p-group, metabelian, nilpotent (class 3), monomial
Aliases: C24.3Q8, C22⋊C8.8C4, (C2×C4).10C42, C4.34(C23⋊C4), C23.43(C4⋊C4), (C22×C4).641D4, (C2×M4(2)).1C4, C24.4C4.7C2, C2.6(C4.C42), C22.6(C8.C4), (C23×C4).194C22, C2.8(C23.9D4), C22.48(C2.C42), (C2×C22⋊C8).6C2, (C22×C4).153(C2×C4), (C2×C4).301(C22⋊C4), SmallGroup(128,30)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C24.3Q8
G = < a,b,c,d,e,f | a2=b2=c2=d2=1, e4=c, f2=bce2, ab=ba, ac=ca, eae-1=faf-1=ad=da, bc=cb, bd=db, be=eb, fbf-1=bcd, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=acde3 >
Subgroups: 200 in 95 conjugacy classes, 36 normal (12 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C8, C2×C4, C2×C4, C2×C4, C23, C23, C23, C2×C8, M4(2), C22×C4, C22×C4, C22×C4, C24, C22⋊C8, C22⋊C8, C22×C8, C2×M4(2), C23×C4, C2×C22⋊C8, C24.4C4, C24.3Q8
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C42, C22⋊C4, C4⋊C4, C2.C42, C23⋊C4, C8.C4, C4.C42, C23.9D4, C24.3Q8
►On 32 points
(2 18)(4 20)(6 22)(8 24)(10 28)(12 30)(14 32)(16 26)
(1 5)(2 6)(3 7)(4 8)(9 27)(10 28)(11 29)(12 30)(13 31)(14 32)(15 25)(16 26)(17 21)(18 22)(19 23)(20 24)
(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 17)(2 18)(3 19)(4 20)(5 21)(6 22)(7 23)(8 24)(9 27)(10 28)(11 29)(12 30)(13 31)(14 32)(15 25)(16 26)
(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 14 3 30 5 10 7 26)(2 13 4 29 6 9 8 25)(11 22 27 24 15 18 31 20)(12 21 28 23 16 17 32 19)
G:=sub<Sym(32)| (2,18)(4,20)(6,22)(8,24)(10,28)(12,30)(14,32)(16,26), (1,5)(2,6)(3,7)(4,8)(9,27)(10,28)(11,29)(12,30)(13,31)(14,32)(15,25)(16,26)(17,21)(18,22)(19,23)(20,24), (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), (1,17)(2,18)(3,19)(4,20)(5,21)(6,22)(7,23)(8,24)(9,27)(10,28)(11,29)(12,30)(13,31)(14,32)(15,25)(16,26), (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), (1,14,3,30,5,10,7,26)(2,13,4,29,6,9,8,25)(11,22,27,24,15,18,31,20)(12,21,28,23,16,17,32,19)>;
G:=Group( (2,18)(4,20)(6,22)(8,24)(10,28)(12,30)(14,32)(16,26), (1,5)(2,6)(3,7)(4,8)(9,27)(10,28)(11,29)(12,30)(13,31)(14,32)(15,25)(16,26)(17,21)(18,22)(19,23)(20,24), (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), (1,17)(2,18)(3,19)(4,20)(5,21)(6,22)(7,23)(8,24)(9,27)(10,28)(11,29)(12,30)(13,31)(14,32)(15,25)(16,26), (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), (1,14,3,30,5,10,7,26)(2,13,4,29,6,9,8,25)(11,22,27,24,15,18,31,20)(12,21,28,23,16,17,32,19) );
G=PermutationGroup([[(2,18),(4,20),(6,22),(8,24),(10,28),(12,30),(14,32),(16,26)], [(1,5),(2,6),(3,7),(4,8),(9,27),(10,28),(11,29),(12,30),(13,31),(14,32),(15,25),(16,26),(17,21),(18,22),(19,23),(20,24)], [(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)], [(1,17),(2,18),(3,19),(4,20),(5,21),(6,22),(7,23),(8,24),(9,27),(10,28),(11,29),(12,30),(13,31),(14,32),(15,25),(16,26)], [(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)], [(1,14,3,30,5,10,7,26),(2,13,4,29,6,9,8,25),(11,22,27,24,15,18,31,20),(12,21,28,23,16,17,32,19)]])
38 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 8A | ··· | 8P | 8Q | 8R | 8S | 8T |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 8 | ··· | 8 | 8 | 8 | 8 | 8 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | ··· | 4 | 8 | 8 | 8 | 8 |
38 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | - | + | |||
| image | C1 | C2 | C2 | C4 | C4 | D4 | Q8 | C8.C4 | C23⋊C4 |
| kernel | C24.3Q8 | C2×C22⋊C8 | C24.4C4 | C22⋊C8 | C2×M4(2) | C22×C4 | C24 | C22 | C4 |
| # reps | 1 | 2 | 1 | 8 | 4 | 3 | 1 | 16 | 2 |
Matrix representation of C24.3Q8 ►in GL4(𝔽17) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 9 | 16 |
| 16 | 0 | 0 | 0 |
| 1 | 1 | 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 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 16 |
| 9 | 0 | 0 | 0 |
| 5 | 2 | 0 | 0 |
| 0 | 0 | 9 | 15 |
| 0 | 0 | 13 | 8 |
| 16 | 15 | 0 | 0 |
| 11 | 1 | 0 | 0 |
| 0 | 0 | 4 | 1 |
| 0 | 0 | 5 | 13 |
G:=sub<GL(4,GF(17))| [1,0,0,0,0,1,0,0,0,0,1,9,0,0,0,16],[16,1,0,0,0,1,0,0,0,0,16,0,0,0,0,16],[16,0,0,0,0,16,0,0,0,0,16,0,0,0,0,16],[1,0,0,0,0,1,0,0,0,0,16,0,0,0,0,16],[9,5,0,0,0,2,0,0,0,0,9,13,0,0,15,8],[16,11,0,0,15,1,0,0,0,0,4,5,0,0,1,13] >;
C24.3Q8 in GAP, Magma, Sage, TeX
C_2^4._3Q_8
% in TeX
G:=Group("C2^4.3Q8"); // GroupNames label
G:=SmallGroup(128,30);
// by ID
G=gap.SmallGroup(128,30);
# by ID
G:=PCGroup([7,-2,2,-2,2,2,-2,2,56,85,120,758,723,184,136,3924,102]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=1,e^4=c,f^2=b*c*e^2,a*b=b*a,a*c=c*a,e*a*e^-1=f*a*f^-1=a*d=d*a,b*c=c*b,b*d=d*b,b*e=e*b,f*b*f^-1=b*c*d,c*d=d*c,c*e=e*c,c*f=f*c,d*e=e*d,d*f=f*d,f*e*f^-1=a*c*d*e^3>;
// generators/relations