p-group, metabelian, nilpotent (class 3), monomial
Aliases: C24.103D4, C4○D4⋊13D4, C4○(D4⋊D4), C4○(C22⋊D8), C4○(Q8⋊D4), D4.39(C2×D4), C4⋊C4.5C23, Q8.39(C2×D4), D4⋊D4⋊51C2, C22⋊D8⋊35C2, Q8⋊D4⋊36C2, C4○(D4.7D4), C4○(C22⋊SD16), (C2×D8)⋊37C22, C22⋊5(C4○D8), C4○(C22⋊Q16), C4.39(C22×D4), C4.129C22≀C2, D4.7D4⋊52C2, C22⋊SD16⋊36C2, C4⋊D4⋊49C22, C22⋊C8⋊58C22, (C2×C4).221C24, (C2×C8).128C23, C22⋊Q16⋊35C2, (C22×C8)⋊14C22, (C2×Q16)⋊38C22, C22.2C22≀C2, (C22×C4).714D4, C23.645(C2×D4), C22⋊Q8⋊61C22, (C2×Q8).18C23, D4⋊C4⋊71C22, C42⋊C2⋊5C22, C22.19C24⋊2C2, Q8⋊C4⋊65C22, (C2×SD16)⋊69C22, (C2×D4).380C23, C23.24D4⋊14C2, C2.6(D8⋊C22), (C23×C4).541C22, (C22×C4).959C23, C22.481(C22×D4), (C22×D4).562C22, (C22×Q8).466C22, (C2×C4○D8)⋊2C2, C2.9(C2×C4○D8), (C2×C4)○(C22⋊D8), (C2×C4)○(D4⋊D4), (C2×C22⋊C8)⋊25C2, (C2×C4)○(Q8⋊D4), (C2×C4).449(C2×D4), (C22×C4○D4)⋊8C2, (C2×C4○D4)⋊1C22, C2.39(C2×C22≀C2), (C2×C4)○(D4.7D4), (C2×C4)○(C22⋊SD16), (C2×C4)○(C22⋊Q16), SmallGroup(128,1734)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C24.103D4
G = < a,b,c,d,e,f | a2=b2=c2=d2=f2=1, e4=d, ab=ba, eae-1=faf=ac=ca, ad=da, bc=cb, fbf=bd=db, be=eb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef=de3 >
Subgroups: 740 in 382 conjugacy classes, 110 normal (42 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, C4⋊C4, C4⋊C4, C2×C8, C2×C8, D8, SD16, Q16, C22×C4, C22×C4, C2×D4, C2×D4, C2×D4, C2×Q8, C2×Q8, C2×Q8, C4○D4, C4○D4, C24, C24, C22⋊C8, C22⋊C8, D4⋊C4, Q8⋊C4, C42⋊C2, C4×D4, C22≀C2, C4⋊D4, C22⋊Q8, C22.D4, C22×C8, C2×D8, C2×SD16, C2×Q16, C4○D8, C23×C4, C23×C4, C22×D4, C22×D4, C22×Q8, C2×C4○D4, C2×C4○D4, C2×C4○D4, C2×C22⋊C8, C23.24D4, C22⋊D8, Q8⋊D4, D4⋊D4, C22⋊SD16, C22⋊Q16, D4.7D4, C22.19C24, C2×C4○D8, C22×C4○D4, C24.103D4
Quotients: C1, C2, C22, D4, C23, C2×D4, C24, C22≀C2, C4○D8, C22×D4, C2×C22≀C2, C2×C4○D8, D8⋊C22, C24.103D4
►On 32 points
(2 25)(4 27)(6 29)(8 31)(9 24)(11 18)(13 20)(15 22)
(1 32)(2 25)(3 26)(4 27)(5 28)(6 29)(7 30)(8 31)(9 20)(10 21)(11 22)(12 23)(13 24)(14 17)(15 18)(16 19)
(1 32)(2 25)(3 26)(4 27)(5 28)(6 29)(7 30)(8 31)(9 24)(10 17)(11 18)(12 19)(13 20)(14 21)(15 22)(16 23)
(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 15)(2 14)(3 13)(4 12)(5 11)(6 10)(7 9)(8 16)(17 29)(18 28)(19 27)(20 26)(21 25)(22 32)(23 31)(24 30)
G:=sub<Sym(32)| (2,25)(4,27)(6,29)(8,31)(9,24)(11,18)(13,20)(15,22), (1,32)(2,25)(3,26)(4,27)(5,28)(6,29)(7,30)(8,31)(9,20)(10,21)(11,22)(12,23)(13,24)(14,17)(15,18)(16,19), (1,32)(2,25)(3,26)(4,27)(5,28)(6,29)(7,30)(8,31)(9,24)(10,17)(11,18)(12,19)(13,20)(14,21)(15,22)(16,23), (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), (1,15)(2,14)(3,13)(4,12)(5,11)(6,10)(7,9)(8,16)(17,29)(18,28)(19,27)(20,26)(21,25)(22,32)(23,31)(24,30)>;
G:=Group( (2,25)(4,27)(6,29)(8,31)(9,24)(11,18)(13,20)(15,22), (1,32)(2,25)(3,26)(4,27)(5,28)(6,29)(7,30)(8,31)(9,20)(10,21)(11,22)(12,23)(13,24)(14,17)(15,18)(16,19), (1,32)(2,25)(3,26)(4,27)(5,28)(6,29)(7,30)(8,31)(9,24)(10,17)(11,18)(12,19)(13,20)(14,21)(15,22)(16,23), (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), (1,15)(2,14)(3,13)(4,12)(5,11)(6,10)(7,9)(8,16)(17,29)(18,28)(19,27)(20,26)(21,25)(22,32)(23,31)(24,30) );
G=PermutationGroup([[(2,25),(4,27),(6,29),(8,31),(9,24),(11,18),(13,20),(15,22)], [(1,32),(2,25),(3,26),(4,27),(5,28),(6,29),(7,30),(8,31),(9,20),(10,21),(11,22),(12,23),(13,24),(14,17),(15,18),(16,19)], [(1,32),(2,25),(3,26),(4,27),(5,28),(6,29),(7,30),(8,31),(9,24),(10,17),(11,18),(12,19),(13,20),(14,21),(15,22),(16,23)], [(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)], [(1,15),(2,14),(3,13),(4,12),(5,11),(6,10),(7,9),(8,16),(17,29),(18,28),(19,27),(20,26),(21,25),(22,32),(23,31),(24,30)]])
38 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | ··· | 2L | 2M | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | ··· | 4M | 4N | 4O | 4P | 8A | ··· | 8H |
| order | 1 | 2 | 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 | ··· | 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 | 1 | 1 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D4 | C4○D8 | D8⋊C22 |
| kernel | C24.103D4 | C2×C22⋊C8 | C23.24D4 | C22⋊D8 | Q8⋊D4 | D4⋊D4 | C22⋊SD16 | C22⋊Q16 | D4.7D4 | C22.19C24 | C2×C4○D8 | C22×C4○D4 | C22×C4 | C4○D4 | C24 | C22 | C2 |
| # reps | 1 | 1 | 2 | 1 | 1 | 2 | 1 | 1 | 2 | 1 | 2 | 1 | 3 | 8 | 1 | 8 | 2 |
Matrix representation of C24.103D4 ►in GL4(𝔽17) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 16 |
| 1 | 0 | 0 | 0 |
| 0 | 16 | 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 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 15 | 0 | 0 | 0 |
| 0 | 8 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 |
| 0 | 8 | 0 | 0 |
| 15 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 |
G:=sub<GL(4,GF(17))| [1,0,0,0,0,1,0,0,0,0,1,0,0,0,0,16],[1,0,0,0,0,16,0,0,0,0,16,0,0,0,0,16],[1,0,0,0,0,1,0,0,0,0,16,0,0,0,0,16],[16,0,0,0,0,16,0,0,0,0,1,0,0,0,0,1],[15,0,0,0,0,8,0,0,0,0,0,1,0,0,1,0],[0,15,0,0,8,0,0,0,0,0,0,1,0,0,1,0] >;
C24.103D4 in GAP, Magma, Sage, TeX
C_2^4._{103}D_4 % in TeX
G:=Group("C2^4.103D4"); // GroupNames label
G:=SmallGroup(128,1734);
// by ID
G=gap.SmallGroup(128,1734);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,758,248,2804,1411,172]);
// 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,e*a*e^-1=f*a*f=a*c=c*a,a*d=d*a,b*c=c*b,f*b*f=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=d*e^3>;
// generators/relations