C2^2xC3^2:4D6 - GroupNames

G = C₂²×C₃²⋊₄D₆ order 432 = 2⁴·3³

Direct product of C₂² and C₃²⋊₄D₆

direct product, metabelian, supersoluble, monomial, A-group

Aliases: C₂²×C₃²⋊₄D₆, C₆²⋊₂₇D₆, C₃³⋊₄C₂⁴, (C₃²×C₆)⋊₄C₂³, C₃²⋊₆(S₃×C₂³), (C₃×C₆²)⋊₁₂C₂², C₆⋊₃(C₂×S₃²), (C₂×C₆)⋊₁₂S₃², C₃⋊₃(C₂²×S₃²), (C₂×C₃⋊S₃)⋊₂₄D₆, C₃⋊S₃⋊₃(C₂²×S₃), (C₃×C₃⋊S₃)⋊₄C₂³, (C₃×C₆)⋊₆(C₂²×S₃), (C₂²×C₃⋊S₃)⋊₁₃S₃, (C₆×C₃⋊S₃)⋊₂₆C₂², (C₂×C₆×C₃⋊S₃)⋊₁₃C₂, SmallGroup(432,769)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₃³ — C₂²×C₃²⋊₄D₆

Chief series

C₁ — C₃ — C₃² — C₃³ — C₃×C₃⋊S₃ — C₃²⋊₄D₆ — C₂×C₃²⋊₄D₆ — C₂²×C₃²⋊₄D₆

Lower central

C₃³ — C₂²×C₃²⋊₄D₆

Upper central

C₁ — C₂²

Generators and relations for C₂²×C₃²⋊₄D₆
G = < a,b,c,d,e,f | a²=b²=c³=d³=e⁶=f²=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, cd=dc, ece^-1=c^-1, cf=fc, ede^-1=fdf=d^-1, fef=e^-1 >

Subgroups: 3448 in 642 conjugacy classes, 135 normal (5 characteristic)
C₁, C₂, C₂, C₃, C₃, C₂², C₂², S₃, C₆, C₆, C₂³, C₃², C₃², D₆, C₂×C₆, C₂×C₆, C₂⁴, C₃×S₃, C₃⋊S₃, C₃×C₆, C₃×C₆, C₂²×S₃, C₂²×C₆, C₃³, S₃², S₃×C₆, C₂×C₃⋊S₃, C₆², C₆², S₃×C₂³, C₃×C₃⋊S₃, C₃²×C₆, C₂×S₃², S₃×C₂×C₆, C₂²×C₃⋊S₃, C₃²⋊₄D₆, C₆×C₃⋊S₃, C₃×C₆², C₂²×S₃², C₂×C₃²⋊₄D₆, C₂×C₆×C₃⋊S₃, C₂²×C₃²⋊₄D₆
Quotients: C₁, C₂, C₂², S₃, C₂³, D₆, C₂⁴, C₂²×S₃, S₃², S₃×C₂³, C₂×S₃², C₃²⋊₄D₆, C₂²×S₃², C₂×C₃²⋊₄D₆, C₂²×C₃²⋊₄D₆

Smallest permutation representation of C₂²×C₃²⋊₄D₆
►On 48 points

Generators in S₄₈

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

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

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

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

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

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

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

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

60 conjugacy classes

class	1	2A	2B	2C	2D	···	2O	3A	3B	3C	3D	···	3H	6A	···	6I	6J	···	6X	6Y	···	6AJ
order	1	2	2	2	2	···	2	3	3	3	3	···	3	6	···	6	6	···	6	6	···	6
size	1	1	1	1	9	···	9	2	2	2	4	···	4	2	···	2	4	···	4	18	···	18

60 irreducible representations

dim	1	1	1	2	2	2	4	4	4	4
type	+	+	+	+	+	+	+	+
image	C₁	C₂	C₂	S₃	D₆	D₆	S₃²	C₂×S₃²	C₃²⋊₄D₆	C₂×C₃²⋊₄D₆
kernel	C₂²×C₃²⋊₄D₆	C₂×C₃²⋊₄D₆	C₂×C₆×C₃⋊S₃	C₂²×C₃⋊S₃	C₂×C₃⋊S₃	C₆²	C₂×C₆	C₆	C₂²	C₂
# reps	1	12	3	3	18	3	3	9	2	6

Matrix representation of C₂²×C₃²⋊₄D₆ ►in GL₆(ℤ)

-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	-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	-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	-1	-1

-1	-1	0	0	0	0
1	0	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
-1	-1	0	0	0	0
0	0	0	-1	0	0
0	0	1	-1	0	0
0	0	0	0	1	0
0	0	0	0	-1	-1

-1	0	0	0	0	0
1	1	0	0	0	0
0	0	-1	1	0	0
0	0	0	1	0	0
0	0	0	0	1	0
0	0	0	0	0	1

G:=sub<GL(6,Integers())| [-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,-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,-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,-1,0,0,0,0,1,-1],[-1,1,0,0,0,0,-1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,-1,0,0,0,0,0,-1,0,0,0,0,0,0,0,1,0,0,0,0,-1,-1,0,0,0,0,0,0,1,-1,0,0,0,0,0,-1],[-1,1,0,0,0,0,0,1,0,0,0,0,0,0,-1,0,0,0,0,0,1,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;

C₂²×C₃²⋊₄D₆ in GAP, Magma, Sage, TeX

C_2^2\times C_3^2\rtimes_4D_6

% in TeX

G:=Group("C2^2xC3^2:4D6");

// GroupNames label

G:=SmallGroup(432,769);

// by ID

G=gap.SmallGroup(432,769);

# by ID

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

// Polycyclic

G:=Group<a,b,c,d,e,f|a^2=b^2=c^3=d^3=e^6=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,b*f=f*b,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 = C22×C32⋊4D6 order 432 = 24·33

Direct product of C22 and C32⋊4D6

G = C₂²×C₃²⋊₄D₆ order 432 = 2⁴·3³

Direct product of C₂² and C₃²⋊₄D₆