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

 Derived series C1 — C32 — C2×He3.C3
 Chief series C1 — C3 — C32 — C3×C9 — He3.C3 — C2×He3.C3
 Lower central C1 — C3 — C32 — C2×He3.C3
 Upper central C1 — C6 — C3×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 >

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 3A 3B 3C 3D 3E 3F 6A 6B 6C 6D 6E 6F 9A ··· 9F 9G 9H 9I 9J 18A ··· 18F 18G 18H 18I 18J order 1 2 3 3 3 3 3 3 6 6 6 6 6 6 9 ··· 9 9 9 9 9 18 ··· 18 18 18 18 18 size 1 1 1 1 3 3 9 9 1 1 3 3 9 9 3 ··· 3 9 9 9 9 3 ··· 3 9 9 9 9

34 irreducible representations

 dim 1 1 1 1 1 1 1 1 3 3 3 3 type + + image C1 C2 C3 C3 C3 C6 C6 C6 He3 C2×He3 He3.C3 C2×He3.C3 kernel C2×He3.C3 He3.C3 C3×C18 C2×He3 C2×3- 1+2 C3×C9 He3 3- 1+2 C6 C3 C2 C1 # reps 1 1 2 2 4 2 2 4 2 2 6 6

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

 18 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1
,
 1 0 0 0 0 0 1 0 0 0 0 1 0 1 0 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 18 12 0 12 12 18 0 18 12 12
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

׿
×
𝔽