p-group, metabelian, nilpotent (class 3), monomial
Aliases: C24.53D4, C4⋊D4⋊2C4, C22⋊Q8⋊2C4, C22.7C4≀C2, C42⋊C2⋊1C4, C4.45(C23⋊C4), C23.491(C2×D4), (C22×C4).661D4, C4○2(C22.SD16), C22.17(C4○D8), C22.SD16⋊23C2, C4○2(C23.31D4), C4⋊D4.130C22, C23.31D4⋊24C2, C22⋊C8.162C22, C23.51(C22⋊C4), (C23×C4).205C22, (C22×C4).623C23, C22.19C24.1C2, C22⋊Q8.135C22, C2.9(C23.24D4), C2.C42.500C22, (C2×C4○D4)⋊1C4, C4⋊C4.2(C2×C4), C2.18(C2×C4≀C2), (C2×C22⋊C8)⋊6C2, (C2×D4).4(C2×C4), (C2×Q8).4(C2×C4), (C4×C22⋊C4)⋊20C2, C2.14(C2×C23⋊C4), (C2×C4).1147(C2×D4), (C22×C4).196(C2×C4), (C2×C4).113(C22×C4), (C2×C4).390(C22⋊C4), C22.177(C2×C22⋊C4), SmallGroup(128,233)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C24.53D4
G = < a,b,c,d,e,f | a2=b2=c2=d2=1, e4=d, f2=b, ab=ba, faf-1=ac=ca, ad=da, eae-1=acd, bc=cb, bd=db, ebe-1=bcd, bf=fb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=bde3 >
Subgroups: 348 in 152 conjugacy classes, 48 normal (34 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C8, C2×C4, C2×C4, D4, Q8, C23, C23, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4, C24, C2.C42, C22⋊C8, C22⋊C8, C2×C42, C2×C22⋊C4, C42⋊C2, C4×D4, C22≀C2, C4⋊D4, C22⋊Q8, C22.D4, C22×C8, C23×C4, C2×C4○D4, C22.SD16, C23.31D4, C4×C22⋊C4, C2×C22⋊C8, C22.19C24, C24.53D4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22⋊C4, C22×C4, C2×D4, C23⋊C4, C4≀C2, C2×C22⋊C4, C4○D8, C2×C23⋊C4, C23.24D4, C2×C4≀C2, C24.53D4
►On 32 points
(1 19)(2 11)(3 21)(4 13)(5 23)(6 15)(7 17)(8 9)(10 31)(12 25)(14 27)(16 29)(18 30)(20 32)(22 26)(24 28)
(1 5)(2 28)(3 7)(4 30)(6 32)(8 26)(9 22)(10 14)(11 24)(12 16)(13 18)(15 20)(17 21)(19 23)(25 29)(27 31)
(1 27)(2 28)(3 29)(4 30)(5 31)(6 32)(7 25)(8 26)(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 17 5 21)(2 11 28 24)(3 14 7 10)(4 18 30 13)(6 15 32 20)(8 22 26 9)(12 31 16 27)(19 25 23 29)
G:=sub<Sym(32)| (1,19)(2,11)(3,21)(4,13)(5,23)(6,15)(7,17)(8,9)(10,31)(12,25)(14,27)(16,29)(18,30)(20,32)(22,26)(24,28), (1,5)(2,28)(3,7)(4,30)(6,32)(8,26)(9,22)(10,14)(11,24)(12,16)(13,18)(15,20)(17,21)(19,23)(25,29)(27,31), (1,27)(2,28)(3,29)(4,30)(5,31)(6,32)(7,25)(8,26)(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), (1,17,5,21)(2,11,28,24)(3,14,7,10)(4,18,30,13)(6,15,32,20)(8,22,26,9)(12,31,16,27)(19,25,23,29)>;
G:=Group( (1,19)(2,11)(3,21)(4,13)(5,23)(6,15)(7,17)(8,9)(10,31)(12,25)(14,27)(16,29)(18,30)(20,32)(22,26)(24,28), (1,5)(2,28)(3,7)(4,30)(6,32)(8,26)(9,22)(10,14)(11,24)(12,16)(13,18)(15,20)(17,21)(19,23)(25,29)(27,31), (1,27)(2,28)(3,29)(4,30)(5,31)(6,32)(7,25)(8,26)(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), (1,17,5,21)(2,11,28,24)(3,14,7,10)(4,18,30,13)(6,15,32,20)(8,22,26,9)(12,31,16,27)(19,25,23,29) );
G=PermutationGroup([[(1,19),(2,11),(3,21),(4,13),(5,23),(6,15),(7,17),(8,9),(10,31),(12,25),(14,27),(16,29),(18,30),(20,32),(22,26),(24,28)], [(1,5),(2,28),(3,7),(4,30),(6,32),(8,26),(9,22),(10,14),(11,24),(12,16),(13,18),(15,20),(17,21),(19,23),(25,29),(27,31)], [(1,27),(2,28),(3,29),(4,30),(5,31),(6,32),(7,25),(8,26),(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)], [(1,17,5,21),(2,11,28,24),(3,14,7,10),(4,18,30,13),(6,15,32,20),(8,22,26,9),(12,31,16,27),(19,25,23,29)]])
38 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | ··· | 4Q | 4R | 4S | 4T | 8A | ··· | 8H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | 4 | 4 | 8 | ··· | 8 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 8 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 8 | 8 | 8 | 4 | ··· | 4 |
38 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | ||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | C4 | D4 | D4 | C4≀C2 | C4○D8 | C23⋊C4 |
| kernel | C24.53D4 | C22.SD16 | C23.31D4 | C4×C22⋊C4 | C2×C22⋊C8 | C22.19C24 | C42⋊C2 | C4⋊D4 | C22⋊Q8 | C2×C4○D4 | C22×C4 | C24 | C22 | C22 | C4 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 3 | 1 | 8 | 8 | 2 |
Matrix representation of C24.53D4 ►in GL4(𝔽17) generated by
| 1 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 0 | 13 |
| 0 | 0 | 4 | 0 |
| 16 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 16 | 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 |
| 0 | 16 | 0 | 0 |
| 4 | 0 | 0 | 0 |
| 0 | 0 | 5 | 12 |
| 0 | 0 | 5 | 5 |
| 4 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 13 | 0 |
| 0 | 0 | 0 | 4 |
G:=sub<GL(4,GF(17))| [1,0,0,0,0,16,0,0,0,0,0,4,0,0,13,0],[16,0,0,0,0,1,0,0,0,0,16,0,0,0,0,16],[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],[0,4,0,0,16,0,0,0,0,0,5,5,0,0,12,5],[4,0,0,0,0,16,0,0,0,0,13,0,0,0,0,4] >;
C24.53D4 in GAP, Magma, Sage, TeX
C_2^4._{53}D_4 % in TeX
G:=Group("C2^4.53D4"); // GroupNames label
G:=SmallGroup(128,233);
// by ID
G=gap.SmallGroup(128,233);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,-2,2,112,141,352,1123,1018,248,1971]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=1,e^4=d,f^2=b,a*b=b*a,f*a*f^-1=a*c=c*a,a*d=d*a,e*a*e^-1=a*c*d,b*c=c*b,b*d=d*b,e*b*e^-1=b*c*d,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*d*e^3>;
// generators/relations