p-group, metabelian, nilpotent (class 2), monomial
Aliases: C16⋊9D4, C22⋊1M5(2), C4⋊C4.9C8, C4⋊C16⋊17C2, C4⋊C8.22C4, (C2×D4).9C8, C2.16(C8×D4), (C4×D4).19C4, C16⋊5C4⋊10C2, (C8×D4).17C2, C8.139(C2×D4), C4.176(C4×D4), C22⋊C4.5C8, C22⋊C16⋊15C2, (C22×C16)⋊13C2, C4.58(C8○D4), C2.7(D4○C16), C22⋊C8.21C4, C23.23(C2×C8), C2.9(C2×M5(2)), C8.102(C4○D4), (C2×M5(2))⋊19C2, (C2×C16).54C22, C42.171(C2×C4), (C4×C8).323C22, (C2×C8).633C23, C22.52(C22×C8), (C22×C8).502C22, (C2×C4).29(C2×C8), (C2×C8).151(C2×C4), (C22×C4).290(C2×C4), (C2×C4).618(C22×C4), SmallGroup(128,900)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C16⋊9D4
G = < a,b,c | a16=b4=c2=1, bab-1=cac=a9, cbc=b-1 >
Subgroups: 116 in 82 conjugacy classes, 50 normal (46 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C8, C8, C2×C4, C2×C4, D4, C23, C16, C16, C42, C22⋊C4, C4⋊C4, C2×C8, C2×C8, C22×C4, C2×D4, C4×C8, C22⋊C8, C4⋊C8, C2×C16, C2×C16, M5(2), C4×D4, C22×C8, C16⋊5C4, C22⋊C16, C4⋊C16, C8×D4, C22×C16, C2×M5(2), C16⋊9D4
Quotients: C1, C2, C4, C22, C8, C2×C4, D4, C23, C2×C8, C22×C4, C2×D4, C4○D4, M5(2), C4×D4, C22×C8, C8○D4, C8×D4, C2×M5(2), D4○C16, C16⋊9D4
►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 31 36 53)(2 24 37 62)(3 17 38 55)(4 26 39 64)(5 19 40 57)(6 28 41 50)(7 21 42 59)(8 30 43 52)(9 23 44 61)(10 32 45 54)(11 25 46 63)(12 18 47 56)(13 27 48 49)(14 20 33 58)(15 29 34 51)(16 22 35 60)
(1 36)(2 45)(3 38)(4 47)(5 40)(6 33)(7 42)(8 35)(9 44)(10 37)(11 46)(12 39)(13 48)(14 41)(15 34)(16 43)(18 26)(20 28)(22 30)(24 32)(50 58)(52 60)(54 62)(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), (1,31,36,53)(2,24,37,62)(3,17,38,55)(4,26,39,64)(5,19,40,57)(6,28,41,50)(7,21,42,59)(8,30,43,52)(9,23,44,61)(10,32,45,54)(11,25,46,63)(12,18,47,56)(13,27,48,49)(14,20,33,58)(15,29,34,51)(16,22,35,60), (1,36)(2,45)(3,38)(4,47)(5,40)(6,33)(7,42)(8,35)(9,44)(10,37)(11,46)(12,39)(13,48)(14,41)(15,34)(16,43)(18,26)(20,28)(22,30)(24,32)(50,58)(52,60)(54,62)(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), (1,31,36,53)(2,24,37,62)(3,17,38,55)(4,26,39,64)(5,19,40,57)(6,28,41,50)(7,21,42,59)(8,30,43,52)(9,23,44,61)(10,32,45,54)(11,25,46,63)(12,18,47,56)(13,27,48,49)(14,20,33,58)(15,29,34,51)(16,22,35,60), (1,36)(2,45)(3,38)(4,47)(5,40)(6,33)(7,42)(8,35)(9,44)(10,37)(11,46)(12,39)(13,48)(14,41)(15,34)(16,43)(18,26)(20,28)(22,30)(24,32)(50,58)(52,60)(54,62)(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)], [(1,31,36,53),(2,24,37,62),(3,17,38,55),(4,26,39,64),(5,19,40,57),(6,28,41,50),(7,21,42,59),(8,30,43,52),(9,23,44,61),(10,32,45,54),(11,25,46,63),(12,18,47,56),(13,27,48,49),(14,20,33,58),(15,29,34,51),(16,22,35,60)], [(1,36),(2,45),(3,38),(4,47),(5,40),(6,33),(7,42),(8,35),(9,44),(10,37),(11,46),(12,39),(13,48),(14,41),(15,34),(16,43),(18,26),(20,28),(22,30),(24,32),(50,58),(52,60),(54,62),(56,64)]])
56 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 8A | ··· | 8H | 8I | 8J | 8K | 8L | 8M | 8N | 8O | 8P | 16A | ··· | 16P | 16Q | ··· | 16X |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 8 | ··· | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 16 | ··· | 16 | 16 | ··· | 16 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 4 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
56 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | ||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | C8 | C8 | C8 | D4 | C4○D4 | C8○D4 | M5(2) | D4○C16 |
| kernel | C16⋊9D4 | C16⋊5C4 | C22⋊C16 | C4⋊C16 | C8×D4 | C22×C16 | C2×M5(2) | C22⋊C8 | C4⋊C8 | C4×D4 | C22⋊C4 | C4⋊C4 | C2×D4 | C16 | C8 | C4 | C22 | C2 |
| # reps | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 4 | 2 | 2 | 8 | 4 | 4 | 2 | 2 | 4 | 8 | 8 |
Matrix representation of C16⋊9D4 ►in GL4(𝔽17) generated by
| 2 | 0 | 0 | 0 |
| 0 | 2 | 0 | 0 |
| 0 | 0 | 9 | 2 |
| 0 | 0 | 3 | 8 |
| 0 | 1 | 0 | 0 |
| 16 | 0 | 0 | 0 |
| 0 | 0 | 10 | 6 |
| 0 | 0 | 3 | 7 |
| 16 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 16 | 0 |
| 0 | 0 | 9 | 1 |
G:=sub<GL(4,GF(17))| [2,0,0,0,0,2,0,0,0,0,9,3,0,0,2,8],[0,16,0,0,1,0,0,0,0,0,10,3,0,0,6,7],[16,0,0,0,0,1,0,0,0,0,16,9,0,0,0,1] >;
C16⋊9D4 in GAP, Magma, Sage, TeX
C_{16}\rtimes_9D_4 % in TeX
G:=Group("C16:9D4"); // GroupNames label
G:=SmallGroup(128,900);
// by ID
G=gap.SmallGroup(128,900);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,-2,-2,112,141,1430,100,102,124]);
// Polycyclic
G:=Group<a,b,c|a^16=b^4=c^2=1,b*a*b^-1=c*a*c=a^9,c*b*c=b^-1>;
// generators/relations