p-group, metabelian, nilpotent (class 3), monomial
Aliases: C24.124D4, C4.62+ 1+4, C8:7D4:4C2, D4:D4:2C2, C8:8D4:26C2, (C2xD8):4C22, C2.D8:7C22, D4.7D4:2C2, C8.18D4:4C2, (C2xQ16):4C22, C4.Q8:34C22, C4:C4.130C23, (C2xC8).153C23, (C2xC4).389C24, C22.4(C4oD8), (C22xC4).171D4, C23.273(C2xD4), D4:C4:44C22, Q8:C4:47C22, (C2xSD16):41C22, (C2xD4).141C23, C23.20D4:2C2, C23.19D4:2C2, (C2xQ8).129C23, C22.19C24:10C2, C4:D4.182C22, C2.70(C23:3D4), C22:C8.191C22, (C23xC4).569C22, (C22xC8).150C22, C22.649(C22xD4), C22:Q8.187C22, C2.50(D8:C22), (C22xC4).1067C23, C42:C2.151C22, C2.40(C2xC4oD8), (C2xC22:C8):30C2, (C2xC4).706(C2xD4), (C2xC4oD4).162C22, SmallGroup(128,1923)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C24.124D4
G = < a,b,c,d,e,f | a2=b2=c2=d2=f2=1, e4=d, ab=ba, faf=ac=ca, ad=da, ae=ea, ebe-1=bc=cb, bd=db, fbf=bcd, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef=de3 >
Subgroups: 452 in 212 conjugacy classes, 86 normal (26 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C8, C2xC4, C2xC4, C2xC4, D4, Q8, C23, C23, C23, C42, C22:C4, C4:C4, C4:C4, C2xC8, C2xC8, C2xC8, D8, SD16, Q16, C22xC4, C22xC4, C22xC4, C2xD4, C2xD4, C2xQ8, C4oD4, C24, C22:C8, D4:C4, Q8:C4, C4.Q8, C2.D8, C42:C2, C4xD4, C22wrC2, C4:D4, C22:Q8, C22.D4, C22xC8, C2xD8, C2xSD16, C2xQ16, C23xC4, C2xC4oD4, C2xC22:C8, D4:D4, D4.7D4, C8:8D4, C8:7D4, C8.18D4, C23.19D4, C23.20D4, C22.19C24, C24.124D4
Quotients: C1, C2, C22, D4, C23, C2xD4, C24, C4oD8, C22xD4, 2+ 1+4, C23:3D4, C2xC4oD8, D8:C22, C24.124D4
(9 22)(10 23)(11 24)(12 17)(13 18)(14 19)(15 20)(16 21)
(1 5)(2 27)(3 7)(4 29)(6 31)(8 25)(9 22)(11 24)(13 18)(15 20)(26 30)(28 32)
(1 30)(2 31)(3 32)(4 25)(5 26)(6 27)(7 28)(8 29)(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 9)(2 16)(3 15)(4 14)(5 13)(6 12)(7 11)(8 10)(17 27)(18 26)(19 25)(20 32)(21 31)(22 30)(23 29)(24 28)
G:=sub<Sym(32)| (9,22)(10,23)(11,24)(12,17)(13,18)(14,19)(15,20)(16,21), (1,5)(2,27)(3,7)(4,29)(6,31)(8,25)(9,22)(11,24)(13,18)(15,20)(26,30)(28,32), (1,30)(2,31)(3,32)(4,25)(5,26)(6,27)(7,28)(8,29)(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), (1,9)(2,16)(3,15)(4,14)(5,13)(6,12)(7,11)(8,10)(17,27)(18,26)(19,25)(20,32)(21,31)(22,30)(23,29)(24,28)>;
G:=Group( (9,22)(10,23)(11,24)(12,17)(13,18)(14,19)(15,20)(16,21), (1,5)(2,27)(3,7)(4,29)(6,31)(8,25)(9,22)(11,24)(13,18)(15,20)(26,30)(28,32), (1,30)(2,31)(3,32)(4,25)(5,26)(6,27)(7,28)(8,29)(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), (1,9)(2,16)(3,15)(4,14)(5,13)(6,12)(7,11)(8,10)(17,27)(18,26)(19,25)(20,32)(21,31)(22,30)(23,29)(24,28) );
G=PermutationGroup([[(9,22),(10,23),(11,24),(12,17),(13,18),(14,19),(15,20),(16,21)], [(1,5),(2,27),(3,7),(4,29),(6,31),(8,25),(9,22),(11,24),(13,18),(15,20),(26,30),(28,32)], [(1,30),(2,31),(3,32),(4,25),(5,26),(6,27),(7,28),(8,29),(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)], [(1,9),(2,16),(3,15),(4,14),(5,13),(6,12),(7,11),(8,10),(17,27),(18,26),(19,25),(20,32),(21,31),(22,30),(23,29),(24,28)]])
32 conjugacy classes
class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 4A | ··· | 4H | 4I | ··· | 4N | 8A | ··· | 8H |
order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 8 | ··· | 8 |
size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 8 | 8 | 2 | ··· | 2 | 8 | ··· | 8 | 4 | ··· | 4 |
32 irreducible representations
dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 4 | 4 |
type | + | + | + | + | + | + | + | + | + | + | + | + | + | ||
image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | C4oD8 | 2+ 1+4 | D8:C22 |
kernel | C24.124D4 | C2xC22:C8 | D4:D4 | D4.7D4 | C8:8D4 | C8:7D4 | C8.18D4 | C23.19D4 | C23.20D4 | C22.19C24 | C22xC4 | C24 | C22 | C4 | C2 |
# reps | 1 | 1 | 2 | 2 | 2 | 1 | 1 | 2 | 2 | 2 | 3 | 1 | 8 | 2 | 2 |
Matrix representation of C24.124D4 ►in GL6(F17)
16 | 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 | 0 | 0 | 1 |
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 | 1 | 16 | 0 |
0 | 0 | 1 | 0 | 0 | 16 |
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 | 1 | 0 | 0 | 0 |
0 | 0 | 0 | 1 | 0 | 0 |
0 | 0 | 0 | 0 | 1 | 0 |
0 | 0 | 0 | 0 | 0 | 1 |
8 | 0 | 0 | 0 | 0 | 0 |
0 | 15 | 0 | 0 | 0 | 0 |
0 | 0 | 1 | 0 | 0 | 15 |
0 | 0 | 0 | 1 | 15 | 0 |
0 | 0 | 0 | 1 | 16 | 0 |
0 | 0 | 1 | 0 | 0 | 16 |
0 | 2 | 0 | 0 | 0 | 0 |
9 | 0 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 1 | 15 | 0 |
0 | 0 | 1 | 0 | 0 | 15 |
0 | 0 | 0 | 0 | 0 | 16 |
0 | 0 | 0 | 0 | 16 | 0 |
G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,1],[16,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,0,1,1,0,0,0,0,0,16,0,0,0,0,0,0,16],[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],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[8,0,0,0,0,0,0,15,0,0,0,0,0,0,1,0,0,1,0,0,0,1,1,0,0,0,0,15,16,0,0,0,15,0,0,16],[0,9,0,0,0,0,2,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,15,0,0,16,0,0,0,15,16,0] >;
C24.124D4 in GAP, Magma, Sage, TeX
C_2^4._{124}D_4
% in TeX
G:=Group("C2^4.124D4");
// GroupNames label
G:=SmallGroup(128,1923);
// by ID
G=gap.SmallGroup(128,1923);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,758,219,675,248,4037,1027,124]);
// 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,f*a*f=a*c=c*a,a*d=d*a,a*e=e*a,e*b*e^-1=b*c=c*b,b*d=d*b,f*b*f=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=d*e^3>;
// generators/relations