C3xC3^2:M4(2) - GroupNames

G = C₃xC₃²:M₄(2) order 432 = 2⁴·3³

Direct product of C₃ and C₃²:M₄(2)

direct product, metabelian, soluble, monomial

Aliases: C₃xC₃²:M₄(2), C₃³:₃M₄(2), (C₃xC₁₂).₉C₁₂, C₃²:₂C₈:₃C₆, C₁₂.₆(C₃²:C₄), (C₃²xC₁₂).₆C₄, C₃²:₃(C₃xM₄(2)), C₄.(C₃xC₃²:C₄), C₂.₄(C₆xC₃²:C₄), (C₄xC₃:S₃).₁₀C₆, (C₆xC₃:S₃).₁₁C₄, (C₂xC₃:S₃).₈C₁₂, C₆.₂₀(C₂xC₃²:C₄), (C₁₂xC₃:S₃).₁₈C₂, (C₃xC₆).₁₁(C₂xC₁₂), (C₃xC₃²:₂C₈):₉C₂, (C₃²xC₆).₉(C₂xC₄), C₃:Dic₃.₁₃(C₂xC₆), (C₃xC₃:Dic₃).₃₈C₂², SmallGroup(432,629)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₃xC₆ — C₃xC₃²:M₄(2)

Chief series

C₁ — C₃² — C₃xC₆ — C₃:Dic₃ — C₃xC₃:Dic₃ — C₃xC₃²:₂C₈ — C₃xC₃²:M₄(2)

Lower central

C₃² — C₃xC₆ — C₃xC₃²:M₄(2)

Upper central

C₁ — C₆ — C₁₂

Generators and relations for C₃xC₃²:M₄(2)
G = < a,b,c,d,e | a³=b³=c³=d⁸=e²=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, dbd^-1=bc^-1, ebe=b^-1, dcd^-1=b^-1c^-1, ece=c^-1, ede=d⁵ >

Subgroups: 364 in 84 conjugacy classes, 24 normal (20 characteristic)
C₁, C₂, C₂, C₃, C₃, C₄, C₄, C₂², S₃, C₆, C₆, C₈, C₂xC₄, C₃², C₃², Dic₃, C₁₂, C₁₂, D₆, C₂xC₆, M₄(2), C₃xS₃, C₃:S₃, C₃xC₆, C₃xC₆, C₂₄, C₄xS₃, C₂xC₁₂, C₃³, C₃xDic₃, C₃:Dic₃, C₃xC₁₂, C₃xC₁₂, S₃xC₆, C₂xC₃:S₃, C₃xM₄(2), C₃xC₃:S₃, C₃²xC₆, C₃²:₂C₈, S₃xC₁₂, C₄xC₃:S₃, C₃xC₃:Dic₃, C₃²xC₁₂, C₆xC₃:S₃, C₃²:M₄(2), C₃xC₃²:₂C₈, C₁₂xC₃:S₃, C₃xC₃²:M₄(2)
Quotients: C₁, C₂, C₃, C₄, C₂², C₆, C₂xC₄, C₁₂, C₂xC₆, M₄(2), C₂xC₁₂, C₃²:C₄, C₃xM₄(2), C₂xC₃²:C₄, C₃xC₃²:C₄, C₃²:M₄(2), C₆xC₃²:C₄, C₃xC₃²:M₄(2)

Smallest permutation representation of C₃xC₃²:M₄(2)
►On 48 points

Generators in S₄₈

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

(2 38 16)(4 10 40)(6 34 12)(8 14 36)(18 31 48)(20 42 25)(22 27 44)(24 46 29)

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

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

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

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

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

54 conjugacy classes

class	1	2A	2B	3A	3B	3C	···	3H	4A	4B	4C	6A	6B	6C	···	6H	6I	6J	8A	8B	8C	8D	12A	12B	12C	···	12N	12O	12P	12Q	12R	24A	···	24H
order	1	2	2	3	3	3	···	3	4	4	4	6	6	6	···	6	6	6	8	8	8	8	12	12	12	···	12	12	12	12	12	24	···	24
size	1	1	18	1	1	4	···	4	2	9	9	1	1	4	···	4	18	18	18	18	18	18	2	2	4	···	4	9	9	9	9	18	···	18

54 irreducible representations

dim	1	1	1	1	1	1	1	1	1	1	2	2	4	4	4	4	4	4
type	+	+	+										+	+
image	C₁	C₂	C₂	C₃	C₄	C₄	C₆	C₆	C₁₂	C₁₂	M₄(2)	C₃xM₄(2)	C₃²:C₄	C₂xC₃²:C₄	C₃xC₃²:C₄	C₃²:M₄(2)	C₆xC₃²:C₄	C₃xC₃²:M₄(2)
kernel	C₃xC₃²:M₄(2)	C₃xC₃²:₂C₈	C₁₂xC₃:S₃	C₃²:M₄(2)	C₃²xC₁₂	C₆xC₃:S₃	C₃²:₂C₈	C₄xC₃:S₃	C₃xC₁₂	C₂xC₃:S₃	C₃³	C₃²	C₁₂	C₆	C₄	C₃	C₂	C₁
# reps	1	2	1	2	2	2	4	2	4	4	2	4	2	2	4	4	4	8

Matrix representation of C₃xC₃²:M₄(2) ►in GL₄(F₇₃) generated by

64	0	0	0
0	64	0	0
0	0	64	0
0	0	0	64

1	0	0	0
0	1	0	0
48	13	8	0
18	20	0	64

64	0	0	0
0	8	0	0
54	0	8	0
0	34	0	64

54	60	0	43
18	14	30	0
59	11	59	18
31	46	60	19

0	27	0	0
46	0	0	0
32	67	0	46
6	32	27	0

G:=sub<GL(4,GF(73))| [64,0,0,0,0,64,0,0,0,0,64,0,0,0,0,64],[1,0,48,18,0,1,13,20,0,0,8,0,0,0,0,64],[64,0,54,0,0,8,0,34,0,0,8,0,0,0,0,64],[54,18,59,31,60,14,11,46,0,30,59,60,43,0,18,19],[0,46,32,6,27,0,67,32,0,0,0,27,0,0,46,0] >;

C₃xC₃²:M₄(2) in GAP, Magma, Sage, TeX

C_3\times C_3^2\rtimes M_4(2)

% in TeX

G:=Group("C3xC3^2:M4(2)");

// GroupNames label

G:=SmallGroup(432,629);

// by ID

G=gap.SmallGroup(432,629);

# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,-3,3,84,365,176,80,14117,362,18822,1203]);

// Polycyclic

G:=Group<a,b,c,d,e|a^3=b^3=c^3=d^8=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,d*b*d^-1=b*c^-1,e*b*e=b^-1,d*c*d^-1=b^-1*c^-1,e*c*e=c^-1,e*d*e=d^5>;

// generators/relations

G = C3xC32:M4(2) order 432 = 24·33

Direct product of C3 and C32:M4(2)

G = C₃xC₃²:M₄(2) order 432 = 2⁴·3³

Direct product of C₃ and C₃²:M₄(2)