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, C2, C4, C4, C22, C22, C22, C8, C8, C2×C4, C2×C4, D4, Q8, C23, C16, C2×C8, C2×C8, M4(2), C22×C4, C2×D4, C2×Q8, C4○D4, C2×C16, C2×C16, M5(2), C22×C8, C2×M4(2), C8○D4, C2×C4○D4, C22×C16, C2×M5(2), D4○C16, C2×C8○D4, C2×D4○C16
Quotients: C1, C2, C4, C22, C8, C2×C4, C23, C2×C8, C22×C4, C24, C22×C8, C23×C4, D4○C16, C23×C8, C2×D4○C16
►On 64 points
(1 40)(2 41)(3 42)(4 43)(5 44)(6 45)(7 46)(8 47)(9 48)(10 33)(11 34)(12 35)(13 36)(14 37)(15 38)(16 39)(17 54)(18 55)(19 56)(20 57)(21 58)(22 59)(23 60)(24 61)(25 62)(26 63)(27 64)(28 49)(29 50)(30 51)(31 52)(32 53)
(1 63 9 55)(2 64 10 56)(3 49 11 57)(4 50 12 58)(5 51 13 59)(6 52 14 60)(7 53 15 61)(8 54 16 62)(17 39 25 47)(18 40 26 48)(19 41 27 33)(20 42 28 34)(21 43 29 35)(22 44 30 36)(23 45 31 37)(24 46 32 38)
(1 55)(2 56)(3 57)(4 58)(5 59)(6 60)(7 61)(8 62)(9 63)(10 64)(11 49)(12 50)(13 51)(14 52)(15 53)(16 54)(17 39)(18 40)(19 41)(20 42)(21 43)(22 44)(23 45)(24 46)(25 47)(26 48)(27 33)(28 34)(29 35)(30 36)(31 37)(32 38)
(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)| (1,40)(2,41)(3,42)(4,43)(5,44)(6,45)(7,46)(8,47)(9,48)(10,33)(11,34)(12,35)(13,36)(14,37)(15,38)(16,39)(17,54)(18,55)(19,56)(20,57)(21,58)(22,59)(23,60)(24,61)(25,62)(26,63)(27,64)(28,49)(29,50)(30,51)(31,52)(32,53), (1,63,9,55)(2,64,10,56)(3,49,11,57)(4,50,12,58)(5,51,13,59)(6,52,14,60)(7,53,15,61)(8,54,16,62)(17,39,25,47)(18,40,26,48)(19,41,27,33)(20,42,28,34)(21,43,29,35)(22,44,30,36)(23,45,31,37)(24,46,32,38), (1,55)(2,56)(3,57)(4,58)(5,59)(6,60)(7,61)(8,62)(9,63)(10,64)(11,49)(12,50)(13,51)(14,52)(15,53)(16,54)(17,39)(18,40)(19,41)(20,42)(21,43)(22,44)(23,45)(24,46)(25,47)(26,48)(27,33)(28,34)(29,35)(30,36)(31,37)(32,38), (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( (1,40)(2,41)(3,42)(4,43)(5,44)(6,45)(7,46)(8,47)(9,48)(10,33)(11,34)(12,35)(13,36)(14,37)(15,38)(16,39)(17,54)(18,55)(19,56)(20,57)(21,58)(22,59)(23,60)(24,61)(25,62)(26,63)(27,64)(28,49)(29,50)(30,51)(31,52)(32,53), (1,63,9,55)(2,64,10,56)(3,49,11,57)(4,50,12,58)(5,51,13,59)(6,52,14,60)(7,53,15,61)(8,54,16,62)(17,39,25,47)(18,40,26,48)(19,41,27,33)(20,42,28,34)(21,43,29,35)(22,44,30,36)(23,45,31,37)(24,46,32,38), (1,55)(2,56)(3,57)(4,58)(5,59)(6,60)(7,61)(8,62)(9,63)(10,64)(11,49)(12,50)(13,51)(14,52)(15,53)(16,54)(17,39)(18,40)(19,41)(20,42)(21,43)(22,44)(23,45)(24,46)(25,47)(26,48)(27,33)(28,34)(29,35)(30,36)(31,37)(32,38), (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([[(1,40),(2,41),(3,42),(4,43),(5,44),(6,45),(7,46),(8,47),(9,48),(10,33),(11,34),(12,35),(13,36),(14,37),(15,38),(16,39),(17,54),(18,55),(19,56),(20,57),(21,58),(22,59),(23,60),(24,61),(25,62),(26,63),(27,64),(28,49),(29,50),(30,51),(31,52),(32,53)], [(1,63,9,55),(2,64,10,56),(3,49,11,57),(4,50,12,58),(5,51,13,59),(6,52,14,60),(7,53,15,61),(8,54,16,62),(17,39,25,47),(18,40,26,48),(19,41,27,33),(20,42,28,34),(21,43,29,35),(22,44,30,36),(23,45,31,37),(24,46,32,38)], [(1,55),(2,56),(3,57),(4,58),(5,59),(6,60),(7,61),(8,62),(9,63),(10,64),(11,49),(12,50),(13,51),(14,52),(15,53),(16,54),(17,39),(18,40),(19,41),(20,42),(21,43),(22,44),(23,45),(24,46),(25,47),(26,48),(27,33),(28,34),(29,35),(30,36),(31,37),(32,38)], [(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