Aliases: 2- 1+4⋊3C6, 2+ 1+4⋊4C6, SL2(𝔽3).11C23, C4○D4.A4, D4.A4⋊6C2, D4.5(C2×A4), Q8.A4⋊6C2, Q8.7(C2×A4), C4.A4⋊8C22, C2.9(C23×A4), C4.11(C22×A4), C2.C25⋊1C3, Q8.4(C22×C6), C22.9(C22×A4), (C2×SL2(𝔽3))⋊3C22, (C2×Q8).(C2×C6), (C2×C4.A4)⋊9C2, (C2×C4○D4)⋊3C6, (C2×C4).14(C2×A4), C4○D4.5(C2×C6), SmallGroup(192,1504)
Series: Derived ►Chief ►Lower central ►Upper central
| Q8 — 2- 1+4⋊3C6 |
Generators and relations for 2- 1+4⋊3C6
G = < a,b,c,d,e | a4=b2=e6=1, c2=d2=a2, bab=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, ebe-1=a2b, dcd-1=a2c, ece-1=cd, ede-1=c >
Subgroups: 573 in 195 conjugacy classes, 49 normal (13 characteristic)
C1, C2, C2, C3, C4, C4, C4, C22, C22, C6, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, C12, C2×C6, C22×C4, C2×D4, C2×Q8, C2×Q8, C4○D4, C4○D4, C4○D4, SL2(𝔽3), C2×C12, C3×D4, C3×Q8, C2×C4○D4, C2×C4○D4, 2+ 1+4, 2+ 1+4, 2- 1+4, 2- 1+4, C2×SL2(𝔽3), C4.A4, C4.A4, C3×C4○D4, C2.C25, C2×C4.A4, Q8.A4, D4.A4, 2- 1+4⋊3C6
Quotients: C1, C2, C3, C22, C6, C23, A4, C2×C6, C2×A4, C22×C6, C22×A4, C23×A4, 2- 1+4⋊3C6
►On 32 points
(1 2 4 3)(5 8 6 7)(9 14 19 16)(10 12 20 17)(11 13 18 15)(21 27 24 30)(22 28 25 31)(23 29 26 32)
(1 7)(2 6)(3 5)(4 8)(9 31)(10 29)(11 27)(12 23)(13 21)(14 25)(15 24)(16 22)(17 26)(18 30)(19 28)(20 32)
(1 20 4 10)(2 17 3 12)(5 23 6 26)(7 32 8 29)(9 11 19 18)(13 16 15 14)(21 22 24 25)(27 28 30 31)
(1 18 4 11)(2 15 3 13)(5 21 6 24)(7 30 8 27)(9 20 19 10)(12 14 17 16)(22 23 25 26)(28 29 31 32)
(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)
G:=sub<Sym(32)| (1,2,4,3)(5,8,6,7)(9,14,19,16)(10,12,20,17)(11,13,18,15)(21,27,24,30)(22,28,25,31)(23,29,26,32), (1,7)(2,6)(3,5)(4,8)(9,31)(10,29)(11,27)(12,23)(13,21)(14,25)(15,24)(16,22)(17,26)(18,30)(19,28)(20,32), (1,20,4,10)(2,17,3,12)(5,23,6,26)(7,32,8,29)(9,11,19,18)(13,16,15,14)(21,22,24,25)(27,28,30,31), (1,18,4,11)(2,15,3,13)(5,21,6,24)(7,30,8,27)(9,20,19,10)(12,14,17,16)(22,23,25,26)(28,29,31,32), (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)>;
G:=Group( (1,2,4,3)(5,8,6,7)(9,14,19,16)(10,12,20,17)(11,13,18,15)(21,27,24,30)(22,28,25,31)(23,29,26,32), (1,7)(2,6)(3,5)(4,8)(9,31)(10,29)(11,27)(12,23)(13,21)(14,25)(15,24)(16,22)(17,26)(18,30)(19,28)(20,32), (1,20,4,10)(2,17,3,12)(5,23,6,26)(7,32,8,29)(9,11,19,18)(13,16,15,14)(21,22,24,25)(27,28,30,31), (1,18,4,11)(2,15,3,13)(5,21,6,24)(7,30,8,27)(9,20,19,10)(12,14,17,16)(22,23,25,26)(28,29,31,32), (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) );
G=PermutationGroup([[(1,2,4,3),(5,8,6,7),(9,14,19,16),(10,12,20,17),(11,13,18,15),(21,27,24,30),(22,28,25,31),(23,29,26,32)], [(1,7),(2,6),(3,5),(4,8),(9,31),(10,29),(11,27),(12,23),(13,21),(14,25),(15,24),(16,22),(17,26),(18,30),(19,28),(20,32)], [(1,20,4,10),(2,17,3,12),(5,23,6,26),(7,32,8,29),(9,11,19,18),(13,16,15,14),(21,22,24,25),(27,28,30,31)], [(1,18,4,11),(2,15,3,13),(5,21,6,24),(7,30,8,27),(9,20,19,10),(12,14,17,16),(22,23,25,26),(28,29,31,32)], [(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)]])
38 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 3A | 3B | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 6A | 6B | 6C | ··· | 6H | 12A | 12B | 12C | 12D | 12E | ··· | 12J |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | ··· | 6 | 12 | 12 | 12 | 12 | 12 | ··· | 12 |
| size | 1 | 1 | 2 | 2 | 2 | 6 | 6 | 6 | 6 | 4 | 4 | 1 | 1 | 2 | 2 | 2 | 6 | 6 | 6 | 6 | 4 | 4 | 8 | ··· | 8 | 4 | 4 | 4 | 4 | 8 | ··· | 8 |
38 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 3 | 3 | 3 | 3 | 4 |
| type | + | + | + | + | + | + | + | + | |||||
| image | C1 | C2 | C2 | C2 | C3 | C6 | C6 | C6 | A4 | C2×A4 | C2×A4 | C2×A4 | 2- 1+4⋊3C6 |
| kernel | 2- 1+4⋊3C6 | C2×C4.A4 | Q8.A4 | D4.A4 | C2.C25 | C2×C4○D4 | 2+ 1+4 | 2- 1+4 | C4○D4 | C2×C4 | D4 | Q8 | C1 |
| # reps | 1 | 3 | 1 | 3 | 2 | 6 | 2 | 6 | 1 | 3 | 3 | 1 | 6 |
Matrix representation of 2- 1+4⋊3C6 ►in GL4(𝔽5) generated by
| 0 | 3 | 0 | 0 |
| 3 | 0 | 0 | 0 |
| 0 | 2 | 0 | 3 |
| 3 | 0 | 3 | 0 |
| 0 | 2 | 0 | 0 |
| 3 | 0 | 0 | 0 |
| 0 | 3 | 0 | 2 |
| 3 | 0 | 3 | 0 |
| 2 | 0 | 2 | 0 |
| 0 | 0 | 0 | 2 |
| 0 | 0 | 3 | 0 |
| 0 | 2 | 0 | 0 |
| 2 | 0 | 0 | 0 |
| 0 | 2 | 0 | 0 |
| 1 | 0 | 3 | 0 |
| 0 | 0 | 0 | 3 |
| 0 | 2 | 0 | 2 |
| 4 | 0 | 2 | 0 |
| 0 | 4 | 0 | 2 |
| 0 | 0 | 4 | 0 |
G:=sub<GL(4,GF(5))| [0,3,0,3,3,0,2,0,0,0,0,3,0,0,3,0],[0,3,0,3,2,0,3,0,0,0,0,3,0,0,2,0],[2,0,0,0,0,0,0,2,2,0,3,0,0,2,0,0],[2,0,1,0,0,2,0,0,0,0,3,0,0,0,0,3],[0,4,0,0,2,0,4,0,0,2,0,4,2,0,2,0] >;
2- 1+4⋊3C6 in GAP, Magma, Sage, TeX
2_-^{1+4}\rtimes_3C_6 % in TeX
G:=Group("ES-(2,2):3C6"); // GroupNames label
G:=SmallGroup(192,1504);
// by ID
G=gap.SmallGroup(192,1504);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-2,2,-2,680,2102,235,172,404,285,124]);
// Polycyclic
G:=Group<a,b,c,d,e|a^4=b^2=e^6=1,c^2=d^2=a^2,b*a*b=a^-1,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,e*b*e^-1=a^2*b,d*c*d^-1=a^2*c,e*c*e^-1=c*d,e*d*e^-1=c>;
// generators/relations