p-group, metabelian, nilpotent (class 3), monomial
Aliases: C24.144D4, (C2×C8)⋊38D4, C4○(C8⋊7D4), C4○(C8⋊8D4), C8⋊7D4⋊41C2, C8⋊8D4⋊55C2, (C23×C8)⋊10C2, C8.118(C2×D4), C4○(C8.18D4), (C2×D8)⋊43C22, C22⋊1(C4○D8), C4⋊C4.19C23, C2.D8⋊46C22, C4.Q8⋊55C22, C8.18D4⋊41C2, (C2×C8).591C23, (C2×C4).254C24, (C2×Q16)⋊43C22, (C2×D4).58C23, C4.148(C22×D4), C23.383(C2×D4), (C22×C4).564D4, (C2×Q8).46C23, C4.211(C4⋊D4), D4⋊C4⋊58C22, C22.19C24⋊6C2, Q8⋊C4⋊59C22, (C2×SD16)⋊76C22, C23.24D4⋊4C2, C23.25D4⋊3C2, C4⋊D4.147C22, C22.34(C4⋊D4), (C23×C4).703C22, (C22×C8).557C22, C22.514(C22×D4), C22⋊Q8.152C22, (C22×C4).1533C23, C42⋊C2.106C22, (C2×C4○D8)⋊7C2, (C2×C4)○(C8⋊8D4), (C2×C4)○(C8⋊7D4), C2.16(C2×C4○D8), C4.21(C2×C4○D4), C2.72(C2×C4⋊D4), (C2×C4)○(C8.18D4), (C2×C4).1426(C2×D4), (C2×C4).700(C4○D4), (C2×C4○D4).123C22, SmallGroup(128,1782)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C24.144D4
G = < a,b,c,d,e,f | a2=b2=c2=d2=1, e4=f2=d, ab=ba, faf-1=ac=ca, ad=da, ae=ea, bc=cb, fbf-1=bd=db, be=eb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=e3 >
Subgroups: 484 in 260 conjugacy classes, 104 normal (22 characteristic)
C1, C2, C2, C2, C4, C4, C4, C22, C22, C22, C8, C8, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, C23, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, C2×C8, D8, SD16, Q16, C22×C4, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4, C24, D4⋊C4, Q8⋊C4, C4.Q8, C2.D8, C42⋊C2, C4×D4, C22≀C2, C4⋊D4, C22⋊Q8, C22.D4, C22×C8, C22×C8, C22×C8, C2×D8, C2×SD16, C2×Q16, C4○D8, C23×C4, C2×C4○D4, C23.24D4, C23.25D4, C8⋊8D4, C8⋊7D4, C8.18D4, C22.19C24, C23×C8, C2×C4○D8, C24.144D4
Quotients: C1, C2, C22, D4, C23, C2×D4, C4○D4, C24, C4⋊D4, C4○D8, C22×D4, C2×C4○D4, C2×C4⋊D4, C2×C4○D8, C24.144D4
►On 32 points
(9 26)(10 27)(11 28)(12 29)(13 30)(14 31)(15 32)(16 25)
(1 23)(2 24)(3 17)(4 18)(5 19)(6 20)(7 21)(8 22)(9 26)(10 27)(11 28)(12 29)(13 30)(14 31)(15 32)(16 25)
(1 19)(2 20)(3 21)(4 22)(5 23)(6 24)(7 17)(8 18)(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 31 5 27)(2 26 6 30)(3 29 7 25)(4 32 8 28)(9 24 13 20)(10 19 14 23)(11 22 15 18)(12 17 16 21)
G:=sub<Sym(32)| (9,26)(10,27)(11,28)(12,29)(13,30)(14,31)(15,32)(16,25), (1,23)(2,24)(3,17)(4,18)(5,19)(6,20)(7,21)(8,22)(9,26)(10,27)(11,28)(12,29)(13,30)(14,31)(15,32)(16,25), (1,19)(2,20)(3,21)(4,22)(5,23)(6,24)(7,17)(8,18)(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), (1,31,5,27)(2,26,6,30)(3,29,7,25)(4,32,8,28)(9,24,13,20)(10,19,14,23)(11,22,15,18)(12,17,16,21)>;
G:=Group( (9,26)(10,27)(11,28)(12,29)(13,30)(14,31)(15,32)(16,25), (1,23)(2,24)(3,17)(4,18)(5,19)(6,20)(7,21)(8,22)(9,26)(10,27)(11,28)(12,29)(13,30)(14,31)(15,32)(16,25), (1,19)(2,20)(3,21)(4,22)(5,23)(6,24)(7,17)(8,18)(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), (1,31,5,27)(2,26,6,30)(3,29,7,25)(4,32,8,28)(9,24,13,20)(10,19,14,23)(11,22,15,18)(12,17,16,21) );
G=PermutationGroup([[(9,26),(10,27),(11,28),(12,29),(13,30),(14,31),(15,32),(16,25)], [(1,23),(2,24),(3,17),(4,18),(5,19),(6,20),(7,21),(8,22),(9,26),(10,27),(11,28),(12,29),(13,30),(14,31),(15,32),(16,25)], [(1,19),(2,20),(3,21),(4,22),(5,23),(6,24),(7,17),(8,18),(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)], [(1,31,5,27),(2,26,6,30),(3,29,7,25),(4,32,8,28),(9,24,13,20),(10,19,14,23),(11,22,15,18),(12,17,16,21)]])
44 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | ··· | 2I | 2J | 2K | 4A | 4B | 4C | 4D | 4E | ··· | 4J | 4K | ··· | 4P | 8A | ··· | 8P |
| order | 1 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | ··· | 4 | 8 | ··· | 8 |
| size | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 8 | 8 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 8 | ··· | 8 | 2 | ··· | 2 |
44 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D4 | C4○D4 | C4○D8 |
| kernel | C24.144D4 | C23.24D4 | C23.25D4 | C8⋊8D4 | C8⋊7D4 | C8.18D4 | C22.19C24 | C23×C8 | C2×C4○D8 | C2×C8 | C22×C4 | C24 | C2×C4 | C22 |
| # reps | 1 | 2 | 1 | 4 | 2 | 2 | 2 | 1 | 1 | 4 | 3 | 1 | 4 | 16 |
Matrix representation of C24.144D4 ►in GL4(𝔽17) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 16 |
| 16 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 16 |
| 1 | 0 | 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 |
| 2 | 0 | 0 | 0 |
| 0 | 8 | 0 | 0 |
| 0 | 0 | 15 | 0 |
| 0 | 0 | 0 | 9 |
| 0 | 8 | 0 | 0 |
| 2 | 0 | 0 | 0 |
| 0 | 0 | 0 | 9 |
| 0 | 0 | 15 | 0 |
G:=sub<GL(4,GF(17))| [1,0,0,0,0,1,0,0,0,0,1,0,0,0,0,16],[16,0,0,0,0,1,0,0,0,0,1,0,0,0,0,16],[1,0,0,0,0,1,0,0,0,0,16,0,0,0,0,16],[16,0,0,0,0,16,0,0,0,0,16,0,0,0,0,16],[2,0,0,0,0,8,0,0,0,0,15,0,0,0,0,9],[0,2,0,0,8,0,0,0,0,0,0,15,0,0,9,0] >;
C24.144D4 in GAP, Magma, Sage, TeX
C_2^4._{144}D_4 % in TeX
G:=Group("C2^4.144D4"); // GroupNames label
G:=SmallGroup(128,1782);
// by ID
G=gap.SmallGroup(128,1782);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,120,758,248,2804,172]);
// 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,f*a*f^-1=a*c=c*a,a*d=d*a,a*e=e*a,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