p-group, metabelian, nilpotent (class 4), monomial
Aliases: C8.9C42, C23.23SD16, (C2×C16)⋊12C4, C8.27(C4⋊C4), (C2×C8).48Q8, C8.C4⋊1C4, C2.D8.8C4, (C2×C4).163D8, (C2×C8).361D4, (C2×C4).32Q16, (C22×C16).5C2, (C2×C4).66SD16, C4.22(C2.D8), C8.40(C22⋊C4), (C22×C4).571D4, C4.50(D4⋊C4), C22.9(C4.Q8), C4.14(Q8⋊C4), C2.3(D8.C4), C4.3(C2.C42), (C22×C8).546C22, C22.9(Q8⋊C4), C23.25D4.1C2, C22.45(D4⋊C4), C2.12(C22.4Q16), (C2×C8).168(C2×C4), (C2×C4).110(C4⋊C4), (C2×C8.C4).2C2, (C2×C4).229(C22⋊C4), SmallGroup(128,114)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C8.9C42
G = < a,b,c | a8=b4=1, c4=a2, bab-1=a-1, ac=ca, cbc-1=a-1b >
Subgroups: 120 in 66 conjugacy classes, 40 normal (32 characteristic)
C1, C2, C2 [×2], C2 [×2], C4 [×4], C4 [×2], C22 [×3], C22 [×2], C8 [×4], C8 [×2], C2×C4 [×6], C2×C4 [×2], C23, C16 [×2], C42, C22⋊C4, C4⋊C4 [×2], C2×C8 [×6], C2×C8, M4(2) [×3], C22×C4, C4.Q8, C2.D8 [×2], C8.C4 [×2], C8.C4, C2×C16 [×2], C2×C16 [×2], C42⋊C2, C22×C8, C2×M4(2), C23.25D4, C2×C8.C4, C22×C16, C8.9C42
Quotients: C1, C2 [×3], C4 [×6], C22, C2×C4 [×3], D4 [×3], Q8, C42, C22⋊C4 [×3], C4⋊C4 [×3], D8, SD16 [×2], Q16, C2.C42, D4⋊C4 [×2], Q8⋊C4 [×2], C4.Q8, C2.D8, C22.4Q16, D8.C4 [×2], C8.9C42
►On 64 points
(1 3 5 7 9 11 13 15)(2 4 6 8 10 12 14 16)(17 27 21 31 25 19 29 23)(18 28 22 32 26 20 30 24)(33 43 37 47 41 35 45 39)(34 44 38 48 42 36 46 40)(49 51 53 55 57 59 61 63)(50 52 54 56 58 60 62 64)
(1 26 63 42)(2 21 64 37)(3 32 49 48)(4 27 50 43)(5 22 51 38)(6 17 52 33)(7 28 53 44)(8 23 54 39)(9 18 55 34)(10 29 56 45)(11 24 57 40)(12 19 58 35)(13 30 59 46)(14 25 60 41)(15 20 61 36)(16 31 62 47)
(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)
G:=sub<Sym(64)| (1,3,5,7,9,11,13,15)(2,4,6,8,10,12,14,16)(17,27,21,31,25,19,29,23)(18,28,22,32,26,20,30,24)(33,43,37,47,41,35,45,39)(34,44,38,48,42,36,46,40)(49,51,53,55,57,59,61,63)(50,52,54,56,58,60,62,64), (1,26,63,42)(2,21,64,37)(3,32,49,48)(4,27,50,43)(5,22,51,38)(6,17,52,33)(7,28,53,44)(8,23,54,39)(9,18,55,34)(10,29,56,45)(11,24,57,40)(12,19,58,35)(13,30,59,46)(14,25,60,41)(15,20,61,36)(16,31,62,47), (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)>;
G:=Group( (1,3,5,7,9,11,13,15)(2,4,6,8,10,12,14,16)(17,27,21,31,25,19,29,23)(18,28,22,32,26,20,30,24)(33,43,37,47,41,35,45,39)(34,44,38,48,42,36,46,40)(49,51,53,55,57,59,61,63)(50,52,54,56,58,60,62,64), (1,26,63,42)(2,21,64,37)(3,32,49,48)(4,27,50,43)(5,22,51,38)(6,17,52,33)(7,28,53,44)(8,23,54,39)(9,18,55,34)(10,29,56,45)(11,24,57,40)(12,19,58,35)(13,30,59,46)(14,25,60,41)(15,20,61,36)(16,31,62,47), (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) );
G=PermutationGroup([(1,3,5,7,9,11,13,15),(2,4,6,8,10,12,14,16),(17,27,21,31,25,19,29,23),(18,28,22,32,26,20,30,24),(33,43,37,47,41,35,45,39),(34,44,38,48,42,36,46,40),(49,51,53,55,57,59,61,63),(50,52,54,56,58,60,62,64)], [(1,26,63,42),(2,21,64,37),(3,32,49,48),(4,27,50,43),(5,22,51,38),(6,17,52,33),(7,28,53,44),(8,23,54,39),(9,18,55,34),(10,29,56,45),(11,24,57,40),(12,19,58,35),(13,30,59,46),(14,25,60,41),(15,20,61,36),(16,31,62,47)], [(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)])
44 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 8A | ··· | 8H | 8I | 8J | 8K | 8L | 16A | ··· | 16P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 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 | 8 | 8 | 8 | 8 | 2 | ··· | 2 | 8 | 8 | 8 | 8 | 2 | ··· | 2 |
44 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | - | + | + | - | ||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | C4 | D4 | Q8 | D4 | D8 | SD16 | Q16 | SD16 | D8.C4 |
| kernel | C8.9C42 | C23.25D4 | C2×C8.C4 | C22×C16 | C2.D8 | C8.C4 | C2×C16 | C2×C8 | C2×C8 | C22×C4 | C2×C4 | C2×C4 | C2×C4 | C23 | C2 |
| # reps | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 2 | 1 | 1 | 2 | 2 | 2 | 2 | 16 |
Matrix representation of C8.9C42 ►in GL3(𝔽17) generated by
| 1 | 0 | 0 |
| 0 | 8 | 0 |
| 0 | 14 | 15 |
| 4 | 0 | 0 |
| 0 | 13 | 15 |
| 0 | 16 | 4 |
| 4 | 0 | 0 |
| 0 | 5 | 0 |
| 0 | 2 | 6 |
G:=sub<GL(3,GF(17))| [1,0,0,0,8,14,0,0,15],[4,0,0,0,13,16,0,15,4],[4,0,0,0,5,2,0,0,6] >;
C8.9C42 in GAP, Magma, Sage, TeX
C_8._9C_4^2
% in TeX
G:=Group("C8.9C4^2"); // GroupNames label
G:=SmallGroup(128,114);
// by ID
G=gap.SmallGroup(128,114);
# by ID
G:=PCGroup([7,-2,2,-2,2,2,-2,-2,56,85,120,758,520,3924,242,4037,124]);
// Polycyclic
G:=Group<a,b,c|a^8=b^4=1,c^4=a^2,b*a*b^-1=a^-1,a*c=c*a,c*b*c^-1=a^-1*b>;
// generators/relations