direct product, metabelian, nilpotent (class 3), monomial, 3-elementary
Aliases: C2×He3.C3, C6.3He3, He3.3C6, 3- 1+2⋊2C6, (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
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