p-group, metabelian, nilpotent (class 4), monomial
Aliases: C8.22SD16, C42.335D4, (C4×C16)⋊7C2, (C2×C4).62D8, (C2×C8).279D4, C8.5Q8⋊2C2, C2.Q32⋊3C2, C2.D16.1C2, C8.52(C4○D4), C4.16(C2×SD16), C2.17(C4○D16), (C2×C8).540C23, (C4×C8).413C22, (C2×C16).74C22, C8.12D4.3C2, C4.9(C4.4D4), (C2×D8).13C22, C22.126(C2×D8), C2.D8.25C22, C2.11(C4.4D8), (C2×Q16).14C22, (C2×C4).808(C2×D4), SmallGroup(128,974)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C8.22SD16
G = < a,b,c | a8=c2=1, b8=a4, ab=ba, cac=a3, cbc=a2b3 >
Subgroups: 184 in 69 conjugacy classes, 32 normal (16 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C8, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C16, C42, C22⋊C4, C4⋊C4, C2×C8, D8, SD16, Q16, C2×D4, C2×Q8, C4×C8, C4.Q8, C2.D8, C2×C16, C4.4D4, C42.C2, C2×D8, C2×SD16, C2×Q16, C4×C16, C2.D16, C2.Q32, C8.12D4, C8.5Q8, C8.22SD16
Quotients: C1, C2, C22, D4, C23, D8, SD16, C2×D4, C4○D4, C4.4D4, C2×D8, C2×SD16, C4.4D8, C4○D16, C8.22SD16
►On 64 points
(1 19 58 35 9 27 50 43)(2 20 59 36 10 28 51 44)(3 21 60 37 11 29 52 45)(4 22 61 38 12 30 53 46)(5 23 62 39 13 31 54 47)(6 24 63 40 14 32 55 48)(7 25 64 41 15 17 56 33)(8 26 49 42 16 18 57 34)
(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)
(2 61)(3 15)(4 59)(5 13)(6 57)(7 11)(8 55)(10 53)(12 51)(14 49)(16 63)(17 37)(18 32)(19 35)(20 30)(21 33)(22 28)(23 47)(24 26)(25 45)(27 43)(29 41)(31 39)(34 40)(36 38)(42 48)(44 46)(50 58)(52 56)(60 64)
G:=sub<Sym(64)| (1,19,58,35,9,27,50,43)(2,20,59,36,10,28,51,44)(3,21,60,37,11,29,52,45)(4,22,61,38,12,30,53,46)(5,23,62,39,13,31,54,47)(6,24,63,40,14,32,55,48)(7,25,64,41,15,17,56,33)(8,26,49,42,16,18,57,34), (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), (2,61)(3,15)(4,59)(5,13)(6,57)(7,11)(8,55)(10,53)(12,51)(14,49)(16,63)(17,37)(18,32)(19,35)(20,30)(21,33)(22,28)(23,47)(24,26)(25,45)(27,43)(29,41)(31,39)(34,40)(36,38)(42,48)(44,46)(50,58)(52,56)(60,64)>;
G:=Group( (1,19,58,35,9,27,50,43)(2,20,59,36,10,28,51,44)(3,21,60,37,11,29,52,45)(4,22,61,38,12,30,53,46)(5,23,62,39,13,31,54,47)(6,24,63,40,14,32,55,48)(7,25,64,41,15,17,56,33)(8,26,49,42,16,18,57,34), (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), (2,61)(3,15)(4,59)(5,13)(6,57)(7,11)(8,55)(10,53)(12,51)(14,49)(16,63)(17,37)(18,32)(19,35)(20,30)(21,33)(22,28)(23,47)(24,26)(25,45)(27,43)(29,41)(31,39)(34,40)(36,38)(42,48)(44,46)(50,58)(52,56)(60,64) );
G=PermutationGroup([[(1,19,58,35,9,27,50,43),(2,20,59,36,10,28,51,44),(3,21,60,37,11,29,52,45),(4,22,61,38,12,30,53,46),(5,23,62,39,13,31,54,47),(6,24,63,40,14,32,55,48),(7,25,64,41,15,17,56,33),(8,26,49,42,16,18,57,34)], [(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)], [(2,61),(3,15),(4,59),(5,13),(6,57),(7,11),(8,55),(10,53),(12,51),(14,49),(16,63),(17,37),(18,32),(19,35),(20,30),(21,33),(22,28),(23,47),(24,26),(25,45),(27,43),(29,41),(31,39),(34,40),(36,38),(42,48),(44,46),(50,58),(52,56),(60,64)]])
38 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 4A | ··· | 4F | 4G | 4H | 4I | 8A | ··· | 8H | 16A | ··· | 16P |
| order | 1 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | 4 | 4 | 8 | ··· | 8 | 16 | ··· | 16 |
| size | 1 | 1 | 1 | 1 | 16 | 2 | ··· | 2 | 16 | 16 | 16 | 2 | ··· | 2 | 2 | ··· | 2 |
38 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | SD16 | C4○D4 | D8 | C4○D16 |
| kernel | C8.22SD16 | C4×C16 | C2.D16 | C2.Q32 | C8.12D4 | C8.5Q8 | C42 | C2×C8 | C8 | C8 | C2×C4 | C2 |
| # reps | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 16 |
Matrix representation of C8.22SD16 ►in GL4(𝔽17) generated by
| 0 | 13 | 0 | 0 |
| 13 | 0 | 0 | 0 |
| 0 | 0 | 0 | 10 |
| 0 | 0 | 12 | 10 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 9 | 2 |
| 0 | 0 | 16 | 11 |
| 1 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 1 | 16 |
G:=sub<GL(4,GF(17))| [0,13,0,0,13,0,0,0,0,0,0,12,0,0,10,10],[0,1,0,0,1,0,0,0,0,0,9,16,0,0,2,11],[1,0,0,0,0,16,0,0,0,0,1,1,0,0,0,16] >;
C8.22SD16 in GAP, Magma, Sage, TeX
C_8._{22}{\rm SD}_{16} % in TeX
G:=Group("C8.22SD16"); // GroupNames label
G:=SmallGroup(128,974);
// by ID
G=gap.SmallGroup(128,974);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,-2,-2,141,568,422,58,1123,360,3924,102,4037,124]);
// Polycyclic
G:=Group<a,b,c|a^8=c^2=1,b^8=a^4,a*b=b*a,c*a*c=a^3,c*b*c=a^2*b^3>;
// generators/relations