p-group, metabelian, nilpotent (class 3), monomial
Aliases: C24.10Q8, M4(2).39D4, C4.139(C4×D4), C22⋊C8.14C4, C4.116C22≀C2, C23.85(C2×Q8), (C22×C4).51Q8, C23.30(C4⋊C4), C4.C42⋊1C2, (C22×C4).682D4, C22⋊2(C8.C4), C22.1(C22⋊Q8), C24.4C4.21C2, (C22×C8).166C22, (C23×C4).251C22, (C22×C4).1352C23, (C22×M4(2)).20C2, C2.11(M4(2).C4), C2.11(C23.8Q8), C4.133(C22.D4), (C2×M4(2)).171C22, (C2×C8).34(C2×C4), (C2×C8.C4)⋊7C2, (C2×C4).53(C4⋊C4), C2.14(C2×C8.C4), (C2×C4).1527(C2×D4), (C2×C22⋊C8).34C2, C22.112(C2×C4⋊C4), (C2×C4).748(C4○D4), (C2×C4).551(C22×C4), (C22×C4).274(C2×C4), SmallGroup(128,587)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C24.10Q8
G = < a,b,c,d,e,f | a2=b2=c2=d2=1, e4=d, f2=e2, ab=ba, faf-1=ac=ca, ad=da, ae=ea, bc=cb, ebe-1=fbf-1=bd=db, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=be3 >
Subgroups: 236 in 139 conjugacy classes, 58 normal (30 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C22, C8, C2×C4, C2×C4, C2×C4, C23, C23, C2×C8, C2×C8, M4(2), M4(2), C22×C4, C22×C4, C24, C22⋊C8, C22⋊C8, C22⋊C8, C8.C4, C22×C8, C22×C8, C2×M4(2), C2×M4(2), C2×M4(2), C23×C4, C4.C42, C2×C22⋊C8, C24.4C4, C2×C8.C4, C22×M4(2), C24.10Q8
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C23, C4⋊C4, C22×C4, C2×D4, C2×Q8, C4○D4, C8.C4, C2×C4⋊C4, C4×D4, C22≀C2, C22⋊Q8, C22.D4, C23.8Q8, C2×C8.C4, M4(2).C4, C24.10Q8
►On 32 points
(9 26)(10 27)(11 28)(12 29)(13 30)(14 31)(15 32)(16 25)
(1 24)(2 21)(3 18)(4 23)(5 20)(6 17)(7 22)(8 19)(9 30)(10 27)(11 32)(12 29)(13 26)(14 31)(15 28)(16 25)
(1 24)(2 17)(3 18)(4 19)(5 20)(6 21)(7 22)(8 23)(9 26)(10 27)(11 28)(12 29)(13 30)(14 31)(15 32)(16 25)
(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 15 3 9 5 11 7 13)(2 31 4 25 6 27 8 29)(10 23 12 17 14 19 16 21)(18 26 20 28 22 30 24 32)
G:=sub<Sym(32)| (9,26)(10,27)(11,28)(12,29)(13,30)(14,31)(15,32)(16,25), (1,24)(2,21)(3,18)(4,23)(5,20)(6,17)(7,22)(8,19)(9,30)(10,27)(11,32)(12,29)(13,26)(14,31)(15,28)(16,25), (1,24)(2,17)(3,18)(4,19)(5,20)(6,21)(7,22)(8,23)(9,26)(10,27)(11,28)(12,29)(13,30)(14,31)(15,32)(16,25), (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), (1,15,3,9,5,11,7,13)(2,31,4,25,6,27,8,29)(10,23,12,17,14,19,16,21)(18,26,20,28,22,30,24,32)>;
G:=Group( (9,26)(10,27)(11,28)(12,29)(13,30)(14,31)(15,32)(16,25), (1,24)(2,21)(3,18)(4,23)(5,20)(6,17)(7,22)(8,19)(9,30)(10,27)(11,32)(12,29)(13,26)(14,31)(15,28)(16,25), (1,24)(2,17)(3,18)(4,19)(5,20)(6,21)(7,22)(8,23)(9,26)(10,27)(11,28)(12,29)(13,30)(14,31)(15,32)(16,25), (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), (1,15,3,9,5,11,7,13)(2,31,4,25,6,27,8,29)(10,23,12,17,14,19,16,21)(18,26,20,28,22,30,24,32) );
G=PermutationGroup([[(9,26),(10,27),(11,28),(12,29),(13,30),(14,31),(15,32),(16,25)], [(1,24),(2,21),(3,18),(4,23),(5,20),(6,17),(7,22),(8,19),(9,30),(10,27),(11,32),(12,29),(13,26),(14,31),(15,28),(16,25)], [(1,24),(2,17),(3,18),(4,19),(5,20),(6,21),(7,22),(8,23),(9,26),(10,27),(11,28),(12,29),(13,30),(14,31),(15,32),(16,25)], [(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)], [(1,15,3,9,5,11,7,13),(2,31,4,25,6,27,8,29),(10,23,12,17,14,19,16,21),(18,26,20,28,22,30,24,32)]])
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 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | - | - | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | D4 | D4 | Q8 | Q8 | C4○D4 | C8.C4 | M4(2).C4 |
| kernel | C24.10Q8 | C4.C42 | C2×C22⋊C8 | C24.4C4 | C2×C8.C4 | C22×M4(2) | C22⋊C8 | M4(2) | C22×C4 | C22×C4 | C24 | C2×C4 | C22 | C2 |
| # reps | 1 | 2 | 1 | 1 | 2 | 1 | 8 | 4 | 2 | 1 | 1 | 4 | 8 | 2 |
Matrix representation of C24.10Q8 ►in GL4(𝔽17) generated by
| 1 | 0 | 0 | 0 |
| 15 | 16 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 16 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 16 |
| 16 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 16 |
| 1 | 0 | 0 | 0 |
| 15 | 16 | 0 | 0 |
| 0 | 0 | 0 | 15 |
| 0 | 0 | 15 | 0 |
| 16 | 16 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 4 | 0 |
G:=sub<GL(4,GF(17))| [1,15,0,0,0,16,0,0,0,0,1,0,0,0,0,1],[16,0,0,0,0,16,0,0,0,0,1,0,0,0,0,16],[16,0,0,0,0,16,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,16,0,0,0,0,16],[1,15,0,0,0,16,0,0,0,0,0,15,0,0,15,0],[16,0,0,0,16,1,0,0,0,0,0,4,0,0,1,0] >;
C24.10Q8 in GAP, Magma, Sage, TeX
C_2^4._{10}Q_8 % in TeX
G:=Group("C2^4.10Q8"); // GroupNames label
G:=SmallGroup(128,587);
// by ID
G=gap.SmallGroup(128,587);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,232,422,2019,1018,248,124]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=1,e^4=d,f^2=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,e*b*e^-1=f*b*f^-1=b*d=d*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=b*e^3>;
// generators/relations