p-group, metabelian, nilpotent (class 3), monomial
Aliases: C24.110D4, (C2xC8):10D4, C4o(C8:2D4), C4o(C8:D4), C8.21(C2xD4), C8:2D4:35C2, C8:D4:57C2, C4o(C8.D4), C8.D4:35C2, (C2xD8):47C22, C4:C4.23C23, C4.Q8:50C22, C2.D8:69C22, (C2xC4).258C24, (C2xC8).250C23, (C2xQ16):52C22, (C2xD4).61C23, C4.152(C22xD4), C23.238(C2xD4), (C22xC4).428D4, (C2xQ8).49C23, C4.212(C4:D4), D4:C4:92C22, C22.19C24:7C2, Q8:C4:96C22, (C2xSD16):55C22, (C22xM4(2)):3C2, C4:D4.149C22, C23.24D4:39C2, C23.25D4:26C2, C22.35(C4:D4), (C23xC4).550C22, (C22xC8).258C22, C22.518(C22xD4), C22:Q8.154C22, C2.14(D8:C22), (C22xC4).1537C23, C42:C2.107C22, (C2xM4(2)).263C22, (C2xC4oD8):16C2, (C2xC4)o(C8:D4), (C2xC4)o(C8:2D4), C4.25(C2xC4oD4), (C2xC4)o(C8.D4), (C2xC4).474(C2xD4), C2.76(C2xC4:D4), (C2xC4).704(C4oD4), (C2xC4oD4).124C22, SmallGroup(128,1786)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C24.110D4
G = < a,b,c,d,e,f | a2=b2=c2=d2=1, e4=f2=d, ab=ba, ac=ca, eae-1=ad=da, faf-1=acd, bc=cb, fbf-1=bd=db, be=eb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=e3 >
Subgroups: 484 in 250 conjugacy classes, 100 normal (22 characteristic)
C1, C2, C2, C2, C4, C4, C4, C22, C22, C22, C8, C8, C2xC4, C2xC4, C2xC4, D4, Q8, C23, C23, C23, C42, C22:C4, C4:C4, C4:C4, C2xC8, C2xC8, C2xC8, M4(2), D8, SD16, Q16, C22xC4, C22xC4, C22xC4, C2xD4, C2xD4, C2xQ8, C4oD4, C24, D4:C4, Q8:C4, C4.Q8, C2.D8, C42:C2, C4xD4, C22wrC2, C4:D4, C22:Q8, C22.D4, C22xC8, C2xM4(2), C2xM4(2), C2xD8, C2xSD16, C2xQ16, C4oD8, C23xC4, C2xC4oD4, C23.24D4, C23.25D4, C8:D4, C8:2D4, C8.D4, C22.19C24, C22xM4(2), C2xC4oD8, C24.110D4
Quotients: C1, C2, C22, D4, C23, C2xD4, C4oD4, C24, C4:D4, C22xD4, C2xC4oD4, C2xC4:D4, D8:C22, C24.110D4
(2 6)(4 8)(9 32)(10 29)(11 26)(12 31)(13 28)(14 25)(15 30)(16 27)(18 22)(20 24)
(1 19)(2 20)(3 21)(4 22)(5 23)(6 24)(7 17)(8 18)(9 28)(10 29)(11 30)(12 31)(13 32)(14 25)(15 26)(16 27)
(1 19)(2 20)(3 21)(4 22)(5 23)(6 24)(7 17)(8 18)(9 32)(10 25)(11 26)(12 27)(13 28)(14 29)(15 30)(16 31)
(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 29 5 25)(2 32 6 28)(3 27 7 31)(4 30 8 26)(9 24 13 20)(10 19 14 23)(11 22 15 18)(12 17 16 21)
G:=sub<Sym(32)| (2,6)(4,8)(9,32)(10,29)(11,26)(12,31)(13,28)(14,25)(15,30)(16,27)(18,22)(20,24), (1,19)(2,20)(3,21)(4,22)(5,23)(6,24)(7,17)(8,18)(9,28)(10,29)(11,30)(12,31)(13,32)(14,25)(15,26)(16,27), (1,19)(2,20)(3,21)(4,22)(5,23)(6,24)(7,17)(8,18)(9,32)(10,25)(11,26)(12,27)(13,28)(14,29)(15,30)(16,31), (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), (1,29,5,25)(2,32,6,28)(3,27,7,31)(4,30,8,26)(9,24,13,20)(10,19,14,23)(11,22,15,18)(12,17,16,21)>;
G:=Group( (2,6)(4,8)(9,32)(10,29)(11,26)(12,31)(13,28)(14,25)(15,30)(16,27)(18,22)(20,24), (1,19)(2,20)(3,21)(4,22)(5,23)(6,24)(7,17)(8,18)(9,28)(10,29)(11,30)(12,31)(13,32)(14,25)(15,26)(16,27), (1,19)(2,20)(3,21)(4,22)(5,23)(6,24)(7,17)(8,18)(9,32)(10,25)(11,26)(12,27)(13,28)(14,29)(15,30)(16,31), (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), (1,29,5,25)(2,32,6,28)(3,27,7,31)(4,30,8,26)(9,24,13,20)(10,19,14,23)(11,22,15,18)(12,17,16,21) );
G=PermutationGroup([[(2,6),(4,8),(9,32),(10,29),(11,26),(12,31),(13,28),(14,25),(15,30),(16,27),(18,22),(20,24)], [(1,19),(2,20),(3,21),(4,22),(5,23),(6,24),(7,17),(8,18),(9,28),(10,29),(11,30),(12,31),(13,32),(14,25),(15,26),(16,27)], [(1,19),(2,20),(3,21),(4,22),(5,23),(6,24),(7,17),(8,18),(9,32),(10,25),(11,26),(12,27),(13,28),(14,29),(15,30),(16,31)], [(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)], [(1,29,5,25),(2,32,6,28),(3,27,7,31),(4,30,8,26),(9,24,13,20),(10,19,14,23),(11,22,15,18),(12,17,16,21)]])
32 conjugacy classes
class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | ··· | 4N | 8A | ··· | 8H |
order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 8 | ··· | 8 |
size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 8 | 8 | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 8 | ··· | 8 | 4 | ··· | 4 |
32 irreducible representations
dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 |
type | + | + | + | + | + | + | + | + | + | + | + | + | ||
image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D4 | C4oD4 | D8:C22 |
kernel | C24.110D4 | C23.24D4 | C23.25D4 | C8:D4 | C8:2D4 | C8.D4 | C22.19C24 | C22xM4(2) | C2xC4oD8 | C2xC8 | C22xC4 | C24 | C2xC4 | C2 |
# reps | 1 | 2 | 1 | 4 | 2 | 2 | 2 | 1 | 1 | 4 | 3 | 1 | 4 | 4 |
Matrix representation of C24.110D4 ►in GL6(F17)
1 | 0 | 0 | 0 | 0 | 0 |
0 | 16 | 0 | 0 | 0 | 0 |
0 | 0 | 1 | 0 | 0 | 0 |
0 | 0 | 0 | 16 | 0 | 0 |
0 | 0 | 0 | 0 | 1 | 0 |
0 | 0 | 0 | 0 | 0 | 16 |
1 | 0 | 0 | 0 | 0 | 0 |
0 | 1 | 0 | 0 | 0 | 0 |
0 | 0 | 1 | 0 | 0 | 0 |
0 | 0 | 0 | 1 | 0 | 0 |
0 | 0 | 0 | 0 | 16 | 0 |
0 | 0 | 0 | 0 | 0 | 16 |
16 | 0 | 0 | 0 | 0 | 0 |
0 | 16 | 0 | 0 | 0 | 0 |
0 | 0 | 1 | 0 | 0 | 0 |
0 | 0 | 0 | 1 | 0 | 0 |
0 | 0 | 0 | 0 | 1 | 0 |
0 | 0 | 0 | 0 | 0 | 1 |
1 | 0 | 0 | 0 | 0 | 0 |
0 | 1 | 0 | 0 | 0 | 0 |
0 | 0 | 16 | 0 | 0 | 0 |
0 | 0 | 0 | 16 | 0 | 0 |
0 | 0 | 0 | 0 | 16 | 0 |
0 | 0 | 0 | 0 | 0 | 16 |
16 | 0 | 0 | 0 | 0 | 0 |
0 | 16 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 1 | 0 | 0 |
0 | 0 | 4 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | 0 | 4 |
0 | 0 | 0 | 0 | 16 | 0 |
0 | 1 | 0 | 0 | 0 | 0 |
1 | 0 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | 0 | 4 |
0 | 0 | 0 | 0 | 16 | 0 |
0 | 0 | 0 | 1 | 0 | 0 |
0 | 0 | 4 | 0 | 0 | 0 |
G:=sub<GL(6,GF(17))| [1,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,16],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,0,4,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,4,0],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,1,0,0,0,0,16,0,0,0,0,4,0,0,0] >;
C24.110D4 in GAP, Magma, Sage, TeX
C_2^4._{110}D_4
% in TeX
G:=Group("C2^4.110D4");
// GroupNames label
G:=SmallGroup(128,1786);
// by ID
G=gap.SmallGroup(128,1786);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,120,758,723,248,2804,172]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=1,e^4=f^2=d,a*b=b*a,a*c=c*a,e*a*e^-1=a*d=d*a,f*a*f^-1=a*c*d,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