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 |
Subgroups: 573 in 195 conjugacy classes, 49 normal (13 characteristic)
C1, C2, C2 [×7], C3, C4, C4 [×3], C4 [×4], C22 [×3], C22 [×9], C6 [×4], C2×C4 [×3], C2×C4 [×19], D4 [×3], D4 [×19], Q8 [×2], Q8 [×6], C23 [×5], C12 [×4], C2×C6 [×3], C22×C4 [×5], C2×D4 [×15], C2×Q8 [×3], C2×Q8 [×4], C4○D4 [×2], C4○D4 [×3], C4○D4 [×25], SL2(𝔽3), C2×C12 [×3], C3×D4 [×3], C3×Q8, C2×C4○D4 [×3], C2×C4○D4 [×4], 2+ (1+4), 2+ (1+4) [×3], 2- (1+4) [×3], 2- (1+4), C2×SL2(𝔽3) [×3], C4.A4, C4.A4 [×3], C3×C4○D4, C2.C25, C2×C4.A4 [×3], Q8.A4, D4.A4 [×3], 2- (1+4)⋊3C6
Quotients:
C1, C2 [×7], C3, C22 [×7], C6 [×7], C23, A4, C2×C6 [×7], C2×A4 [×7], C22×C6, C22×A4 [×7], C23×A4, 2- (1+4)⋊3C6
Generators and relations
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 >
►On 32 points
(1 2 3 4)(5 8 6 7)(9 16 13 20)(10 17 14 18)(11 15 12 19)(21 29 24 32)(22 30 25 27)(23 31 26 28)
(1 5)(2 7)(3 6)(4 8)(9 26)(10 24)(11 22)(12 25)(13 23)(14 21)(15 27)(16 31)(17 29)(18 32)(19 30)(20 28)
(1 16 3 20)(2 13 4 9)(5 31 6 28)(7 23 8 26)(10 12 14 11)(15 17 19 18)(21 22 24 25)(27 29 30 32)
(1 17 3 18)(2 14 4 10)(5 29 6 32)(7 21 8 24)(9 11 13 12)(15 20 19 16)(22 23 25 26)(27 28 30 31)
(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)| (1,2,3,4)(5,8,6,7)(9,16,13,20)(10,17,14,18)(11,15,12,19)(21,29,24,32)(22,30,25,27)(23,31,26,28), (1,5)(2,7)(3,6)(4,8)(9,26)(10,24)(11,22)(12,25)(13,23)(14,21)(15,27)(16,31)(17,29)(18,32)(19,30)(20,28), (1,16,3,20)(2,13,4,9)(5,31,6,28)(7,23,8,26)(10,12,14,11)(15,17,19,18)(21,22,24,25)(27,29,30,32), (1,17,3,18)(2,14,4,10)(5,29,6,32)(7,21,8,24)(9,11,13,12)(15,20,19,16)(22,23,25,26)(27,28,30,31), (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( (1,2,3,4)(5,8,6,7)(9,16,13,20)(10,17,14,18)(11,15,12,19)(21,29,24,32)(22,30,25,27)(23,31,26,28), (1,5)(2,7)(3,6)(4,8)(9,26)(10,24)(11,22)(12,25)(13,23)(14,21)(15,27)(16,31)(17,29)(18,32)(19,30)(20,28), (1,16,3,20)(2,13,4,9)(5,31,6,28)(7,23,8,26)(10,12,14,11)(15,17,19,18)(21,22,24,25)(27,29,30,32), (1,17,3,18)(2,14,4,10)(5,29,6,32)(7,21,8,24)(9,11,13,12)(15,20,19,16)(22,23,25,26)(27,28,30,31), (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([(1,2,3,4),(5,8,6,7),(9,16,13,20),(10,17,14,18),(11,15,12,19),(21,29,24,32),(22,30,25,27),(23,31,26,28)], [(1,5),(2,7),(3,6),(4,8),(9,26),(10,24),(11,22),(12,25),(13,23),(14,21),(15,27),(16,31),(17,29),(18,32),(19,30),(20,28)], [(1,16,3,20),(2,13,4,9),(5,31,6,28),(7,23,8,26),(10,12,14,11),(15,17,19,18),(21,22,24,25),(27,29,30,32)], [(1,17,3,18),(2,14,4,10),(5,29,6,32),(7,21,8,24),(9,11,13,12),(15,20,19,16),(22,23,25,26),(27,28,30,31)], [(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)])
Matrix representation ►G ⊆ 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] >;
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 |
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