₂xC₆, C₂²xC₃:S₃, C₃xC₃:Dic₃, C₆xC₃:S₃, C₆xC₃:S₃, C₃xC₆², C₂xD₆:S₃, C₂xC₃:D₁₂, C₃³:₉D₄, C₆xC₃:Dic₃, C₂xC₆xC₃:S₃, C₂xC₃³:₉D₄
Quotients: C₁, C₂, C₂², S₃, D₄, C₂³, D₆, C₂xD₄, D₁₂, C₃:D₄, C₂²xS₃, S₃², C₂xD₁₂, C₂xC₃:D₄, D₆:S₃, C₃:D₁₂, C₂xS₃², C₃²:₄D₆, C₂xD₆:S₃, C₂xC₃:D₁₂, C₃³:₉D₄, C₂xC₃²:₄D₆, C₂xC₃³:₉D₄

Smallest permutation representation of C₂xC₃³:₉D₄
►On 48 points

Generators in S₄₈

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

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

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

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

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

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

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

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

54 conjugacy classes

class	1	2A	2B	2C	2D	2E	2F	2G	3A	3B	3C	3D	···	3H	4A	4B	6A	···	6I	6J	···	6X	6Y	···	6AF	12A	12B	12C	12D
order	1	2	2	2	2	2	2	2	3	3	3	3	···	3	4	4	6	···	6	6	···	6	6	···	6	12	12	12	12
size	1	1	1	1	18	18	18	18	2	2	2	4	···	4	18	18	2	···	2	4	···	4	18	···	18	18	18	18	18

54 irreducible representations

dim	1	1	1	1	2	2	2	2	2	2	2	2	4	4	4	4	4	4	4
type	+	+	+	+	+	+	+	+	+	+	+		+	-	+	+
image	C₁	C₂	C₂	C₂	S₃	S₃	D₄	D₆	D₆	D₆	D₁₂	C₃:D₄	S₃²	D₆:S₃	C₃:D₁₂	C₂xS₃²	C₃²:₄D₆	C₃³:₉D₄	C₂xC₃²:₄D₆
kernel	C₂xC₃³:₉D₄	C₃³:₉D₄	C₆xC₃:Dic₃	C₂xC₆xC₃:S₃	C₂xC₃:Dic₃	C₂²xC₃:S₃	C₃²xC₆	C₃:Dic₃	C₂xC₃:S₃	C₆²	C₃xC₆	C₃xC₆	C₂xC₆	C₆	C₆	C₆	C₂²	C₂	C₂
# reps	1	4	1	2	1	2	2	2	4	3	4	8	3	2	4	3	2	4	2

Matrix representation of C₂xC₃³:₉D₄ ►in GL₆(F₁₃)

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	12	0
0	0	0	0	0	12

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

0	1	0	0	0	0
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

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	12	1
0	0	0	0	12	0

1	0	0	0	0	0
12	12	0	0	0	0
0	0	1	0	0	0
0	0	0	1	0	0
0	0	0	0	4	11
0	0	0	0	2	9

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

G:=sub<GL(6,GF(13))| [12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,12,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,12,0,0,0,0,1,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,12,0,0,0,0,1,0],[1,12,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,2,0,0,0,0,11,9],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,12,0,0,0,0,12,0] >;

C₂xC₃³:₉D₄ in GAP, Magma, Sage, TeX

C_2\times C_3^3\rtimes_9D_4

% in TeX

G:=Group("C2xC3^3:9D4");

// GroupNames label

G:=SmallGroup(432,694);

// by ID

G=gap.SmallGroup(432,694);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,141,1124,571,2028,14118]);

// Polycyclic

G:=Group<a,b,c,d,e,f|a^2=b^3=c^3=d^3=e^4=f^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,b*c=c*b,b*d=d*b,b*e=e*b,f*b*f=b^-1,c*d=d*c,e*c*e^-1=c^-1,c*f=f*c,e*d*e^-1=f*d*f=d^-1,f*e*f=e^-1>;

// generators/relations

G = C2xC33:9D4 order 432 = 24·33

Direct product of C2 and C33:9D4

G = C₂xC₃³:₉D₄ order 432 = 2⁴·3³

Direct product of C₂ and C₃³:₉D₄