direct product, p-group, metabelian, nilpotent (class 2), monomial
Aliases: C2×D4○C16, C8.24C24, C16.14C23, M5(2)⋊16C22, D4○(C2×C16), C16○(C2×D4), Q8○(C2×C16), C16○(C2×Q8), C8○(D4○C16), C4○(D4○C16), C4○D4.6C8, C8○D4.7C4, D4.8(C2×C8), Q8.9(C2×C8), C16○2(C4○D4), C16○(D4○C16), C16○2(C8○D4), (C2×D4).13C8, (C2×Q8).12C8, (C22×C16)⋊15C2, (C2×C16)⋊22C22, (C2×C16)○2M5(2), C16○2(C2×M5(2)), C16○2(C2×M4(2)), M4(2)○2(C2×C16), C4.63(C23×C4), C8.50(C22×C4), C2.11(C23×C8), C4.22(C22×C8), C23.24(C2×C8), (C2×M5(2))⋊22C2, (C2×C8).617C23, C8○D4.19C22, C22.4(C22×C8), (C2×M4(2)).38C4, M4(2).35(C2×C4), (C22×C8).587C22, C4○D4○(C2×C16), C16○(C2×C4○D4), C16○(C2×C8○D4), (C2×Q8)○(C2×C16), (C2×C16)○(C8○D4), (C2×C4).57(C2×C8), (C2×C8).198(C2×C4), C4○D4.38(C2×C4), (C2×C4○D4).35C4, (C2×C8○D4).24C2, (C2×C16)○(C2×M5(2)), (C2×C16)○(C2×M4(2)), (C22×C4).421(C2×C4), (C2×C4).476(C22×C4), (C2×C16)○(C2×C8○D4), (C2×C16)○(C2×C4○D4), SmallGroup(128,2138)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C2×D4○C16
G = < a,b,c,d | a2=b4=c2=1, d8=b2, ab=ba, ac=ca, ad=da, cbc=b-1, bd=db, cd=dc >
Subgroups: 196 in 184 conjugacy classes, 172 normal (14 characteristic)
C1, C2, C2 [×2], C2 [×6], C4 [×2], C4 [×6], C22, C22 [×6], C22 [×6], C8 [×2], C8 [×6], C2×C4, C2×C4 [×15], D4 [×12], Q8 [×4], C23 [×3], C16 [×8], C2×C8, C2×C8 [×15], M4(2) [×12], C22×C4 [×3], C2×D4 [×3], C2×Q8, C4○D4 [×8], C2×C16, C2×C16 [×15], M5(2) [×12], C22×C8 [×3], C2×M4(2) [×3], C8○D4 [×8], C2×C4○D4, C22×C16 [×3], C2×M5(2) [×3], D4○C16 [×8], C2×C8○D4, C2×D4○C16
Quotients: C1, C2 [×15], C4 [×8], C22 [×35], C8 [×8], C2×C4 [×28], C23 [×15], C2×C8 [×28], C22×C4 [×14], C24, C22×C8 [×14], C23×C4, D4○C16 [×2], C23×C8, C2×D4○C16
►On 64 points
(1 35)(2 36)(3 37)(4 38)(5 39)(6 40)(7 41)(8 42)(9 43)(10 44)(11 45)(12 46)(13 47)(14 48)(15 33)(16 34)(17 58)(18 59)(19 60)(20 61)(21 62)(22 63)(23 64)(24 49)(25 50)(26 51)(27 52)(28 53)(29 54)(30 55)(31 56)(32 57)
(1 51 9 59)(2 52 10 60)(3 53 11 61)(4 54 12 62)(5 55 13 63)(6 56 14 64)(7 57 15 49)(8 58 16 50)(17 34 25 42)(18 35 26 43)(19 36 27 44)(20 37 28 45)(21 38 29 46)(22 39 30 47)(23 40 31 48)(24 41 32 33)
(1 59)(2 60)(3 61)(4 62)(5 63)(6 64)(7 49)(8 50)(9 51)(10 52)(11 53)(12 54)(13 55)(14 56)(15 57)(16 58)(17 34)(18 35)(19 36)(20 37)(21 38)(22 39)(23 40)(24 41)(25 42)(26 43)(27 44)(28 45)(29 46)(30 47)(31 48)(32 33)
(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)| (1,35)(2,36)(3,37)(4,38)(5,39)(6,40)(7,41)(8,42)(9,43)(10,44)(11,45)(12,46)(13,47)(14,48)(15,33)(16,34)(17,58)(18,59)(19,60)(20,61)(21,62)(22,63)(23,64)(24,49)(25,50)(26,51)(27,52)(28,53)(29,54)(30,55)(31,56)(32,57), (1,51,9,59)(2,52,10,60)(3,53,11,61)(4,54,12,62)(5,55,13,63)(6,56,14,64)(7,57,15,49)(8,58,16,50)(17,34,25,42)(18,35,26,43)(19,36,27,44)(20,37,28,45)(21,38,29,46)(22,39,30,47)(23,40,31,48)(24,41,32,33), (1,59)(2,60)(3,61)(4,62)(5,63)(6,64)(7,49)(8,50)(9,51)(10,52)(11,53)(12,54)(13,55)(14,56)(15,57)(16,58)(17,34)(18,35)(19,36)(20,37)(21,38)(22,39)(23,40)(24,41)(25,42)(26,43)(27,44)(28,45)(29,46)(30,47)(31,48)(32,33), (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( (1,35)(2,36)(3,37)(4,38)(5,39)(6,40)(7,41)(8,42)(9,43)(10,44)(11,45)(12,46)(13,47)(14,48)(15,33)(16,34)(17,58)(18,59)(19,60)(20,61)(21,62)(22,63)(23,64)(24,49)(25,50)(26,51)(27,52)(28,53)(29,54)(30,55)(31,56)(32,57), (1,51,9,59)(2,52,10,60)(3,53,11,61)(4,54,12,62)(5,55,13,63)(6,56,14,64)(7,57,15,49)(8,58,16,50)(17,34,25,42)(18,35,26,43)(19,36,27,44)(20,37,28,45)(21,38,29,46)(22,39,30,47)(23,40,31,48)(24,41,32,33), (1,59)(2,60)(3,61)(4,62)(5,63)(6,64)(7,49)(8,50)(9,51)(10,52)(11,53)(12,54)(13,55)(14,56)(15,57)(16,58)(17,34)(18,35)(19,36)(20,37)(21,38)(22,39)(23,40)(24,41)(25,42)(26,43)(27,44)(28,45)(29,46)(30,47)(31,48)(32,33), (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([(1,35),(2,36),(3,37),(4,38),(5,39),(6,40),(7,41),(8,42),(9,43),(10,44),(11,45),(12,46),(13,47),(14,48),(15,33),(16,34),(17,58),(18,59),(19,60),(20,61),(21,62),(22,63),(23,64),(24,49),(25,50),(26,51),(27,52),(28,53),(29,54),(30,55),(31,56),(32,57)], [(1,51,9,59),(2,52,10,60),(3,53,11,61),(4,54,12,62),(5,55,13,63),(6,56,14,64),(7,57,15,49),(8,58,16,50),(17,34,25,42),(18,35,26,43),(19,36,27,44),(20,37,28,45),(21,38,29,46),(22,39,30,47),(23,40,31,48),(24,41,32,33)], [(1,59),(2,60),(3,61),(4,62),(5,63),(6,64),(7,49),(8,50),(9,51),(10,52),(11,53),(12,54),(13,55),(14,56),(15,57),(16,58),(17,34),(18,35),(19,36),(20,37),(21,38),(22,39),(23,40),(24,41),(25,42),(26,43),(27,44),(28,45),(29,46),(30,47),(31,48),(32,33)], [(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)])
80 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | ··· | 2I | 4A | 4B | 4C | 4D | 4E | ··· | 4J | 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 | 2 |
| type | + | + | + | + | + | |||||||
| image | C1 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | C8 | C8 | C8 | D4○C16 |
| kernel | C2×D4○C16 | C22×C16 | C2×M5(2) | D4○C16 | C2×C8○D4 | C2×M4(2) | C8○D4 | C2×C4○D4 | C2×D4 | C2×Q8 | C4○D4 | C2 |
| # reps | 1 | 3 | 3 | 8 | 1 | 6 | 8 | 2 | 12 | 4 | 16 | 16 |
Matrix representation of C2×D4○C16 ►in GL3(𝔽17) generated by
| 16 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 0 | 1 |
| 1 | 0 | 0 |
| 0 | 0 | 16 |
| 0 | 1 | 0 |
| 16 | 0 | 0 |
| 0 | 0 | 16 |
| 0 | 16 | 0 |
| 16 | 0 | 0 |
| 0 | 14 | 0 |
| 0 | 0 | 14 |
G:=sub<GL(3,GF(17))| [16,0,0,0,1,0,0,0,1],[1,0,0,0,0,1,0,16,0],[16,0,0,0,0,16,0,16,0],[16,0,0,0,14,0,0,0,14] >;
C2×D4○C16 in GAP, Magma, Sage, TeX
C_2\times D_4\circ C_{16} % in TeX
G:=Group("C2xD4oC16"); // GroupNames label
G:=SmallGroup(128,2138);
// by ID
G=gap.SmallGroup(128,2138);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,-2,-2,112,723,102,124]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^4=c^2=1,d^8=b^2,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c=b^-1,b*d=d*b,c*d=d*c>;
// generators/relations