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, C2, C4, C4, C4, C22, C22, C22, C8, C8, C2×C4, C2×C4, C23, C16, C42, C22⋊C4, C4⋊C4, C2×C8, C2×C8, M4(2), C22×C4, C4×C8, C8⋊C4, C2×C16, C2×C16, M5(2), C42⋊C2, C22×C8, C2×M4(2), C4×C16, C16⋊5C4, C8○2M4(2), C22×C16, C2×M5(2), C16○2M5(2)
Quotients: C1, C2, C4, C22, C8, C2×C4, C23, C42, C2×C8, C22×C4, C4×C8, C2×C42, C22×C8, C2×C4×C8, D4○C16, 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 40 55 22 13 36 51 18 9 48 63 30 5 44 59 26)(2 41 56 23 14 37 52 19 10 33 64 31 6 45 60 27)(3 42 57 24 15 38 53 20 11 34 49 32 7 46 61 28)(4 43 58 25 16 39 54 21 12 35 50 17 8 47 62 29)
(17 25)(18 26)(19 27)(20 28)(21 29)(22 30)(23 31)(24 32)(33 41)(34 42)(35 43)(36 44)(37 45)(38 46)(39 47)(40 48)
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), (1,40,55,22,13,36,51,18,9,48,63,30,5,44,59,26)(2,41,56,23,14,37,52,19,10,33,64,31,6,45,60,27)(3,42,57,24,15,38,53,20,11,34,49,32,7,46,61,28)(4,43,58,25,16,39,54,21,12,35,50,17,8,47,62,29), (17,25)(18,26)(19,27)(20,28)(21,29)(22,30)(23,31)(24,32)(33,41)(34,42)(35,43)(36,44)(37,45)(38,46)(39,47)(40,48)>;
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), (1,40,55,22,13,36,51,18,9,48,63,30,5,44,59,26)(2,41,56,23,14,37,52,19,10,33,64,31,6,45,60,27)(3,42,57,24,15,38,53,20,11,34,49,32,7,46,61,28)(4,43,58,25,16,39,54,21,12,35,50,17,8,47,62,29), (17,25)(18,26)(19,27)(20,28)(21,29)(22,30)(23,31)(24,32)(33,41)(34,42)(35,43)(36,44)(37,45)(38,46)(39,47)(40,48) );
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)], [(1,40,55,22,13,36,51,18,9,48,63,30,5,44,59,26),(2,41,56,23,14,37,52,19,10,33,64,31,6,45,60,27),(3,42,57,24,15,38,53,20,11,34,49,32,7,46,61,28),(4,43,58,25,16,39,54,21,12,35,50,17,8,47,62,29)], [(17,25),(18,26),(19,27),(20,28),(21,29),(22,30),(23,31),(24,32),(33,41),(34,42),(35,43),(36,44),(37,45),(38,46),(39,47),(40,48)]])
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