p-group, metabelian, nilpotent (class 3), monomial
Aliases: C24.66D4, C22⋊3C4≀C2, C4○D4.41D4, D4⋊5(C22⋊C4), (C22×D4)⋊18C4, Q8⋊5(C22⋊C4), (C22×Q8)⋊15C4, C4.112C22≀C2, (C2×C42)⋊10C22, C22.9C22≀C2, C23.544(C2×D4), (C22×C4).676D4, C24.4C4⋊25C2, C2.14(C24⋊3C4), (C2×M4(2))⋊37C22, (C23×C4).234C22, C23.196(C22⋊C4), (C22×C4).1323C23, C2.38(C42⋊C22), (C2×C4≀C2)⋊9C2, C2.38(C2×C4≀C2), (C2×C4○D4)⋊12C4, C4.4(C2×C22⋊C4), (C4×C22⋊C4)⋊22C2, (C2×C4).974(C2×D4), (C2×D4).202(C2×C4), (C2×Q8).185(C2×C4), (C22×C4○D4).5C2, (C2×C4).361(C22×C4), (C22×C4).263(C2×C4), (C2×C4).121(C22⋊C4), (C2×C4○D4).254C22, C22.242(C2×C22⋊C4), SmallGroup(128,521)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C24.66D4
G = < a,b,c,d,e,f | a2=b2=c2=d2=1, e4=d, f2=db=ebe-1=bd, ab=ba, eae-1=faf-1=ac=ca, ad=da, bc=cb, bf=fb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=be3 >
Subgroups: 620 in 309 conjugacy classes, 70 normal (26 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C8, C2×C4, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, C23, C42, C22⋊C4, C2×C8, M4(2), C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C2×Q8, C4○D4, C4○D4, C24, C24, C2.C42, C22⋊C8, C4≀C2, C2×C42, C2×C22⋊C4, C2×M4(2), C23×C4, C23×C4, C22×D4, C22×D4, C22×Q8, C2×C4○D4, C2×C4○D4, C4×C22⋊C4, C24.4C4, C2×C4≀C2, C22×C4○D4, C24.66D4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22⋊C4, C22×C4, C2×D4, C4≀C2, C2×C22⋊C4, C22≀C2, C24⋊3C4, C2×C4≀C2, C42⋊C22, C24.66D4
►On 32 points
(2 32)(4 26)(6 28)(8 30)(10 22)(12 24)(14 18)(16 20)
(1 13)(2 10)(3 15)(4 12)(5 9)(6 14)(7 11)(8 16)(17 31)(18 28)(19 25)(20 30)(21 27)(22 32)(23 29)(24 26)
(1 31)(2 32)(3 25)(4 26)(5 27)(6 28)(7 29)(8 30)(9 21)(10 22)(11 23)(12 24)(13 17)(14 18)(15 19)(16 20)
(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 32 9 18)(2 21 14 31)(3 26 11 20)(4 23 16 25)(5 28 13 22)(6 17 10 27)(7 30 15 24)(8 19 12 29)
G:=sub<Sym(32)| (2,32)(4,26)(6,28)(8,30)(10,22)(12,24)(14,18)(16,20), (1,13)(2,10)(3,15)(4,12)(5,9)(6,14)(7,11)(8,16)(17,31)(18,28)(19,25)(20,30)(21,27)(22,32)(23,29)(24,26), (1,31)(2,32)(3,25)(4,26)(5,27)(6,28)(7,29)(8,30)(9,21)(10,22)(11,23)(12,24)(13,17)(14,18)(15,19)(16,20), (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), (1,32,9,18)(2,21,14,31)(3,26,11,20)(4,23,16,25)(5,28,13,22)(6,17,10,27)(7,30,15,24)(8,19,12,29)>;
G:=Group( (2,32)(4,26)(6,28)(8,30)(10,22)(12,24)(14,18)(16,20), (1,13)(2,10)(3,15)(4,12)(5,9)(6,14)(7,11)(8,16)(17,31)(18,28)(19,25)(20,30)(21,27)(22,32)(23,29)(24,26), (1,31)(2,32)(3,25)(4,26)(5,27)(6,28)(7,29)(8,30)(9,21)(10,22)(11,23)(12,24)(13,17)(14,18)(15,19)(16,20), (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), (1,32,9,18)(2,21,14,31)(3,26,11,20)(4,23,16,25)(5,28,13,22)(6,17,10,27)(7,30,15,24)(8,19,12,29) );
G=PermutationGroup([[(2,32),(4,26),(6,28),(8,30),(10,22),(12,24),(14,18),(16,20)], [(1,13),(2,10),(3,15),(4,12),(5,9),(6,14),(7,11),(8,16),(17,31),(18,28),(19,25),(20,30),(21,27),(22,32),(23,29),(24,26)], [(1,31),(2,32),(3,25),(4,26),(5,27),(6,28),(7,29),(8,30),(9,21),(10,22),(11,23),(12,24),(13,17),(14,18),(15,19),(16,20)], [(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)], [(1,32,9,18),(2,21,14,31),(3,26,11,20),(4,23,16,25),(5,28,13,22),(6,17,10,27),(7,30,15,24),(8,19,12,29)]])
38 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | ··· | 2L | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | ··· | 4U | 8A | 8B | 8C | 8D |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 8 | 8 | 8 | 8 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 8 | 8 | 8 | 8 |
38 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | |||||
| image | C1 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | D4 | D4 | D4 | C4≀C2 | C42⋊C22 |
| kernel | C24.66D4 | C4×C22⋊C4 | C24.4C4 | C2×C4≀C2 | C22×C4○D4 | C22×D4 | C22×Q8 | C2×C4○D4 | C22×C4 | C4○D4 | C24 | C22 | C2 |
| # reps | 1 | 1 | 1 | 4 | 1 | 2 | 2 | 4 | 3 | 8 | 1 | 8 | 2 |
Matrix representation of C24.66D4 ►in GL4(𝔽17) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 2 | 16 |
| 0 | 4 | 0 | 0 |
| 13 | 0 | 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 |
| 16 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 11 | 11 | 0 | 0 |
| 11 | 6 | 0 | 0 |
| 0 | 0 | 1 | 16 |
| 0 | 0 | 2 | 16 |
| 11 | 6 | 0 | 0 |
| 11 | 11 | 0 | 0 |
| 0 | 0 | 1 | 16 |
| 0 | 0 | 0 | 16 |
G:=sub<GL(4,GF(17))| [1,0,0,0,0,1,0,0,0,0,1,2,0,0,0,16],[0,13,0,0,4,0,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,16,0,0,0,0,16],[16,0,0,0,0,16,0,0,0,0,1,0,0,0,0,1],[11,11,0,0,11,6,0,0,0,0,1,2,0,0,16,16],[11,11,0,0,6,11,0,0,0,0,1,0,0,0,16,16] >;
C24.66D4 in GAP, Magma, Sage, TeX
C_2^4._{66}D_4 % in TeX
G:=Group("C2^4.66D4"); // GroupNames label
G:=SmallGroup(128,521);
// by ID
G=gap.SmallGroup(128,521);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,422,2019,1018,248,124]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=1,e^4=d,f^2=d*b=e*b*e^-1=b*d,a*b=b*a,e*a*e^-1=f*a*f^-1=a*c=c*a,a*d=d*a,b*c=c*b,b*f=f*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