C6xHe3:C2 - GroupNames

G = C₆xHe₃:C₂ order 324 = 2²·3⁴

Direct product of C₆ and He₃:C₂

direct product, non-abelian, supersoluble, monomial

Aliases: C₆xHe₃:C₂, C₃³:₁₁D₆, (C₆xHe₃):₄C₂, He₃:₆(C₂xC₆), (C₂xHe₃):₅C₆, (C₃²xC₆):₅S₃, C₃²:₄(S₃xC₆), (C₃xHe₃):₉C₂², (C₃xC₆):₂(C₃xS₃), C₃.₆(C₆xC₃:S₃), C₆.₁₄(C₃xC₃:S₃), (C₃xC₆).₂₆(C₃:S₃), C₃².₁₃(C₂xC₃:S₃), SmallGroup(324,145)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₃ — He₃ — C₆xHe₃:C₂

Chief series

C₁ — C₃ — C₃² — He₃ — C₃xHe₃ — C₃xHe₃:C₂ — C₆xHe₃:C₂

Lower central

He₃ — C₆xHe₃:C₂

Upper central

C₁ — C₃xC₆

Generators and relations for C₆xHe₃:C₂
G = < a,b,c,d,e | a⁶=b³=c³=d³=e²=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, dbd^-1=bc^-1, ebe=b^-1, cd=dc, ce=ec, ede=d^-1 >

Subgroups: 592 in 184 conjugacy classes, 38 normal (12 characteristic)
C₁, C₂, C₂, C₃, C₃, C₃, C₂², S₃, C₆, C₆, C₆, C₃², C₃², C₃², D₆, C₂xC₆, C₃xS₃, C₃xC₆, C₃xC₆, C₃xC₆, He₃, He₃, C₃³, S₃xC₆, C₆², He₃:C₂, C₂xHe₃, C₂xHe₃, S₃xC₃², C₃²xC₆, C₃xHe₃, C₂xHe₃:C₂, S₃xC₃xC₆, C₃xHe₃:C₂, C₆xHe₃, C₆xHe₃:C₂
Quotients: C₁, C₂, C₃, C₂², S₃, C₆, D₆, C₂xC₆, C₃xS₃, C₃:S₃, S₃xC₆, C₂xC₃:S₃, He₃:C₂, C₃xC₃:S₃, C₂xHe₃:C₂, C₆xC₃:S₃, C₃xHe₃:C₂, C₆xHe₃:C₂

Smallest permutation representation of C₆xHe₃:C₂
►On 54 points

Generators in S₅₄

(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)

(7 42 33)(8 37 34)(9 38 35)(10 39 36)(11 40 31)(12 41 32)(13 52 21)(14 53 22)(15 54 23)(16 49 24)(17 50 19)(18 51 20)

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

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

(7 14)(8 15)(9 16)(10 17)(11 18)(12 13)(19 39)(20 40)(21 41)(22 42)(23 37)(24 38)(31 51)(32 52)(33 53)(34 54)(35 49)(36 50)

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

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

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

60 conjugacy classes

class	1	2A	2B	2C	3A	···	3H	3I	···	3T	6A	···	6H	6I	···	6T	6U	···	6AJ
order	1	2	2	2	3	···	3	3	···	3	6	···	6	6	···	6	6	···	6
size	1	1	9	9	1	···	1	6	···	6	1	···	1	6	···	6	9	···	9

60 irreducible representations

dim	1	1	1	1	1	1	2	2	2	2	3	3
type	+	+	+				+	+
image	C₁	C₂	C₂	C₃	C₆	C₆	S₃	D₆	C₃xS₃	S₃xC₆	He₃:C₂	C₂xHe₃:C₂
kernel	C₆xHe₃:C₂	C₃xHe₃:C₂	C₆xHe₃	C₂xHe₃:C₂	He₃:C₂	C₂xHe₃	C₃²xC₆	C₃³	C₃xC₆	C₃²	C₆	C₃
# reps	1	2	1	2	4	2	4	4	8	8	12	12

Matrix representation of C₆xHe₃:C₂ ►in GL₅(F₇)

3	0	0	0	0
0	3	0	0	0
0	0	1	0	0
0	0	0	1	0
0	0	0	0	1

6	6	0	0	0
1	0	0	0	0
0	0	1	0	0
0	0	0	2	0
0	0	0	0	4

1	0	0	0	0
0	1	0	0	0
0	0	2	0	0
0	0	0	2	0
0	0	0	0	2

1	0	0	0	0
0	1	0	0	0
0	0	0	1	0
0	0	0	0	1
0	0	1	0	0

1	0	0	0	0
6	6	0	0	0
0	0	1	0	0
0	0	0	0	1
0	0	0	1	0

G:=sub<GL(5,GF(7))| [3,0,0,0,0,0,3,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[6,1,0,0,0,6,0,0,0,0,0,0,1,0,0,0,0,0,2,0,0,0,0,0,4],[1,0,0,0,0,0,1,0,0,0,0,0,2,0,0,0,0,0,2,0,0,0,0,0,2],[1,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,1,0],[1,6,0,0,0,0,6,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,1,0] >;

C₆xHe₃:C₂ in GAP, Magma, Sage, TeX

C_6\times {\rm He}_3\rtimes C_2

% in TeX

G:=Group("C6xHe3:C2");

// GroupNames label

G:=SmallGroup(324,145);

// by ID

G=gap.SmallGroup(324,145);

# by ID

G:=PCGroup([6,-2,-2,-3,-3,-3,-3,579,2164,382]);

// Polycyclic

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

// generators/relations

G = C6xHe3:C2 order 324 = 22·34

Direct product of C6 and He3:C2

G = C₆xHe₃:C₂ order 324 = 2²·3⁴

Direct product of C₆ and He₃:C₂