p-group, metabelian, nilpotent (class 4), monomial
Aliases: C8○D16, C8○Q32, C8○SD32, D16⋊6C4, Q32⋊6C4, C8.23D8, SD32⋊5C4, C42.332D4, C8○D8⋊1C2, (C4×C16)⋊10C2, C8○(C4○D16), D8.4(C2×C4), C4.92(C2×D8), C4.31(C4×D4), C2.19(C4×D8), C16.14(C2×C4), C4○D16.5C2, (C2×C8).289D4, Q16.4(C2×C4), C8○(C8.4Q8), C8.4Q8⋊8C2, C8○(D8.C4), D8.C4⋊7C2, C8.58(C4○D4), C8.41(C22×C4), (C4×C8).422C22, (C2×C8).580C23, (C2×C16).95C22, C4○D8.14C22, C22.1(C4○D8), C8.C4.16C22, (C2×C4).774(C2×D4), SmallGroup(128,910)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C8○D16
G = < a,b,c | a8=c2=1, b8=a4, ab=ba, ac=ca, cbc=a4b7 >
Subgroups: 148 in 73 conjugacy classes, 38 normal (24 characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C8, C2×C4, C2×C4, D4, Q8, C16, C16, C42, C2×C8, C2×C8, M4(2), D8, SD16, Q16, C4○D4, C4×C8, C4≀C2, C8.C4, C2×C16, D16, SD32, Q32, C8○D4, C4○D8, C4×C16, D8.C4, C8.4Q8, C8○D8, C4○D16, C8○D16
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, D8, C22×C4, C2×D4, C4○D4, C4×D4, C2×D8, C4○D8, C4×D8, C8○D16
►On 32 points
(1 15 13 11 9 7 5 3)(2 16 14 12 10 8 6 4)(17 19 21 23 25 27 29 31)(18 20 22 24 26 28 30 32)
(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)
(1 22)(2 21)(3 20)(4 19)(5 18)(6 17)(7 32)(8 31)(9 30)(10 29)(11 28)(12 27)(13 26)(14 25)(15 24)(16 23)
G:=sub<Sym(32)| (1,15,13,11,9,7,5,3)(2,16,14,12,10,8,6,4)(17,19,21,23,25,27,29,31)(18,20,22,24,26,28,30,32), (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), (1,22)(2,21)(3,20)(4,19)(5,18)(6,17)(7,32)(8,31)(9,30)(10,29)(11,28)(12,27)(13,26)(14,25)(15,24)(16,23)>;
G:=Group( (1,15,13,11,9,7,5,3)(2,16,14,12,10,8,6,4)(17,19,21,23,25,27,29,31)(18,20,22,24,26,28,30,32), (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), (1,22)(2,21)(3,20)(4,19)(5,18)(6,17)(7,32)(8,31)(9,30)(10,29)(11,28)(12,27)(13,26)(14,25)(15,24)(16,23) );
G=PermutationGroup([[(1,15,13,11,9,7,5,3),(2,16,14,12,10,8,6,4),(17,19,21,23,25,27,29,31),(18,20,22,24,26,28,30,32)], [(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)], [(1,22),(2,21),(3,20),(4,19),(5,18),(6,17),(7,32),(8,31),(9,30),(10,29),(11,28),(12,27),(13,26),(14,25),(15,24),(16,23)]])
44 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 4A | 4B | 4C | ··· | 4G | 4H | 4I | 8A | 8B | 8C | 8D | 8E | ··· | 8J | 8K | 8L | 8M | 8N | 16A | ··· | 16P |
| order | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | ··· | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 8 | ··· | 8 | 8 | 8 | 8 | 8 | 16 | ··· | 16 |
| size | 1 | 1 | 2 | 8 | 8 | 1 | 1 | 2 | ··· | 2 | 8 | 8 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 8 | 8 | 8 | 8 | 2 | ··· | 2 |
44 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | ||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | D4 | D4 | D8 | C4○D4 | C4○D8 | C8○D16 |
| kernel | C8○D16 | C4×C16 | D8.C4 | C8.4Q8 | C8○D8 | C4○D16 | D16 | SD32 | Q32 | C42 | C2×C8 | C8 | C8 | C22 | C1 |
| # reps | 1 | 1 | 2 | 1 | 2 | 1 | 2 | 4 | 2 | 1 | 1 | 4 | 2 | 4 | 16 |
Matrix representation of C8○D16 ►in GL2(𝔽17) generated by
| 2 | 0 |
| 0 | 2 |
| 3 | 0 |
| 0 | 6 |
| 0 | 1 |
| 1 | 0 |
G:=sub<GL(2,GF(17))| [2,0,0,2],[3,0,0,6],[0,1,1,0] >;
C8○D16 in GAP, Magma, Sage, TeX
C_8\circ D_{16} % in TeX
G:=Group("C8oD16"); // GroupNames label
G:=SmallGroup(128,910);
// by ID
G=gap.SmallGroup(128,910);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,-2,-2,112,141,100,1123,570,360,172,4037,2028,124]);
// Polycyclic
G:=Group<a,b,c|a^8=c^2=1,b^8=a^4,a*b=b*a,a*c=c*a,c*b*c=a^4*b^7>;
// generators/relations