p-group, metabelian, nilpotent (class 3), monomial
Aliases: C24.116D4, C4.Q8:8C22, (C2xC8).33C23, C2.D8:19C22, C4:C4.391C23, (C2xC4).291C24, C24.4C4:8C2, (C2xD4).80C23, C23.242(C2xD4), (C22xC4).442D4, (C2xQ8).68C23, D4:C4:19C22, Q8:C4:21C22, C22:C8.15C22, M4(2):C4:23C2, C23.36D4:10C2, C23.19D4:12C2, C23.20D4:12C2, C4:D4.156C22, (C23xC4).561C22, C22.551(C22xD4), C22:Q8.161C22, C2.22(D8:C22), (C22xC4).1007C23, C22.19C24.19C2, (C2xM4(2)).73C22, C42:C2.317C22, C4.127(C22.D4), C22.41(C22.D4), C4.101(C2xC4oD4), (C2xC4).1218(C2xD4), (C2xC42:C2):46C2, (C2xC4).486(C4oD4), (C2xC4:C4).928C22, (C2xC4oD4).138C22, C2.56(C2xC22.D4), SmallGroup(128,1825)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C24.116D4
G = < a,b,c,d,e,f | a2=b2=c2=d2=f2=1, e4=d, ab=ba, eae-1=ac=ca, ad=da, faf=acd, bc=cb, ebe-1=fbf=bd=db, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef=cde3 >
Subgroups: 388 in 207 conjugacy classes, 92 normal (18 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C8, C2xC4, C2xC4, C2xC4, D4, Q8, C23, C23, C23, C42, C22:C4, C4:C4, C4:C4, C2xC8, M4(2), C22xC4, C22xC4, C22xC4, C2xD4, C2xD4, C2xQ8, C4oD4, C24, C22:C8, D4:C4, Q8:C4, C4.Q8, C2.D8, C2xC42, C2xC22:C4, C2xC4:C4, C42:C2, C42:C2, C42:C2, C4xD4, C22wrC2, C4:D4, C22:Q8, C22.D4, C2xM4(2), C23xC4, C2xC4oD4, C24.4C4, C23.36D4, M4(2):C4, C23.19D4, C23.20D4, C2xC42:C2, C22.19C24, C24.116D4
Quotients: C1, C2, C22, D4, C23, C2xD4, C4oD4, C24, C22.D4, C22xD4, C2xC4oD4, C2xC22.D4, D8:C22, C24.116D4
(2 29)(4 31)(6 25)(8 27)(9 18)(10 14)(11 20)(12 16)(13 22)(15 24)(17 21)(19 23)
(1 28)(2 25)(3 30)(4 27)(5 32)(6 29)(7 26)(8 31)(9 18)(10 23)(11 20)(12 17)(13 22)(14 19)(15 24)(16 21)
(1 28)(2 29)(3 30)(4 31)(5 32)(6 25)(7 26)(8 27)(9 22)(10 23)(11 24)(12 17)(13 18)(14 19)(15 20)(16 21)
(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 22)(2 16)(3 20)(4 14)(5 18)(6 12)(7 24)(8 10)(9 28)(11 26)(13 32)(15 30)(17 25)(19 31)(21 29)(23 27)
G:=sub<Sym(32)| (2,29)(4,31)(6,25)(8,27)(9,18)(10,14)(11,20)(12,16)(13,22)(15,24)(17,21)(19,23), (1,28)(2,25)(3,30)(4,27)(5,32)(6,29)(7,26)(8,31)(9,18)(10,23)(11,20)(12,17)(13,22)(14,19)(15,24)(16,21), (1,28)(2,29)(3,30)(4,31)(5,32)(6,25)(7,26)(8,27)(9,22)(10,23)(11,24)(12,17)(13,18)(14,19)(15,20)(16,21), (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), (1,22)(2,16)(3,20)(4,14)(5,18)(6,12)(7,24)(8,10)(9,28)(11,26)(13,32)(15,30)(17,25)(19,31)(21,29)(23,27)>;
G:=Group( (2,29)(4,31)(6,25)(8,27)(9,18)(10,14)(11,20)(12,16)(13,22)(15,24)(17,21)(19,23), (1,28)(2,25)(3,30)(4,27)(5,32)(6,29)(7,26)(8,31)(9,18)(10,23)(11,20)(12,17)(13,22)(14,19)(15,24)(16,21), (1,28)(2,29)(3,30)(4,31)(5,32)(6,25)(7,26)(8,27)(9,22)(10,23)(11,24)(12,17)(13,18)(14,19)(15,20)(16,21), (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), (1,22)(2,16)(3,20)(4,14)(5,18)(6,12)(7,24)(8,10)(9,28)(11,26)(13,32)(15,30)(17,25)(19,31)(21,29)(23,27) );
G=PermutationGroup([[(2,29),(4,31),(6,25),(8,27),(9,18),(10,14),(11,20),(12,16),(13,22),(15,24),(17,21),(19,23)], [(1,28),(2,25),(3,30),(4,27),(5,32),(6,29),(7,26),(8,31),(9,18),(10,23),(11,20),(12,17),(13,22),(14,19),(15,24),(16,21)], [(1,28),(2,29),(3,30),(4,31),(5,32),(6,25),(7,26),(8,27),(9,22),(10,23),(11,24),(12,17),(13,18),(14,19),(15,20),(16,21)], [(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)], [(1,22),(2,16),(3,20),(4,14),(5,18),(6,12),(7,24),(8,10),(9,28),(11,26),(13,32),(15,30),(17,25),(19,31),(21,29),(23,27)]])
32 conjugacy classes
class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 4A | ··· | 4H | 4I | ··· | 4P | 4Q | 4R | 4S | 8A | 8B | 8C | 8D |
order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 |
size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 8 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | 8 | 8 | 8 | 8 | 8 | 8 |
32 irreducible representations
dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 4 |
type | + | + | + | + | + | + | + | + | + | + | ||
image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | C4oD4 | D8:C22 |
kernel | C24.116D4 | C24.4C4 | C23.36D4 | M4(2):C4 | C23.19D4 | C23.20D4 | C2xC42:C2 | C22.19C24 | C22xC4 | C24 | C2xC4 | C2 |
# reps | 1 | 1 | 2 | 2 | 4 | 4 | 1 | 1 | 3 | 1 | 8 | 4 |
Matrix representation of C24.116D4 ►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 | 1 |
0 | 0 | 0 | 0 | 16 | 16 |
0 | 0 | 0 | 0 | 0 | 1 |
1 | 0 | 0 | 0 | 0 | 0 |
0 | 1 | 0 | 0 | 0 | 0 |
0 | 0 | 1 | 0 | 0 | 1 |
0 | 0 | 0 | 1 | 0 | 16 |
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 |
0 | 4 | 0 | 0 | 0 | 0 |
4 | 0 | 0 | 0 | 0 | 0 |
0 | 0 | 4 | 0 | 0 | 10 |
0 | 0 | 13 | 0 | 1 | 7 |
0 | 0 | 4 | 13 | 0 | 4 |
0 | 0 | 9 | 0 | 0 | 13 |
0 | 16 | 0 | 0 | 0 | 0 |
16 | 0 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 1 | 1 | 0 |
0 | 0 | 0 | 16 | 0 | 0 |
0 | 0 | 1 | 1 | 0 | 0 |
0 | 0 | 0 | 15 | 0 | 1 |
G:=sub<GL(6,GF(17))| [1,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,1,16,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,1,16,0,16],[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],[0,4,0,0,0,0,4,0,0,0,0,0,0,0,4,13,4,9,0,0,0,0,13,0,0,0,0,1,0,0,0,0,10,7,4,13],[0,16,0,0,0,0,16,0,0,0,0,0,0,0,0,0,1,0,0,0,1,16,1,15,0,0,1,0,0,0,0,0,0,0,0,1] >;
C24.116D4 in GAP, Magma, Sage, TeX
C_2^4._{116}D_4
% in TeX
G:=Group("C2^4.116D4");
// GroupNames label
G:=SmallGroup(128,1825);
// by ID
G=gap.SmallGroup(128,1825);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,456,758,100,2019,4037,1027,124]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=f^2=1,e^4=d,a*b=b*a,e*a*e^-1=a*c=c*a,a*d=d*a,f*a*f=a*c*d,b*c=c*b,e*b*e^-1=f*b*f=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=c*d*e^3>;
// generators/relations