p-group, metabelian, nilpotent (class 2), monomial
Aliases: C16○2M5(2), C16○2M4(2), C8.18C42, M5(2)⋊13C4, M4(2).5C8, C16○(C4⋊C4), (C4×C16)⋊15C2, (C2×C16)⋊17C4, C4⋊C4.12C8, C8.12(C2×C8), C4.10(C4×C8), C16○(C8⋊C4), C16○(C22⋊C4), C16.26(C2×C4), C16○(C16⋊5C4), C16⋊5C4⋊15C2, C22⋊C4.7C8, M4(2)○(C2×C16), C16○(C2×M5(2)), C16○(C2×M4(2)), (C2×C16)○M5(2), C8⋊C4.21C4, C2.1(D4○C16), C8.67(C22×C4), (C2×C4).74C42, C4.36(C22×C8), C22.10(C4×C8), C23.20(C2×C8), C4.37(C2×C42), C16○(C42⋊C2), (C2×C8).622C23, (C4×C8).434C22, (C22×C16).17C2, C42.244(C2×C4), C16○(C8○2M4(2)), C42⋊C2.38C4, (C2×C16).105C22, (C2×M4(2)).37C4, (C2×M5(2)).28C2, C22.25(C22×C8), C8○2M4(2).25C2, (C22×C8).576C22, C2.12(C2×C4×C8), C8⋊C4○(C2×C16), (C2×C4).40(C2×C8), (C2×C8).246(C2×C4), (C2×C16)○(C16⋊5C4), (C2×C16)○(C42⋊C2), (C22×C4).407(C2×C4), (C2×C4).607(C22×C4), (C2×C16)○(C8○2M4(2)), SmallGroup(128,840)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C16○2M5(2)
G = < a,b,c | a16=c2=1, b4=a12, ab=ba, ac=ca, cbc=a8b >
Subgroups: 100 in 92 conjugacy classes, 84 normal (22 characteristic)
C1, C2, C2 [×2], C2 [×2], C4 [×2], C4 [×2], C4 [×4], C22, C22 [×2], C22 [×2], C8 [×2], C8 [×6], C2×C4 [×2], C2×C4 [×8], C23, C16 [×8], C42 [×2], C22⋊C4 [×2], C4⋊C4 [×2], C2×C8 [×2], C2×C8 [×6], M4(2) [×4], C22×C4, C4×C8 [×2], C8⋊C4 [×2], C2×C16 [×2], C2×C16 [×6], M5(2) [×4], C42⋊C2, C22×C8, C2×M4(2), C4×C16 [×2], C16⋊5C4 [×2], C8○2M4(2), C22×C16, C2×M5(2), C16○2M5(2)
Quotients: C1, C2 [×7], C4 [×12], C22 [×7], C8 [×8], C2×C4 [×18], C23, C42 [×4], C2×C8 [×12], C22×C4 [×3], C4×C8 [×4], C2×C42, C22×C8 [×2], C2×C4×C8, D4○C16 [×2], C16○2M5(2)
►On 64 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)(33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48)(49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64)
(1 50 42 28 13 62 38 24 9 58 34 20 5 54 46 32)(2 51 43 29 14 63 39 25 10 59 35 21 6 55 47 17)(3 52 44 30 15 64 40 26 11 60 36 22 7 56 48 18)(4 53 45 31 16 49 41 27 12 61 37 23 8 57 33 19)
(17 25)(18 26)(19 27)(20 28)(21 29)(22 30)(23 31)(24 32)(49 57)(50 58)(51 59)(52 60)(53 61)(54 62)(55 63)(56 64)
G:=sub<Sym(64)| (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)(33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64), (1,50,42,28,13,62,38,24,9,58,34,20,5,54,46,32)(2,51,43,29,14,63,39,25,10,59,35,21,6,55,47,17)(3,52,44,30,15,64,40,26,11,60,36,22,7,56,48,18)(4,53,45,31,16,49,41,27,12,61,37,23,8,57,33,19), (17,25)(18,26)(19,27)(20,28)(21,29)(22,30)(23,31)(24,32)(49,57)(50,58)(51,59)(52,60)(53,61)(54,62)(55,63)(56,64)>;
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)(33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64), (1,50,42,28,13,62,38,24,9,58,34,20,5,54,46,32)(2,51,43,29,14,63,39,25,10,59,35,21,6,55,47,17)(3,52,44,30,15,64,40,26,11,60,36,22,7,56,48,18)(4,53,45,31,16,49,41,27,12,61,37,23,8,57,33,19), (17,25)(18,26)(19,27)(20,28)(21,29)(22,30)(23,31)(24,32)(49,57)(50,58)(51,59)(52,60)(53,61)(54,62)(55,63)(56,64) );
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),(33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48),(49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64)], [(1,50,42,28,13,62,38,24,9,58,34,20,5,54,46,32),(2,51,43,29,14,63,39,25,10,59,35,21,6,55,47,17),(3,52,44,30,15,64,40,26,11,60,36,22,7,56,48,18),(4,53,45,31,16,49,41,27,12,61,37,23,8,57,33,19)], [(17,25),(18,26),(19,27),(20,28),(21,29),(22,30),(23,31),(24,32),(49,57),(50,58),(51,59),(52,60),(53,61),(54,62),(55,63),(56,64)])
80 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | ··· | 4N | 8A | ··· | 8H | 8I | ··· | 8T | 16A | ··· | 16P | 16Q | ··· | 16AN |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 8 | ··· | 8 | 8 | ··· | 8 | 16 | ··· | 16 | 16 | ··· | 16 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 |
80 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 |
| type | + | + | + | + | + | + | |||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | C4 | C4 | C8 | C8 | C8 | D4○C16 |
| kernel | C16○2M5(2) | C4×C16 | C16⋊5C4 | C8○2M4(2) | C22×C16 | C2×M5(2) | C8⋊C4 | C2×C16 | M5(2) | C42⋊C2 | C2×M4(2) | C22⋊C4 | C4⋊C4 | M4(2) | C2 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 4 | 8 | 8 | 2 | 2 | 8 | 8 | 16 | 16 |
Matrix representation of C16○2M5(2) ►in GL3(𝔽17) generated by
| 1 | 0 | 0 |
| 0 | 6 | 0 |
| 0 | 0 | 6 |
| 13 | 0 | 0 |
| 0 | 0 | 14 |
| 0 | 3 | 0 |
| 16 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 0 | 16 |
G:=sub<GL(3,GF(17))| [1,0,0,0,6,0,0,0,6],[13,0,0,0,0,3,0,14,0],[16,0,0,0,1,0,0,0,16] >;
C16○2M5(2) in GAP, Magma, Sage, TeX
C_{16}\circ_2M_5(2) % in TeX
G:=Group("C16o2M5(2)"); // GroupNames label
G:=SmallGroup(128,840);
// by ID
G=gap.SmallGroup(128,840);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,-2,-2,56,120,723,136,124]);
// Polycyclic
G:=Group<a,b,c|a^16=c^2=1,b^4=a^12,a*b=b*a,a*c=c*a,c*b*c=a^8*b>;
// generators/relations