C6xC3.A4 - GroupNames

G = C₆xC₃.A₄ order 216 = 2³·3³

Direct product of C₆ and C₃.A₄

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

Aliases: C₆xC₃.A₄, C₆².₆C₆, (C₂²xC₆):C₉, (C₂xC₆):₂C₁₈, C₃.₂(C₆xA₄), C₆.₅(C₃xA₄), (C₃xC₆).₅A₄, C₂³:₂(C₃xC₉), C₂²:₂(C₃xC₁₈), C₃².₃(C₂xA₄), (C₂xC₆²).₁C₃, (C₂²xC₆).₃C₃², (C₂xC₆).₃(C₃xC₆), SmallGroup(216,105)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂² — C₆xC₃.A₄

Chief series

C₁ — C₂² — C₂xC₆ — C₆² — C₃xC₃.A₄ — C₆xC₃.A₄

Lower central

C₂² — C₆xC₃.A₄

Upper central

C₁ — C₃xC₆

Generators and relations for C₆xC₃.A₄
G = < a,b,c,d,e | a⁶=b³=c²=d²=1, e³=b, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, ece^-1=cd=dc, ede^-1=c >

Subgroups: 136 in 64 conjugacy classes, 32 normal (14 characteristic)
C₁, C₂, C₂, C₃, C₃, C₂², C₂², C₆, C₆, C₆, C₂³, C₉, C₃², C₂xC₆, C₂xC₆, C₂xC₆, C₁₈, C₃xC₆, C₃xC₆, C₂²xC₆, C₂²xC₆, C₃xC₉, C₃.A₄, C₆², C₆², C₃xC₁₈, C₂xC₃.A₄, C₂xC₆², C₃xC₃.A₄, C₆xC₃.A₄
Quotients: C₁, C₂, C₃, C₆, C₉, C₃², A₄, C₁₈, C₃xC₆, C₂xA₄, C₃xC₉, C₃.A₄, C₃xA₄, C₃xC₁₈, C₂xC₃.A₄, C₆xA₄, C₃xC₃.A₄, C₆xC₃.A₄

Smallest permutation representation of C₆xC₃.A₄
►On 54 points

Generators in S₅₄

(1 33 24 52 15 41)(2 34 25 53 16 42)(3 35 26 54 17 43)(4 36 27 46 18 44)(5 28 19 47 10 45)(6 29 20 48 11 37)(7 30 21 49 12 38)(8 31 22 50 13 39)(9 32 23 51 14 40)

(1 4 7)(2 5 8)(3 6 9)(10 13 16)(11 14 17)(12 15 18)(19 22 25)(20 23 26)(21 24 27)(28 31 34)(29 32 35)(30 33 36)(37 40 43)(38 41 44)(39 42 45)(46 49 52)(47 50 53)(48 51 54)

(2 53)(3 54)(5 47)(6 48)(8 50)(9 51)(10 28)(11 29)(13 31)(14 32)(16 34)(17 35)(19 45)(20 37)(22 39)(23 40)(25 42)(26 43)

(1 52)(3 54)(4 46)(6 48)(7 49)(9 51)(11 29)(12 30)(14 32)(15 33)(17 35)(18 36)(20 37)(21 38)(23 40)(24 41)(26 43)(27 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 49 50 51 52 53 54)

G:=sub<Sym(54)| (1,33,24,52,15,41)(2,34,25,53,16,42)(3,35,26,54,17,43)(4,36,27,46,18,44)(5,28,19,47,10,45)(6,29,20,48,11,37)(7,30,21,49,12,38)(8,31,22,50,13,39)(9,32,23,51,14,40), (1,4,7)(2,5,8)(3,6,9)(10,13,16)(11,14,17)(12,15,18)(19,22,25)(20,23,26)(21,24,27)(28,31,34)(29,32,35)(30,33,36)(37,40,43)(38,41,44)(39,42,45)(46,49,52)(47,50,53)(48,51,54), (2,53)(3,54)(5,47)(6,48)(8,50)(9,51)(10,28)(11,29)(13,31)(14,32)(16,34)(17,35)(19,45)(20,37)(22,39)(23,40)(25,42)(26,43), (1,52)(3,54)(4,46)(6,48)(7,49)(9,51)(11,29)(12,30)(14,32)(15,33)(17,35)(18,36)(20,37)(21,38)(23,40)(24,41)(26,43)(27,44), (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)(46,47,48,49,50,51,52,53,54)>;

G:=Group( (1,33,24,52,15,41)(2,34,25,53,16,42)(3,35,26,54,17,43)(4,36,27,46,18,44)(5,28,19,47,10,45)(6,29,20,48,11,37)(7,30,21,49,12,38)(8,31,22,50,13,39)(9,32,23,51,14,40), (1,4,7)(2,5,8)(3,6,9)(10,13,16)(11,14,17)(12,15,18)(19,22,25)(20,23,26)(21,24,27)(28,31,34)(29,32,35)(30,33,36)(37,40,43)(38,41,44)(39,42,45)(46,49,52)(47,50,53)(48,51,54), (2,53)(3,54)(5,47)(6,48)(8,50)(9,51)(10,28)(11,29)(13,31)(14,32)(16,34)(17,35)(19,45)(20,37)(22,39)(23,40)(25,42)(26,43), (1,52)(3,54)(4,46)(6,48)(7,49)(9,51)(11,29)(12,30)(14,32)(15,33)(17,35)(18,36)(20,37)(21,38)(23,40)(24,41)(26,43)(27,44), (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)(46,47,48,49,50,51,52,53,54) );

G=PermutationGroup([[(1,33,24,52,15,41),(2,34,25,53,16,42),(3,35,26,54,17,43),(4,36,27,46,18,44),(5,28,19,47,10,45),(6,29,20,48,11,37),(7,30,21,49,12,38),(8,31,22,50,13,39),(9,32,23,51,14,40)], [(1,4,7),(2,5,8),(3,6,9),(10,13,16),(11,14,17),(12,15,18),(19,22,25),(20,23,26),(21,24,27),(28,31,34),(29,32,35),(30,33,36),(37,40,43),(38,41,44),(39,42,45),(46,49,52),(47,50,53),(48,51,54)], [(2,53),(3,54),(5,47),(6,48),(8,50),(9,51),(10,28),(11,29),(13,31),(14,32),(16,34),(17,35),(19,45),(20,37),(22,39),(23,40),(25,42),(26,43)], [(1,52),(3,54),(4,46),(6,48),(7,49),(9,51),(11,29),(12,30),(14,32),(15,33),(17,35),(18,36),(20,37),(21,38),(23,40),(24,41),(26,43),(27,44)], [(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),(46,47,48,49,50,51,52,53,54)]])

C₆xC₃.A₄ is a maximal subgroup of C₆².₁₀Dic₃

72 conjugacy classes

class	1	2A	2B	2C	3A	···	3H	6A	···	6H	6I	···	6X	9A	···	9R	18A	···	18R
order	1	2	2	2	3	···	3	6	···	6	6	···	6	9	···	9	18	···	18
size	1	1	3	3	1	···	1	1	···	1	3	···	3	4	···	4	4	···	4

72 irreducible representations

dim	1	1	1	1	1	1	1	1	3	3	3	3	3	3
type	+	+							+	+
image	C₁	C₂	C₃	C₃	C₆	C₆	C₉	C₁₈	A₄	C₂xA₄	C₃.A₄	C₃xA₄	C₂xC₃.A₄	C₆xA₄
kernel	C₆xC₃.A₄	C₃xC₃.A₄	C₂xC₃.A₄	C₂xC₆²	C₃.A₄	C₆²	C₂²xC₆	C₂xC₆	C₃xC₆	C₃²	C₆	C₆	C₃	C₃
# reps	1	1	6	2	6	2	18	18	1	1	6	2	6	2

Matrix representation of C₆xC₃.A₄ ►in GL₅(F₁₉)

12	0	0	0	0
0	18	0	0	0
0	0	18	0	0
0	0	0	18	0
0	0	0	0	18

1	0	0	0	0
0	7	0	0	0
0	0	1	0	0
0	0	0	1	0
0	0	0	0	1

1	0	0	0	0
0	1	0	0	0
0	0	1	0	0
0	0	0	18	0
0	0	0	0	18

1	0	0	0	0
0	1	0	0	0
0	0	18	0	0
0	0	0	18	0
0	0	0	0	1

1	0	0	0	0
0	9	0	0	0
0	0	0	1	0
0	0	0	0	1
0	0	1	0	0

G:=sub<GL(5,GF(19))| [12,0,0,0,0,0,18,0,0,0,0,0,18,0,0,0,0,0,18,0,0,0,0,0,18],[1,0,0,0,0,0,7,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,18,0,0,0,0,0,18],[1,0,0,0,0,0,1,0,0,0,0,0,18,0,0,0,0,0,18,0,0,0,0,0,1],[1,0,0,0,0,0,9,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,1,0] >;

C₆xC₃.A₄ in GAP, Magma, Sage, TeX

C_6\times C_3.A_4

% in TeX

G:=Group("C6xC3.A4");

// GroupNames label

G:=SmallGroup(216,105);

// by ID

G=gap.SmallGroup(216,105);

# by ID

G:=PCGroup([6,-2,-3,-3,-3,-2,2,115,1630,2927]);

// Polycyclic

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

// generators/relations

G = C6xC3.A4 order 216 = 23·33

Direct product of C6 and C3.A4

G = C₆xC₃.A₄ order 216 = 2³·3³

Direct product of C₆ and C₃.A₄