C2xC3^3:9D4 - GroupNames

G = C₂×C₃³⋊₉D₄ order 432 = 2⁴·3³

Direct product of C₂ and C₃³⋊₉D₄

direct product, metabelian, supersoluble, monomial

Aliases: C₂×C₃³⋊₉D₄, C₆².₉₇D₆, (C₃×C₆)⋊₆D₁₂, (C₃²×C₆)⋊₉D₄, C₃³⋊₂₇(C₂×D₄), C₃⋊Dic₃⋊₂₂D₆, C₆⋊₃(C₃⋊D₁₂), C₆⋊₁(D₆⋊S₃), C₃²⋊₁₁(C₂×D₁₂), (C₃×C₆²).₃₅C₂², (C₃²×C₆).₇₂C₂³, C₂².₆(C₃²⋊₄D₆), (C₂×C₆).₆₄S₃², C₆.₁₀₁(C₂×S₃²), (C₂×C₃⋊S₃)⋊₂₂D₆, C₃⋊₂(C₂×D₆⋊S₃), C₃⋊₄(C₂×C₃⋊D₁₂), (C₃×C₆)⋊₈(C₃⋊D₄), (C₆×C₃⋊Dic₃)⋊₉C₂, (C₆×C₃⋊S₃)⋊₁₉C₂², (C₂²×C₃⋊S₃)⋊₁₁S₃, (C₂×C₃⋊Dic₃)⋊₁₄S₃, C₃²⋊₁₄(C₂×C₃⋊D₄), C₂.₈(C₂×C₃²⋊₄D₆), (C₃×C₆).₁₂₂(C₂²×S₃), (C₃×C₃⋊Dic₃)⋊₁₆C₂², (C₂×C₆×C₃⋊S₃)⋊₆C₂, SmallGroup(432,694)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₃²×C₆ — C₂×C₃³⋊₉D₄

Chief series

C₁ — C₃ — C₃² — C₃³ — C₃²×C₆ — C₆×C₃⋊S₃ — C₃³⋊₉D₄ — C₂×C₃³⋊₉D₄

Lower central

C₃³ — C₃²×C₆ — C₂×C₃³⋊₉D₄

Upper central

C₁ — C₂²

Generators and relations for C₂×C₃³⋊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, C₂×C₄, D₄, C₂³, C₃², C₃², C₃², Dic₃, C₁₂, D₆, C₂×C₆, C₂×C₆, C₂×C₆, C₂×D₄, C₃×S₃, C₃⋊S₃, C₃×C₆, C₃×C₆, C₃×C₆, D₁₂, C₂×Dic₃, C₃⋊D₄, C₂×C₁₂, C₂²×S₃, C₂²×C₆, C₃³, C₃×Dic₃, C₃⋊Dic₃, S₃×C₆, C₂×C₃⋊S₃, C₂×C₃⋊S₃, C₆², C₆², C₆², C₂×D₁₂, C₂×C₃⋊D₄, C₃×C₃⋊S₃, C₃²×C₆, C₃²×C₆, D₆⋊S₃, C₃⋊D₁₂, C₆×Dic₃, C₂×C₃⋊Dic₃, S₃×C₂×C₆, C₂²×C₃⋊S₃, C₃×C₃⋊Dic₃, C₆×C₃⋊S₃, C₆×C₃⋊S₃, C₃×C₆², C₂×D₆⋊S₃, C₂×C₃⋊D₁₂, C₃³⋊₉D₄, C₆×C₃⋊Dic₃, C₂×C₆×C₃⋊S₃, C₂×C₃³⋊₉D₄
Quotients: C₁, C₂, C₂², S₃, D₄, C₂³, D₆, C₂×D₄, D₁₂, C₃⋊D₄, C₂²×S₃, S₃², C₂×D₁₂, C₂×C₃⋊D₄, D₆⋊S₃, C₃⋊D₁₂, C₂×S₃², C₃²⋊₄D₆, C₂×D₆⋊S₃, C₂×C₃⋊D₁₂, C₃³⋊₉D₄, C₂×C₃²⋊₄D₆, C₂×C₃³⋊₉D₄

Smallest permutation representation of C₂×C₃³⋊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	C₂×S₃²	C₃²⋊₄D₆	C₃³⋊₉D₄	C₂×C₃²⋊₄D₆
kernel	C₂×C₃³⋊₉D₄	C₃³⋊₉D₄	C₆×C₃⋊Dic₃	C₂×C₆×C₃⋊S₃	C₂×C₃⋊Dic₃	C₂²×C₃⋊S₃	C₃²×C₆	C₃⋊Dic₃	C₂×C₃⋊S₃	C₆²	C₃×C₆	C₃×C₆	C₂×C₆	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 C₂×C₃³⋊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] >;

C₂×C₃³⋊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 = C2×C33⋊9D4 order 432 = 24·33

Direct product of C2 and C33⋊9D4

G = C₂×C₃³⋊₉D₄ order 432 = 2⁴·3³

Direct product of C₂ and C₃³⋊₉D₄