p-group, metabelian, nilpotent (class 4), monomial
Aliases: M5(2)⋊3C4, C23.22SD16, M5(2).16C22, (C2×C16)⋊2C4, C4○(C8.Q8), C8.Q8⋊7C2, C16.2(C2×C4), C8.22(C4⋊C4), (C2×C8).19Q8, C8.12(C2×Q8), (C2×C8).127D4, C8.59(C22×C4), C4.66(C2×SD16), (C2×C4).56SD16, C4.18(C4.Q8), (C2×C8).230C23, (C22×C4).340D4, (C2×M5(2)).5C2, C22.6(C4.Q8), C22.20(C2×SD16), C4.Q8.122C22, C8.C4.13C22, (C22×C8).239C22, C23.25D4.15C2, C4.49(C2×C4⋊C4), (C2×C8).90(C2×C4), C2.12(C2×C4.Q8), (C2×C4).57(C4⋊C4), (C2×C4).275(C2×D4), (C2×C8.C4).23C2, SmallGroup(128,887)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for M5(2)⋊3C4
G = < a,b,c | a16=b2=c4=1, bab=a9, cac-1=a3, cbc-1=a8b >
Subgroups: 124 in 72 conjugacy classes, 50 normal (32 characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C8, C2×C4, C2×C4, C23, C16, C42, C22⋊C4, C4⋊C4, C2×C8, C2×C8, M4(2), C22×C4, C4.Q8, C2.D8, C8.C4, C8.C4, C2×C16, M5(2), C42⋊C2, C22×C8, C2×M4(2), C8.Q8, C23.25D4, C2×C8.C4, C2×M5(2), M5(2)⋊3C4
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C23, C4⋊C4, SD16, C22×C4, C2×D4, C2×Q8, C4.Q8, C2×C4⋊C4, C2×SD16, C2×C4.Q8, M5(2)⋊3C4
►On 32 points
(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)(2 26)(3 19)(4 28)(5 21)(6 30)(7 23)(8 32)(9 25)(10 18)(11 27)(12 20)(13 29)(14 22)(15 31)(16 24)
(2 12 10 4)(3 7)(5 13)(6 8 14 16)(11 15)(17 25)(18 20 26 28)(19 31)(22 32 30 24)(23 27)
G:=sub<Sym(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)(2,26)(3,19)(4,28)(5,21)(6,30)(7,23)(8,32)(9,25)(10,18)(11,27)(12,20)(13,29)(14,22)(15,31)(16,24), (2,12,10,4)(3,7)(5,13)(6,8,14,16)(11,15)(17,25)(18,20,26,28)(19,31)(22,32,30,24)(23,27)>;
G:=Group( (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)(2,26)(3,19)(4,28)(5,21)(6,30)(7,23)(8,32)(9,25)(10,18)(11,27)(12,20)(13,29)(14,22)(15,31)(16,24), (2,12,10,4)(3,7)(5,13)(6,8,14,16)(11,15)(17,25)(18,20,26,28)(19,31)(22,32,30,24)(23,27) );
G=PermutationGroup([[(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),(2,26),(3,19),(4,28),(5,21),(6,30),(7,23),(8,32),(9,25),(10,18),(11,27),(12,20),(13,29),(14,22),(15,31),(16,24)], [(2,12,10,4),(3,7),(5,13),(6,8,14,16),(11,15),(17,25),(18,20,26,28),(19,31),(22,32,30,24),(23,27)]])
32 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 8I | 8J | 16A | ··· | 16H |
| order | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 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 | 1 | 1 | 2 | 2 | 2 | 8 | 8 | 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 | C4 | C4 | D4 | Q8 | D4 | SD16 | SD16 | M5(2)⋊3C4 |
| kernel | M5(2)⋊3C4 | C8.Q8 | C23.25D4 | C2×C8.C4 | C2×M5(2) | C2×C16 | M5(2) | C2×C8 | C2×C8 | C22×C4 | C2×C4 | C23 | C1 |
| # reps | 1 | 4 | 1 | 1 | 1 | 4 | 4 | 1 | 2 | 1 | 6 | 2 | 4 |
Matrix representation of M5(2)⋊3C4 ►in GL4(𝔽17) generated by
| 13 | 13 | 4 | 8 |
| 0 | 0 | 4 | 0 |
| 14 | 14 | 0 | 0 |
| 15 | 1 | 4 | 4 |
| 0 | 13 | 0 | 0 |
| 4 | 0 | 0 | 0 |
| 4 | 4 | 13 | 9 |
| 0 | 13 | 4 | 4 |
| 1 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 5 | 5 | 7 | 7 |
| 4 | 3 | 5 | 10 |
G:=sub<GL(4,GF(17))| [13,0,14,15,13,0,14,1,4,4,0,4,8,0,0,4],[0,4,4,0,13,0,4,13,0,0,13,4,0,0,9,4],[1,0,5,4,0,16,5,3,0,0,7,5,0,0,7,10] >;
M5(2)⋊3C4 in GAP, Magma, Sage, TeX
M_5(2)\rtimes_3C_4
% in TeX
G:=Group("M5(2):3C4"); // GroupNames label
G:=SmallGroup(128,887);
// by ID
G=gap.SmallGroup(128,887);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,-2,-2,112,141,64,352,1123,136,1411,4037,124]);
// Polycyclic
G:=Group<a,b,c|a^16=b^2=c^4=1,b*a*b=a^9,c*a*c^-1=a^3,c*b*c^-1=a^8*b>;
// generators/relations