p-group, metabelian, nilpotent (class 3), monomial
Aliases: C24.48D4, (C2×C4).9C42, C22.11C4≀C2, (C22×C4).5Q8, (C2×M4(2))⋊1C4, C4.33(C23⋊C4), C23.14(C4⋊C4), (C22×C4).640D4, C2.C42⋊10C4, C24.4C4.6C2, C2.10(C42⋊6C4), (C23×C4).193C22, C2.7(C23.9D4), C23.141(C22⋊C4), C22.47(C2.C42), (C2×C4).71(C4⋊C4), (C4×C22⋊C4).9C2, (C22×C4).152(C2×C4), (C2×C4).373(C22⋊C4), SmallGroup(128,29)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C24.48D4
G = < a,b,c,d,e,f | a2=b2=c2=d2=1, e4=c, f2=abd, ab=ba, eae-1=ac=ca, ad=da, faf-1=acd, bc=cb, bd=db, ebe-1=fbf-1=bcd, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=acde3 >
Subgroups: 280 in 123 conjugacy classes, 36 normal (12 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C8, C2×C4, C2×C4, C2×C4, C23, C23, C23, C42, C22⋊C4, C2×C8, M4(2), C22×C4, C22×C4, C22×C4, C24, C2.C42, C22⋊C8, C2×C42, C2×C22⋊C4, C2×M4(2), C23×C4, C4×C22⋊C4, C24.4C4, C24.48D4
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C42, C22⋊C4, C4⋊C4, C2.C42, C23⋊C4, C4≀C2, C42⋊6C4, C23.9D4, C24.48D4
►On 32 points
(1 10)(2 15)(3 12)(4 9)(5 14)(6 11)(7 16)(8 13)(17 26)(18 31)(19 28)(20 25)(21 30)(22 27)(23 32)(24 29)
(1 14)(2 32)(3 16)(4 26)(5 10)(6 28)(7 12)(8 30)(9 17)(11 19)(13 21)(15 23)(18 27)(20 29)(22 31)(24 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 18)(2 19)(3 20)(4 21)(5 22)(6 23)(7 24)(8 17)(9 30)(10 31)(11 32)(12 25)(13 26)(14 27)(15 28)(16 29)
(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 14 22 31)(2 4 6 8)(3 16 24 25)(5 10 18 27)(7 12 20 29)(9 28 13 32)(11 30 15 26)(17 19 21 23)
G:=sub<Sym(32)| (1,10)(2,15)(3,12)(4,9)(5,14)(6,11)(7,16)(8,13)(17,26)(18,31)(19,28)(20,25)(21,30)(22,27)(23,32)(24,29), (1,14)(2,32)(3,16)(4,26)(5,10)(6,28)(7,12)(8,30)(9,17)(11,19)(13,21)(15,23)(18,27)(20,29)(22,31)(24,25), (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), (1,18)(2,19)(3,20)(4,21)(5,22)(6,23)(7,24)(8,17)(9,30)(10,31)(11,32)(12,25)(13,26)(14,27)(15,28)(16,29), (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), (1,14,22,31)(2,4,6,8)(3,16,24,25)(5,10,18,27)(7,12,20,29)(9,28,13,32)(11,30,15,26)(17,19,21,23)>;
G:=Group( (1,10)(2,15)(3,12)(4,9)(5,14)(6,11)(7,16)(8,13)(17,26)(18,31)(19,28)(20,25)(21,30)(22,27)(23,32)(24,29), (1,14)(2,32)(3,16)(4,26)(5,10)(6,28)(7,12)(8,30)(9,17)(11,19)(13,21)(15,23)(18,27)(20,29)(22,31)(24,25), (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), (1,18)(2,19)(3,20)(4,21)(5,22)(6,23)(7,24)(8,17)(9,30)(10,31)(11,32)(12,25)(13,26)(14,27)(15,28)(16,29), (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), (1,14,22,31)(2,4,6,8)(3,16,24,25)(5,10,18,27)(7,12,20,29)(9,28,13,32)(11,30,15,26)(17,19,21,23) );
G=PermutationGroup([[(1,10),(2,15),(3,12),(4,9),(5,14),(6,11),(7,16),(8,13),(17,26),(18,31),(19,28),(20,25),(21,30),(22,27),(23,32),(24,29)], [(1,14),(2,32),(3,16),(4,26),(5,10),(6,28),(7,12),(8,30),(9,17),(11,19),(13,21),(15,23),(18,27),(20,29),(22,31),(24,25)], [(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)], [(1,18),(2,19),(3,20),(4,21),(5,22),(6,23),(7,24),(8,17),(9,30),(10,31),(11,32),(12,25),(13,26),(14,27),(15,28),(16,29)], [(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)], [(1,14,22,31),(2,4,6,8),(3,16,24,25),(5,10,18,27),(7,12,20,29),(9,28,13,32),(11,30,15,26),(17,19,21,23)]])
38 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | ··· | 4Y | 8A | 8B | 8C | 8D |
| order | 1 | 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 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 8 | 8 | 8 | 8 |
38 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | - | + | + | |||
| image | C1 | C2 | C2 | C4 | C4 | D4 | Q8 | D4 | C4≀C2 | C23⋊C4 |
| kernel | C24.48D4 | C4×C22⋊C4 | C24.4C4 | C2.C42 | C2×M4(2) | C22×C4 | C22×C4 | C24 | C22 | C4 |
| # reps | 1 | 2 | 1 | 8 | 4 | 2 | 1 | 1 | 16 | 2 |
Matrix representation of C24.48D4 ►in GL4(𝔽17) generated by
| 16 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 2 |
| 0 | 0 | 0 | 16 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 16 | 15 |
| 0 | 0 | 0 | 1 |
| 16 | 0 | 0 | 0 |
| 0 | 16 | 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 |
| 0 | 13 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 16 | 12 |
| 0 | 0 | 1 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 13 | 0 | 0 |
| 0 | 0 | 1 | 2 |
| 0 | 0 | 16 | 16 |
G:=sub<GL(4,GF(17))| [16,0,0,0,0,1,0,0,0,0,1,0,0,0,2,16],[1,0,0,0,0,1,0,0,0,0,16,0,0,0,15,1],[16,0,0,0,0,16,0,0,0,0,16,0,0,0,0,16],[16,0,0,0,0,16,0,0,0,0,1,0,0,0,0,1],[0,1,0,0,13,0,0,0,0,0,16,1,0,0,12,1],[1,0,0,0,0,13,0,0,0,0,1,16,0,0,2,16] >;
C24.48D4 in GAP, Magma, Sage, TeX
C_2^4._{48}D_4 % in TeX
G:=Group("C2^4.48D4"); // GroupNames label
G:=SmallGroup(128,29);
// by ID
G=gap.SmallGroup(128,29);
# by ID
G:=PCGroup([7,-2,2,-2,2,2,-2,2,56,85,120,758,723,184,136,3924]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=1,e^4=c,f^2=a*b*d,a*b=b*a,e*a*e^-1=a*c=c*a,a*d=d*a,f*a*f^-1=a*c*d,b*c=c*b,b*d=d*b,e*b*e^-1=f*b*f^-1=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^-1=a*c*d*e^3>;
// generators/relations