p-group, metabelian, nilpotent (class 4), monomial
Aliases: C23.21SD16, M5(2).21C22, C4○D8.5C4, C4.66(C2×D8), C8.98(C2×D4), D8.13(C2×C4), (C2×C8).125D4, (C2×C4).144D8, C4○(M5(2)⋊C2), M5(2)⋊C2⋊9C2, C4○(C8.17D4), C8.17D4⋊9C2, C8.10(C22×C4), Q16.13(C2×C4), C8.12(C22⋊C4), (C2×M5(2))⋊18C2, (C2×C8).228C23, (C2×C4).107SD16, (C22×C4).338D4, C4.40(D4⋊C4), (C2×D8).154C22, C22.18(C2×SD16), C22.9(D4⋊C4), C8.C4.11C22, (C22×C8).237C22, (C2×Q16).149C22, (C2×C8).87(C2×C4), (C2×C4○D8).13C2, (C2×C4).273(C2×D4), C4.60(C2×C22⋊C4), (C2×C8.C4)⋊20C2, C2.38(C2×D4⋊C4), (C2×C4).153(C22⋊C4), SmallGroup(128,880)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C23.21SD16
G = < a,b,c,d,e | a2=b2=c2=e2=1, d8=c, ab=ba, eae=ac=ca, ad=da, dbd-1=bc=cb, be=eb, 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), M5(2), C22×C8, C2×M4(2), C2×D8, C2×SD16, C2×Q16, C4○D8, C4○D8, C2×C4○D4, M5(2)⋊C2, C8.17D4, C2×C8.C4, C2×M5(2), C2×C4○D8, C23.21SD16
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, C23.21SD16
►On 32 points
(1 24)(2 25)(3 26)(4 27)(5 28)(6 29)(7 30)(8 31)(9 32)(10 17)(11 18)(12 19)(13 20)(14 21)(15 22)(16 23)
(1 9)(3 11)(5 13)(7 15)(18 26)(20 28)(22 30)(24 32)
(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 12)(3 15)(4 10)(5 13)(6 8)(7 11)(14 16)(17 19)(18 22)(21 31)(23 29)(24 32)(25 27)(26 30)
G:=sub<Sym(32)| (1,24)(2,25)(3,26)(4,27)(5,28)(6,29)(7,30)(8,31)(9,32)(10,17)(11,18)(12,19)(13,20)(14,21)(15,22)(16,23), (1,9)(3,11)(5,13)(7,15)(18,26)(20,28)(22,30)(24,32), (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), (2,12)(3,15)(4,10)(5,13)(6,8)(7,11)(14,16)(17,19)(18,22)(21,31)(23,29)(24,32)(25,27)(26,30)>;
G:=Group( (1,24)(2,25)(3,26)(4,27)(5,28)(6,29)(7,30)(8,31)(9,32)(10,17)(11,18)(12,19)(13,20)(14,21)(15,22)(16,23), (1,9)(3,11)(5,13)(7,15)(18,26)(20,28)(22,30)(24,32), (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), (2,12)(3,15)(4,10)(5,13)(6,8)(7,11)(14,16)(17,19)(18,22)(21,31)(23,29)(24,32)(25,27)(26,30) );
G=PermutationGroup([[(1,24),(2,25),(3,26),(4,27),(5,28),(6,29),(7,30),(8,31),(9,32),(10,17),(11,18),(12,19),(13,20),(14,21),(15,22),(16,23)], [(1,9),(3,11),(5,13),(7,15),(18,26),(20,28),(22,30),(24,32)], [(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)], [(2,12),(3,15),(4,10),(5,13),(6,8),(7,11),(14,16),(17,19),(18,22),(21,31),(23,29),(24,32),(25,27),(26,30)]])
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 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | D4 | D4 | D8 | SD16 | SD16 | C23.21SD16 |
| kernel | C23.21SD16 | M5(2)⋊C2 | C8.17D4 | C2×C8.C4 | C2×M5(2) | C2×C4○D8 | C4○D8 | C2×C8 | C22×C4 | C2×C4 | C2×C4 | C23 | C1 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 8 | 3 | 1 | 4 | 2 | 2 | 4 |
Matrix representation of C23.21SD16 ►in GL4(𝔽17) generated by
| 0 | 4 | 0 | 0 |
| 13 | 0 | 0 | 0 |
| 1 | 4 | 13 | 8 |
| 11 | 10 | 13 | 4 |
| 16 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 8 | 6 | 1 | 0 |
| 14 | 8 | 0 | 1 |
| 16 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 16 |
| 10 | 5 | 16 | 2 |
| 13 | 14 | 16 | 0 |
| 3 | 9 | 7 | 9 |
| 11 | 11 | 11 | 3 |
| 0 | 16 | 0 | 0 |
| 16 | 0 | 0 | 0 |
| 2 | 15 | 6 | 11 |
| 8 | 9 | 3 | 11 |
G:=sub<GL(4,GF(17))| [0,13,1,11,4,0,4,10,0,0,13,13,0,0,8,4],[16,0,8,14,0,16,6,8,0,0,1,0,0,0,0,1],[16,0,0,0,0,16,0,0,0,0,16,0,0,0,0,16],[10,13,3,11,5,14,9,11,16,16,7,11,2,0,9,3],[0,16,2,8,16,0,15,9,0,0,6,3,0,0,11,11] >;
C23.21SD16 in GAP, Magma, Sage, TeX
C_2^3._{21}{\rm SD}_{16} % in TeX
G:=Group("C2^3.21SD16"); // GroupNames label
G:=SmallGroup(128,880);
// by ID
G=gap.SmallGroup(128,880);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,-2,-2,112,141,352,1123,1466,136,1411,172,4037,2028,124]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^2=c^2=e^2=1,d^8=c,a*b=b*a,e*a*e=a*c=c*a,a*d=d*a,d*b*d^-1=b*c=c*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e=b*d^3>;
// generators/relations