Copied to
clipboard

G = C2×He3.C3order 162 = 2·34

Direct product of C2 and He3.C3

direct product, metabelian, nilpotent (class 3), monomial, 3-elementary

Aliases: C2×He3.C3, C6.3He3, He3.3C6, 3- 1+22C6, (C3×C9)⋊9C6, (C3×C18)⋊2C3, (C2×He3).C3, C3.3(C2×He3), (C3×C6).2C32, C32.2(C3×C6), (C2×3- 1+2)⋊2C3, SmallGroup(162,29)

Series: Derived Chief Lower central Upper central

C1C32 — C2×He3.C3
C1C3C32C3×C9He3.C3 — C2×He3.C3
C1C3C32 — C2×He3.C3
C1C6C3×C6 — C2×He3.C3

Generators and relations for C2×He3.C3
 G = < a,b,c,d,e | a2=b3=c3=d3=1, e3=c, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, dbd-1=bc-1, be=eb, cd=dc, ce=ec, ede-1=bc-1d >

3C3
9C3
3C6
9C6
3C9
3C9
3C32
3C9
3C18
3C18
3C3×C6
3C18

Smallest permutation representation of C2×He3.C3
On 54 points
Generators in S54
(1 10)(2 11)(3 12)(4 13)(5 14)(6 15)(7 16)(8 17)(9 18)(19 43)(20 44)(21 45)(22 37)(23 38)(24 39)(25 40)(26 41)(27 42)(28 48)(29 49)(30 50)(31 51)(32 52)(33 53)(34 54)(35 46)(36 47)
(1 44 28)(2 45 29)(3 37 30)(4 38 31)(5 39 32)(6 40 33)(7 41 34)(8 42 35)(9 43 36)(10 20 48)(11 21 49)(12 22 50)(13 23 51)(14 24 52)(15 25 53)(16 26 54)(17 27 46)(18 19 47)
(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 45 32)(3 30 43)(5 39 35)(6 33 37)(8 42 29)(9 36 40)(11 21 52)(12 50 19)(14 24 46)(15 53 22)(17 27 49)(18 47 25)(20 23 26)(28 34 31)(38 41 44)(48 54 51)
(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)| (1,10)(2,11)(3,12)(4,13)(5,14)(6,15)(7,16)(8,17)(9,18)(19,43)(20,44)(21,45)(22,37)(23,38)(24,39)(25,40)(26,41)(27,42)(28,48)(29,49)(30,50)(31,51)(32,52)(33,53)(34,54)(35,46)(36,47), (1,44,28)(2,45,29)(3,37,30)(4,38,31)(5,39,32)(6,40,33)(7,41,34)(8,42,35)(9,43,36)(10,20,48)(11,21,49)(12,22,50)(13,23,51)(14,24,52)(15,25,53)(16,26,54)(17,27,46)(18,19,47), (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), (2,45,32)(3,30,43)(5,39,35)(6,33,37)(8,42,29)(9,36,40)(11,21,52)(12,50,19)(14,24,46)(15,53,22)(17,27,49)(18,47,25)(20,23,26)(28,34,31)(38,41,44)(48,54,51), (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( (1,10)(2,11)(3,12)(4,13)(5,14)(6,15)(7,16)(8,17)(9,18)(19,43)(20,44)(21,45)(22,37)(23,38)(24,39)(25,40)(26,41)(27,42)(28,48)(29,49)(30,50)(31,51)(32,52)(33,53)(34,54)(35,46)(36,47), (1,44,28)(2,45,29)(3,37,30)(4,38,31)(5,39,32)(6,40,33)(7,41,34)(8,42,35)(9,43,36)(10,20,48)(11,21,49)(12,22,50)(13,23,51)(14,24,52)(15,25,53)(16,26,54)(17,27,46)(18,19,47), (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), (2,45,32)(3,30,43)(5,39,35)(6,33,37)(8,42,29)(9,36,40)(11,21,52)(12,50,19)(14,24,46)(15,53,22)(17,27,49)(18,47,25)(20,23,26)(28,34,31)(38,41,44)(48,54,51), (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([[(1,10),(2,11),(3,12),(4,13),(5,14),(6,15),(7,16),(8,17),(9,18),(19,43),(20,44),(21,45),(22,37),(23,38),(24,39),(25,40),(26,41),(27,42),(28,48),(29,49),(30,50),(31,51),(32,52),(33,53),(34,54),(35,46),(36,47)], [(1,44,28),(2,45,29),(3,37,30),(4,38,31),(5,39,32),(6,40,33),(7,41,34),(8,42,35),(9,43,36),(10,20,48),(11,21,49),(12,22,50),(13,23,51),(14,24,52),(15,25,53),(16,26,54),(17,27,46),(18,19,47)], [(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)], [(2,45,32),(3,30,43),(5,39,35),(6,33,37),(8,42,29),(9,36,40),(11,21,52),(12,50,19),(14,24,46),(15,53,22),(17,27,49),(18,47,25),(20,23,26),(28,34,31),(38,41,44),(48,54,51)], [(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)]])

C2×He3.C3 is a maximal subgroup of   He3.C12  He3.Dic3  He3.3Dic3

34 conjugacy classes

class 1  2 3A3B3C3D3E3F6A6B6C6D6E6F9A···9F9G9H9I9J18A···18F18G18H18I18J
order123333336666669···9999918···1818181818
size111133991133993···399993···39999

34 irreducible representations

dim111111113333
type++
imageC1C2C3C3C3C6C6C6He3C2×He3He3.C3C2×He3.C3
kernelC2×He3.C3He3.C3C3×C18C2×He3C2×3- 1+2C3×C9He33- 1+2C6C3C2C1
# reps112242242266

Matrix representation of C2×He3.C3 in GL4(𝔽19) generated by

18000
0100
0010
0001
,
1000
0010
0001
0100
,
1000
0700
0070
0007
,
7000
0100
00110
0007
,
1000
0121812
0121218
0181212
G:=sub<GL(4,GF(19))| [18,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,1,0,0,0,0,1,0],[1,0,0,0,0,7,0,0,0,0,7,0,0,0,0,7],[7,0,0,0,0,1,0,0,0,0,11,0,0,0,0,7],[1,0,0,0,0,12,12,18,0,18,12,12,0,12,18,12] >;

C2×He3.C3 in GAP, Magma, Sage, TeX

C_2\times {\rm He}_3.C_3
% in TeX

G:=Group("C2xHe3.C3");
// GroupNames label

G:=SmallGroup(162,29);
// by ID

G=gap.SmallGroup(162,29);
# by ID

G:=PCGroup([5,-2,-3,-3,-3,-3,187,147,728]);
// Polycyclic

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

Export

Subgroup lattice of C2×He3.C3 in TeX

׿
×
𝔽