direct product, p-group, metabelian, nilpotent (class 2), monomial
Aliases: C2×M5(2), C4○M5(2), C8○M5(2), C16⋊4C22, C23.3C8, C8.16C23, (C2×C16)⋊8C2, (C2×C4).6C8, C8.21(C2×C4), C4.10(C2×C8), (C2×C8).14C4, C2.6(C22×C8), C22.11(C2×C8), C4.35(C22×C4), (C22×C8).16C2, (C22×C4).17C4, (C2×C8).104C22, (C2×C4).76(C2×C4), SmallGroup(64,184)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C2×M5(2)
G = < a,b,c | a2=b16=c2=1, ab=ba, ac=ca, cbc=b9 >
Smallest permutation representation of C2×M5(2)
►On 32 points
(1 25)(2 26)(3 27)(4 28)(5 29)(6 30)(7 31)(8 32)(9 17)(10 18)(11 19)(12 20)(13 21)(14 22)(15 23)(16 24)
(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 10)(4 12)(6 14)(8 16)(18 26)(20 28)(22 30)(24 32)
G:=sub<Sym(32)| (1,25)(2,26)(3,27)(4,28)(5,29)(6,30)(7,31)(8,32)(9,17)(10,18)(11,19)(12,20)(13,21)(14,22)(15,23)(16,24), (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), (2,10)(4,12)(6,14)(8,16)(18,26)(20,28)(22,30)(24,32)>;
G:=Group( (1,25)(2,26)(3,27)(4,28)(5,29)(6,30)(7,31)(8,32)(9,17)(10,18)(11,19)(12,20)(13,21)(14,22)(15,23)(16,24), (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), (2,10)(4,12)(6,14)(8,16)(18,26)(20,28)(22,30)(24,32) );
G=PermutationGroup([[(1,25),(2,26),(3,27),(4,28),(5,29),(6,30),(7,31),(8,32),(9,17),(10,18),(11,19),(12,20),(13,21),(14,22),(15,23),(16,24)], [(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)], [(2,10),(4,12),(6,14),(8,16),(18,26),(20,28),(22,30),(24,32)]])
C2×M5(2) is a maximal subgroup of
C42.2C8 C42.7C8 M5(2)⋊C4 M4(2).C8 M5(2)⋊7C4 C8.11C42 C23.9D8 C8.13C42 C8.C42 C8.2C42 M5(2).C4 C8.4C42 C23.C16 C16○2M5(2) C8.23C42 C24.5C8 (C2×D4).5C8 M5(2).19C22 M5(2)⋊12C22 C23.39D8 C23.40D8 C23.41D8 C23.20SD16 C23.13D8 C23.21SD16 C4⋊M5(2) C4⋊C4.7C8 M4(2).1C8 M5(2)⋊3C4 M5(2)⋊1C4 M5(2).1C4 C16⋊9D4 C16⋊6D4 C16⋊D4 C16.D4 C16⋊2D4 Q8○M5(2) D16⋊C22 D5⋊M5(2)
C2×M5(2) is a maximal quotient of
C24.5C8 C4⋊M5(2) C42.13C8 C42.6C8 C16⋊9D4 C16⋊6D4 C16⋊4Q8 D5⋊M5(2)
40 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 8A | ··· | 8H | 8I | 8J | 8K | 8L | 16A | ··· | 16P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 8 | ··· | 8 | 8 | 8 | 8 | 8 | 16 | ··· | 16 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 2 | ··· | 2 |
40 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 |
| type | + | + | + | + | |||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | C8 | C8 | M5(2) |
| kernel | C2×M5(2) | C2×C16 | M5(2) | C22×C8 | C2×C8 | C22×C4 | C2×C4 | C23 | C2 |
| # reps | 1 | 2 | 4 | 1 | 6 | 2 | 12 | 4 | 8 |
Matrix representation of C2×M5(2) ►in GL3(𝔽17) generated by
| 16 | 0 | 0 |
| 0 | 16 | 0 |
| 0 | 0 | 16 |
| 8 | 0 | 0 |
| 0 | 15 | 15 |
| 0 | 15 | 2 |
| 16 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 15 | 16 |
G:=sub<GL(3,GF(17))| [16,0,0,0,16,0,0,0,16],[8,0,0,0,15,15,0,15,2],[16,0,0,0,1,15,0,0,16] >;
C2×M5(2) in GAP, Magma, Sage, TeX
C_2\times M_5(2)
% in TeX
G:=Group("C2xM5(2)"); // GroupNames label
G:=SmallGroup(64,184);
// by ID
G=gap.SmallGroup(64,184);
# by ID
G:=PCGroup([6,-2,2,2,-2,-2,-2,48,409,69,88]);
// Polycyclic
G:=Group<a,b,c|a^2=b^16=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^9>;
// generators/relations
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