p-group, metabelian, nilpotent (class 4), monomial
Aliases: C23.20SD16, C4○D8.4C4, C4.89(C2×D8), D8.10(C2×C4), (C2×D8).13C4, (C2×C8).120D4, C8.115(C2×D4), (C2×C4).140D8, (C2×C16)⋊12C22, D8.C4⋊5C2, C8.7(C22⋊C4), C8.34(C22×C4), (C2×Q16).13C4, Q16.10(C2×C4), (C2×C4).50SD16, (C2×M5(2))⋊14C2, (C2×C8).577C23, C8.C4⋊8C22, C4○D8.13C22, (C22×C4).333D4, C4.38(D4⋊C4), C22.2(C2×SD16), (C22×C8).232C22, C22.31(D4⋊C4), (C2×C8).82(C2×C4), (C2×C4○D8).10C2, (C2×C4).761(C2×D4), C4.55(C2×C22⋊C4), (C2×C8.C4)⋊18C2, C2.33(C2×D4⋊C4), (C2×C4).151(C22⋊C4), SmallGroup(128,875)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C23.20SD16
G = < a,b,c,d,e | a2=b2=c2=e2=1, d8=c, ab=ba, dad-1=eae=ac=ca, ebe=bc=cb, bd=db, cd=dc, ce=ec, ede=bd3 >
Subgroups: 244 in 110 conjugacy classes, 50 normal (34 characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C8, C2×C4, C2×C4, D4, Q8, C23, C23, C16, C2×C8, C2×C8, M4(2), D8, D8, SD16, Q16, Q16, C22×C4, C22×C4, C2×D4, C2×Q8, C4○D4, C8.C4, C8.C4, C2×C16, M5(2), C22×C8, C2×M4(2), C2×D8, C2×SD16, C2×Q16, C4○D8, C4○D8, C2×C4○D4, D8.C4, C2×C8.C4, C2×M5(2), C2×C4○D8, C23.20SD16
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22⋊C4, D8, SD16, C22×C4, C2×D4, D4⋊C4, C2×C22⋊C4, C2×D8, C2×SD16, C2×D4⋊C4, C23.20SD16
►On 32 points
(1 30)(2 23)(3 32)(4 25)(5 18)(6 27)(7 20)(8 29)(9 22)(10 31)(11 24)(12 17)(13 26)(14 19)(15 28)(16 21)
(1 30)(2 31)(3 32)(4 17)(5 18)(6 19)(7 20)(8 21)(9 22)(10 23)(11 24)(12 25)(13 26)(14 27)(15 28)(16 29)
(1 9)(2 10)(3 11)(4 12)(5 13)(6 14)(7 15)(8 16)(17 25)(18 26)(19 27)(20 28)(21 29)(22 30)(23 31)(24 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)
(2 17)(3 7)(4 23)(5 13)(6 29)(8 19)(10 25)(11 15)(12 31)(14 21)(16 27)(20 24)(22 30)(28 32)
G:=sub<Sym(32)| (1,30)(2,23)(3,32)(4,25)(5,18)(6,27)(7,20)(8,29)(9,22)(10,31)(11,24)(12,17)(13,26)(14,19)(15,28)(16,21), (1,30)(2,31)(3,32)(4,17)(5,18)(6,19)(7,20)(8,21)(9,22)(10,23)(11,24)(12,25)(13,26)(14,27)(15,28)(16,29), (1,9)(2,10)(3,11)(4,12)(5,13)(6,14)(7,15)(8,16)(17,25)(18,26)(19,27)(20,28)(21,29)(22,30)(23,31)(24,32), (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), (2,17)(3,7)(4,23)(5,13)(6,29)(8,19)(10,25)(11,15)(12,31)(14,21)(16,27)(20,24)(22,30)(28,32)>;
G:=Group( (1,30)(2,23)(3,32)(4,25)(5,18)(6,27)(7,20)(8,29)(9,22)(10,31)(11,24)(12,17)(13,26)(14,19)(15,28)(16,21), (1,30)(2,31)(3,32)(4,17)(5,18)(6,19)(7,20)(8,21)(9,22)(10,23)(11,24)(12,25)(13,26)(14,27)(15,28)(16,29), (1,9)(2,10)(3,11)(4,12)(5,13)(6,14)(7,15)(8,16)(17,25)(18,26)(19,27)(20,28)(21,29)(22,30)(23,31)(24,32), (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), (2,17)(3,7)(4,23)(5,13)(6,29)(8,19)(10,25)(11,15)(12,31)(14,21)(16,27)(20,24)(22,30)(28,32) );
G=PermutationGroup([[(1,30),(2,23),(3,32),(4,25),(5,18),(6,27),(7,20),(8,29),(9,22),(10,31),(11,24),(12,17),(13,26),(14,19),(15,28),(16,21)], [(1,30),(2,31),(3,32),(4,17),(5,18),(6,19),(7,20),(8,21),(9,22),(10,23),(11,24),(12,25),(13,26),(14,27),(15,28),(16,29)], [(1,9),(2,10),(3,11),(4,12),(5,13),(6,14),(7,15),(8,16),(17,25),(18,26),(19,27),(20,28),(21,29),(22,30),(23,31),(24,32)], [(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)], [(2,17),(3,7),(4,23),(5,13),(6,29),(8,19),(10,25),(11,15),(12,31),(14,21),(16,27),(20,24),(22,30),(28,32)]])
32 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 8I | 8J | 16A | ··· | 16H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 16 | ··· | 16 |
| size | 1 | 1 | 2 | 2 | 2 | 8 | 8 | 1 | 1 | 2 | 2 | 2 | 8 | 8 | 2 | 2 | 2 | 2 | 4 | 4 | 8 | 8 | 8 | 8 | 4 | ··· | 4 |
32 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | ||||||
| image | C1 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | D4 | D4 | D8 | SD16 | SD16 | C23.20SD16 |
| kernel | C23.20SD16 | D8.C4 | C2×C8.C4 | C2×M5(2) | C2×C4○D8 | C2×D8 | C2×Q16 | C4○D8 | C2×C8 | C22×C4 | C2×C4 | C2×C4 | C23 | C1 |
| # reps | 1 | 4 | 1 | 1 | 1 | 2 | 2 | 4 | 3 | 1 | 4 | 2 | 2 | 4 |
Matrix representation of C23.20SD16 ►in GL4(𝔽17) generated by
| 0 | 13 | 0 | 0 |
| 4 | 0 | 0 | 0 |
| 15 | 16 | 0 | 13 |
| 16 | 2 | 4 | 0 |
| 0 | 13 | 0 | 0 |
| 4 | 0 | 0 | 0 |
| 3 | 16 | 0 | 4 |
| 1 | 3 | 13 | 0 |
| 16 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 16 |
| 3 | 0 | 0 | 15 |
| 9 | 13 | 15 | 0 |
| 15 | 14 | 4 | 8 |
| 2 | 6 | 0 | 14 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 7 | 7 | 14 | 3 |
| 16 | 16 | 3 | 3 |
G:=sub<GL(4,GF(17))| [0,4,15,16,13,0,16,2,0,0,0,4,0,0,13,0],[0,4,3,1,13,0,16,3,0,0,0,13,0,0,4,0],[16,0,0,0,0,16,0,0,0,0,16,0,0,0,0,16],[3,9,15,2,0,13,14,6,0,15,4,0,15,0,8,14],[0,1,7,16,1,0,7,16,0,0,14,3,0,0,3,3] >;
C23.20SD16 in GAP, Magma, Sage, TeX
C_2^3._{20}{\rm SD}_{16} % in TeX
G:=Group("C2^3.20SD16"); // GroupNames label
G:=SmallGroup(128,875);
// by ID
G=gap.SmallGroup(128,875);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,-2,-2,112,141,723,1123,570,360,172,4037,2028,124]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^2=c^2=e^2=1,d^8=c,a*b=b*a,d*a*d^-1=e*a*e=a*c=c*a,e*b*e=b*c=c*b,b*d=d*b,c*d=d*c,c*e=e*c,e*d*e=b*d^3>;
// generators/relations