direct product, metabelian, nilpotent (class 2), monomial
Aliases: C22xHe3, C62:2C3, C3.1C62, (C3xC6):2C6, C6.4(C3xC6), C32:3(C2xC6), (C2xC6).6C32, SmallGroup(108,30)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C22xHe3
G = < a,b,c,d,e | a2=b2=c3=d3=e3=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ece-1=cd-1, de=ed >
Subgroups: 95 in 55 conjugacy classes, 35 normal (6 characteristic)
C1, C2, C3, C3, C22, C6, C6, C32, C2xC6, C2xC6, C3xC6, He3, C62, C2xHe3, C22xHe3
Quotients: C1, C2, C3, C22, C6, C32, C2xC6, C3xC6, He3, C62, C2xHe3, C22xHe3
►On 36 points
(1 25)(2 26)(3 27)(4 21)(5 19)(6 20)(7 24)(8 22)(9 23)(10 28)(11 29)(12 30)(13 31)(14 32)(15 33)(16 34)(17 35)(18 36)
(1 16)(2 17)(3 18)(4 30)(5 28)(6 29)(7 33)(8 31)(9 32)(10 19)(11 20)(12 21)(13 22)(14 23)(15 24)(25 34)(26 35)(27 36)
(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)
(1 14 10)(2 15 11)(3 13 12)(4 36 8)(5 34 9)(6 35 7)(16 23 19)(17 24 20)(18 22 21)(25 32 28)(26 33 29)(27 31 30)
(1 3 11)(2 14 13)(4 7 5)(6 34 36)(8 35 9)(10 12 15)(16 18 20)(17 23 22)(19 21 24)(25 27 29)(26 32 31)(28 30 33)
G:=sub<Sym(36)| (1,25)(2,26)(3,27)(4,21)(5,19)(6,20)(7,24)(8,22)(9,23)(10,28)(11,29)(12,30)(13,31)(14,32)(15,33)(16,34)(17,35)(18,36), (1,16)(2,17)(3,18)(4,30)(5,28)(6,29)(7,33)(8,31)(9,32)(10,19)(11,20)(12,21)(13,22)(14,23)(15,24)(25,34)(26,35)(27,36), (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), (1,14,10)(2,15,11)(3,13,12)(4,36,8)(5,34,9)(6,35,7)(16,23,19)(17,24,20)(18,22,21)(25,32,28)(26,33,29)(27,31,30), (1,3,11)(2,14,13)(4,7,5)(6,34,36)(8,35,9)(10,12,15)(16,18,20)(17,23,22)(19,21,24)(25,27,29)(26,32,31)(28,30,33)>;
G:=Group( (1,25)(2,26)(3,27)(4,21)(5,19)(6,20)(7,24)(8,22)(9,23)(10,28)(11,29)(12,30)(13,31)(14,32)(15,33)(16,34)(17,35)(18,36), (1,16)(2,17)(3,18)(4,30)(5,28)(6,29)(7,33)(8,31)(9,32)(10,19)(11,20)(12,21)(13,22)(14,23)(15,24)(25,34)(26,35)(27,36), (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), (1,14,10)(2,15,11)(3,13,12)(4,36,8)(5,34,9)(6,35,7)(16,23,19)(17,24,20)(18,22,21)(25,32,28)(26,33,29)(27,31,30), (1,3,11)(2,14,13)(4,7,5)(6,34,36)(8,35,9)(10,12,15)(16,18,20)(17,23,22)(19,21,24)(25,27,29)(26,32,31)(28,30,33) );
G=PermutationGroup([[(1,25),(2,26),(3,27),(4,21),(5,19),(6,20),(7,24),(8,22),(9,23),(10,28),(11,29),(12,30),(13,31),(14,32),(15,33),(16,34),(17,35),(18,36)], [(1,16),(2,17),(3,18),(4,30),(5,28),(6,29),(7,33),(8,31),(9,32),(10,19),(11,20),(12,21),(13,22),(14,23),(15,24),(25,34),(26,35),(27,36)], [(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)], [(1,14,10),(2,15,11),(3,13,12),(4,36,8),(5,34,9),(6,35,7),(16,23,19),(17,24,20),(18,22,21),(25,32,28),(26,33,29),(27,31,30)], [(1,3,11),(2,14,13),(4,7,5),(6,34,36),(8,35,9),(10,12,15),(16,18,20),(17,23,22),(19,21,24),(25,27,29),(26,32,31),(28,30,33)]])
C22xHe3 is a maximal subgroup of
He3:6D4 He3:7D4 He3.A4 He3:A4 He3:2A4 He3.2A4
44 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 3C | ··· | 3J | 6A | ··· | 6F | 6G | ··· | 6AD |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 3 | ··· | 3 | 6 | ··· | 6 | 6 | ··· | 6 |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 3 | ··· | 3 | 1 | ··· | 1 | 3 | ··· | 3 |
44 irreducible representations
| dim | 1 | 1 | 1 | 1 | 3 | 3 |
| type | + | + | ||||
| image | C1 | C2 | C3 | C6 | He3 | C2xHe3 |
| kernel | C22xHe3 | C2xHe3 | C62 | C3xC6 | C22 | C2 |
| # reps | 1 | 3 | 8 | 24 | 2 | 6 |
Matrix representation of C22xHe3 ►in GL4(F7) generated by
| 6 | 0 | 0 | 0 |
| 0 | 6 | 0 | 0 |
| 0 | 0 | 6 | 0 |
| 0 | 0 | 0 | 6 |
| 1 | 0 | 0 | 0 |
| 0 | 6 | 0 | 0 |
| 0 | 0 | 6 | 0 |
| 0 | 0 | 0 | 6 |
| 2 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 5 | 3 | 3 |
| 0 | 0 | 0 | 4 |
| 1 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 |
| 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 4 |
| 2 | 0 | 0 | 0 |
| 0 | 6 | 5 | 5 |
| 0 | 1 | 0 | 0 |
| 0 | 3 | 4 | 1 |
G:=sub<GL(4,GF(7))| [6,0,0,0,0,6,0,0,0,0,6,0,0,0,0,6],[1,0,0,0,0,6,0,0,0,0,6,0,0,0,0,6],[2,0,0,0,0,0,5,0,0,1,3,0,0,0,3,4],[1,0,0,0,0,4,0,0,0,0,4,0,0,0,0,4],[2,0,0,0,0,6,1,3,0,5,0,4,0,5,0,1] >;
C22xHe3 in GAP, Magma, Sage, TeX
C_2^2\times {\rm He}_3 % in TeX
G:=Group("C2^2xHe3"); // GroupNames label
G:=SmallGroup(108,30);
// by ID
G=gap.SmallGroup(108,30);
# by ID
G:=PCGroup([5,-2,-2,-3,-3,-3,253]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^2=c^3=d^3=e^3=1,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,c*d=d*c,e*c*e^-1=c*d^-1,d*e=e*d>;
// generators/relations