p-group, metabelian, nilpotent (class 4), monomial
Aliases: C23.9D8, C8.12C42, (C2×C16)⋊6C4, C4.Q8⋊3C4, C8.2(C4⋊C4), (C2×C8).3Q8, (C2×C8).82D4, (C2×C4).7Q16, (C2×C4).94SD16, C4.15(C4.Q8), C8.35(C22⋊C4), C4.6(Q8⋊C4), (C22×C4).190D4, (C2×M5(2)).12C2, C22.10(C2.D8), C4.5(C2.C42), (C22×C8).203C22, C23.25D4.8C2, C22.22(D4⋊C4), C2.14(C22.4Q16), (C2×C8).50(C2×C4), (C2×C4).112(C4⋊C4), (C2×C4).63(C22⋊C4), SmallGroup(128,116)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C23.9D8
G = < a,b,c,d,e | a2=b2=c2=1, d8=c, e2=abc, ab=ba, eae-1=ac=ca, ad=da, dbd-1=ebe-1=bc=cb, cd=dc, ce=ec, ede-1=ad7 >
Subgroups: 136 in 66 conjugacy classes, 38 normal (16 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C22, C8, C8, C2×C4, C2×C4, C2×C4, C23, C16, C42, C22⋊C4, C4⋊C4, C2×C8, C2×C8, C22×C4, C4.Q8, C2.D8, C2×C16, M5(2), C42⋊C2, C22×C8, C23.25D4, C2×M5(2), C23.9D8
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C42, C22⋊C4, C4⋊C4, D8, SD16, Q16, C2.C42, D4⋊C4, Q8⋊C4, C4.Q8, C2.D8, C22.4Q16, C23.9D8
►On 32 points
(1 19)(2 20)(3 21)(4 22)(5 23)(6 24)(7 25)(8 26)(9 27)(10 28)(11 29)(12 30)(13 31)(14 32)(15 17)(16 18)
(1 19)(2 28)(3 21)(4 30)(5 23)(6 32)(7 25)(8 18)(9 27)(10 20)(11 29)(12 22)(13 31)(14 24)(15 17)(16 26)
(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)
(1 17 9 25)(2 14)(3 31 11 23)(4 12)(5 29 13 21)(6 10)(7 27 15 19)(18 26)(20 24)(28 32)
G:=sub<Sym(32)| (1,19)(2,20)(3,21)(4,22)(5,23)(6,24)(7,25)(8,26)(9,27)(10,28)(11,29)(12,30)(13,31)(14,32)(15,17)(16,18), (1,19)(2,28)(3,21)(4,30)(5,23)(6,32)(7,25)(8,18)(9,27)(10,20)(11,29)(12,22)(13,31)(14,24)(15,17)(16,26), (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), (1,17,9,25)(2,14)(3,31,11,23)(4,12)(5,29,13,21)(6,10)(7,27,15,19)(18,26)(20,24)(28,32)>;
G:=Group( (1,19)(2,20)(3,21)(4,22)(5,23)(6,24)(7,25)(8,26)(9,27)(10,28)(11,29)(12,30)(13,31)(14,32)(15,17)(16,18), (1,19)(2,28)(3,21)(4,30)(5,23)(6,32)(7,25)(8,18)(9,27)(10,20)(11,29)(12,22)(13,31)(14,24)(15,17)(16,26), (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), (1,17,9,25)(2,14)(3,31,11,23)(4,12)(5,29,13,21)(6,10)(7,27,15,19)(18,26)(20,24)(28,32) );
G=PermutationGroup([[(1,19),(2,20),(3,21),(4,22),(5,23),(6,24),(7,25),(8,26),(9,27),(10,28),(11,29),(12,30),(13,31),(14,32),(15,17),(16,18)], [(1,19),(2,28),(3,21),(4,30),(5,23),(6,32),(7,25),(8,18),(9,27),(10,20),(11,29),(12,22),(13,31),(14,24),(15,17),(16,26)], [(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)], [(1,17,9,25),(2,14),(3,31,11,23),(4,12),(5,29,13,21),(6,10),(7,27,15,19),(18,26),(20,24),(28,32)]])
32 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 4A | 4B | 4C | 4D | 4E | 4F | ··· | 4M | 8A | 8B | 8C | 8D | 8E | 8F | 16A | ··· | 16H |
| order | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 8 | 8 | 8 | 8 | 8 | 8 | 16 | ··· | 16 |
| size | 1 | 1 | 2 | 2 | 2 | 1 | 1 | 2 | 2 | 2 | 8 | ··· | 8 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | ··· | 4 |
32 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | - | + | - | + | ||||
| image | C1 | C2 | C2 | C4 | C4 | D4 | Q8 | D4 | SD16 | Q16 | D8 | C23.9D8 |
| kernel | C23.9D8 | C23.25D4 | C2×M5(2) | C4.Q8 | C2×C16 | C2×C8 | C2×C8 | C22×C4 | C2×C4 | C2×C4 | C23 | C1 |
| # reps | 1 | 2 | 1 | 8 | 4 | 2 | 1 | 1 | 4 | 2 | 2 | 4 |
Matrix representation of C23.9D8 ►in GL4(𝔽17) generated by
| 0 | 4 | 0 | 0 |
| 13 | 0 | 0 | 0 |
| 0 | 0 | 0 | 4 |
| 0 | 0 | 13 | 0 |
| 0 | 4 | 0 | 0 |
| 13 | 0 | 0 | 0 |
| 0 | 0 | 0 | 13 |
| 0 | 0 | 4 | 0 |
| 16 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 16 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 14 | 3 | 0 | 0 |
| 14 | 14 | 0 | 0 |
| 5 | 12 | 0 | 0 |
| 12 | 12 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 |
G:=sub<GL(4,GF(17))| [0,13,0,0,4,0,0,0,0,0,0,13,0,0,4,0],[0,13,0,0,4,0,0,0,0,0,0,4,0,0,13,0],[16,0,0,0,0,16,0,0,0,0,16,0,0,0,0,16],[0,0,14,14,0,0,3,14,1,0,0,0,0,1,0,0],[5,12,0,0,12,12,0,0,0,0,0,1,0,0,1,0] >;
C23.9D8 in GAP, Magma, Sage, TeX
C_2^3._9D_8
% in TeX
G:=Group("C2^3.9D8"); // GroupNames label
G:=SmallGroup(128,116);
// by ID
G=gap.SmallGroup(128,116);
# by ID
G:=PCGroup([7,-2,2,-2,2,2,-2,-2,56,85,120,758,184,1018,1684,242,4037,124]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^2=c^2=1,d^8=c,e^2=a*b*c,a*b=b*a,e*a*e^-1=a*c=c*a,a*d=d*a,d*b*d^-1=e*b*e^-1=b*c=c*b,c*d=d*c,c*e=e*c,e*d*e^-1=a*d^7>;
// generators/relations