p-group, metabelian, nilpotent (class 3), monomial
Aliases: C24.19Q8, (C2×C8).356D4, (C23×C8).17C2, (C22×C8).35C4, C23.79(C2×Q8), (C22×C4).78Q8, C8.44(C22⋊C4), C23.69(C4⋊C4), (C22×C4).546D4, C4.182(C4⋊D4), C4.C42⋊18C2, C22⋊1(C8.C4), C4.55(C42⋊C2), C24.4C4.16C2, (C23×C4).672C22, (C22×C8).549C22, C22.22(C22⋊Q8), (C22×C4).1332C23, C2.19(C23.7Q8), (C2×M4(2)).154C22, (C2×C8.C4)⋊1C2, (C2×C4).85(C4⋊C4), (C2×C8).210(C2×C4), C4.89(C2×C22⋊C4), C22.93(C2×C4⋊C4), C2.11(C2×C8.C4), (C2×C4).1515(C2×D4), (C2×C4).554(C4○D4), (C22×C4).482(C2×C4), (C2×C4).530(C22×C4), SmallGroup(128,542)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C24.19Q8
G = < a,b,c,d,e,f | a2=b2=c2=d2=1, e4=d, f2=bde2, ab=ba, faf-1=ac=ca, ad=da, ae=ea, bc=cb, fbf-1=bd=db, be=eb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=e3 >
Subgroups: 236 in 146 conjugacy classes, 64 normal (18 characteristic)
C1, C2, C2, C2, C4, C4, C4, C22, C22, C22, C8, C8, C2×C4, C2×C4, C2×C4, C23, C23, C23, C2×C8, C2×C8, C2×C8, M4(2), C22×C4, C22×C4, C22×C4, C24, C22⋊C8, C8.C4, C22×C8, C22×C8, C22×C8, C2×M4(2), C23×C4, C4.C42, C24.4C4, C2×C8.C4, C23×C8, C24.19Q8
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C23, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C2×Q8, C4○D4, C8.C4, C2×C22⋊C4, C2×C4⋊C4, C42⋊C2, C4⋊D4, C22⋊Q8, C23.7Q8, C2×C8.C4, C24.19Q8
►On 32 points
(9 28)(10 29)(11 30)(12 31)(13 32)(14 25)(15 26)(16 27)
(1 17)(2 18)(3 19)(4 20)(5 21)(6 22)(7 23)(8 24)(9 28)(10 29)(11 30)(12 31)(13 32)(14 25)(15 26)(16 27)
(1 21)(2 22)(3 23)(4 24)(5 17)(6 18)(7 19)(8 20)(9 28)(10 29)(11 30)(12 31)(13 32)(14 25)(15 26)(16 27)
(1 5)(2 6)(3 7)(4 8)(9 13)(10 14)(11 15)(12 16)(17 21)(18 22)(19 23)(20 24)(25 29)(26 30)(27 31)(28 32)
(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)
(1 30 23 9 5 26 19 13)(2 25 24 12 6 29 20 16)(3 28 17 15 7 32 21 11)(4 31 18 10 8 27 22 14)
G:=sub<Sym(32)| (9,28)(10,29)(11,30)(12,31)(13,32)(14,25)(15,26)(16,27), (1,17)(2,18)(3,19)(4,20)(5,21)(6,22)(7,23)(8,24)(9,28)(10,29)(11,30)(12,31)(13,32)(14,25)(15,26)(16,27), (1,21)(2,22)(3,23)(4,24)(5,17)(6,18)(7,19)(8,20)(9,28)(10,29)(11,30)(12,31)(13,32)(14,25)(15,26)(16,27), (1,5)(2,6)(3,7)(4,8)(9,13)(10,14)(11,15)(12,16)(17,21)(18,22)(19,23)(20,24)(25,29)(26,30)(27,31)(28,32), (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), (1,30,23,9,5,26,19,13)(2,25,24,12,6,29,20,16)(3,28,17,15,7,32,21,11)(4,31,18,10,8,27,22,14)>;
G:=Group( (9,28)(10,29)(11,30)(12,31)(13,32)(14,25)(15,26)(16,27), (1,17)(2,18)(3,19)(4,20)(5,21)(6,22)(7,23)(8,24)(9,28)(10,29)(11,30)(12,31)(13,32)(14,25)(15,26)(16,27), (1,21)(2,22)(3,23)(4,24)(5,17)(6,18)(7,19)(8,20)(9,28)(10,29)(11,30)(12,31)(13,32)(14,25)(15,26)(16,27), (1,5)(2,6)(3,7)(4,8)(9,13)(10,14)(11,15)(12,16)(17,21)(18,22)(19,23)(20,24)(25,29)(26,30)(27,31)(28,32), (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), (1,30,23,9,5,26,19,13)(2,25,24,12,6,29,20,16)(3,28,17,15,7,32,21,11)(4,31,18,10,8,27,22,14) );
G=PermutationGroup([[(9,28),(10,29),(11,30),(12,31),(13,32),(14,25),(15,26),(16,27)], [(1,17),(2,18),(3,19),(4,20),(5,21),(6,22),(7,23),(8,24),(9,28),(10,29),(11,30),(12,31),(13,32),(14,25),(15,26),(16,27)], [(1,21),(2,22),(3,23),(4,24),(5,17),(6,18),(7,19),(8,20),(9,28),(10,29),(11,30),(12,31),(13,32),(14,25),(15,26),(16,27)], [(1,5),(2,6),(3,7),(4,8),(9,13),(10,14),(11,15),(12,16),(17,21),(18,22),(19,23),(20,24),(25,29),(26,30),(27,31),(28,32)], [(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)], [(1,30,23,9,5,26,19,13),(2,25,24,12,6,29,20,16),(3,28,17,15,7,32,21,11),(4,31,18,10,8,27,22,14)]])
44 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | ··· | 2I | 4A | 4B | 4C | 4D | 4E | ··· | 4J | 8A | ··· | 8P | 8Q | ··· | 8X |
| order | 1 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 8 | ··· | 8 | 8 | ··· | 8 |
| size | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 2 | ··· | 2 | 8 | ··· | 8 |
44 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | - | - | |||
| image | C1 | C2 | C2 | C2 | C2 | C4 | D4 | D4 | Q8 | Q8 | C4○D4 | C8.C4 |
| kernel | C24.19Q8 | C4.C42 | C24.4C4 | C2×C8.C4 | C23×C8 | C22×C8 | C2×C8 | C22×C4 | C22×C4 | C24 | C2×C4 | C22 |
| # reps | 1 | 2 | 2 | 2 | 1 | 8 | 4 | 2 | 1 | 1 | 4 | 16 |
Matrix representation of C24.19Q8 ►in GL4(𝔽17) generated by
| 1 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 15 | 16 |
| 1 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 16 |
| 16 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 16 |
| 16 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 8 | 0 | 0 | 0 |
| 0 | 2 | 0 | 0 |
| 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 16 |
| 0 | 1 | 0 | 0 |
| 4 | 0 | 0 | 0 |
| 0 | 0 | 16 | 16 |
| 0 | 0 | 2 | 1 |
G:=sub<GL(4,GF(17))| [1,0,0,0,0,16,0,0,0,0,1,15,0,0,0,16],[1,0,0,0,0,16,0,0,0,0,16,0,0,0,0,16],[16,0,0,0,0,16,0,0,0,0,16,0,0,0,0,16],[16,0,0,0,0,16,0,0,0,0,1,0,0,0,0,1],[8,0,0,0,0,2,0,0,0,0,16,0,0,0,0,16],[0,4,0,0,1,0,0,0,0,0,16,2,0,0,16,1] >;
C24.19Q8 in GAP, Magma, Sage, TeX
C_2^4._{19}Q_8 % in TeX
G:=Group("C2^4.19Q8"); // GroupNames label
G:=SmallGroup(128,542);
// by ID
G=gap.SmallGroup(128,542);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,64,422,2019,248,124]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=1,e^4=d,f^2=b*d*e^2,a*b=b*a,f*a*f^-1=a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,f*b*f^-1=b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,c*f=f*c,d*e=e*d,d*f=f*d,f*e*f^-1=e^3>;
// generators/relations