direct product, p-group, metabelian, nilpotent (class 2), monomial
Aliases: D4×C2×C8, C42.679C23, C8○(C8×D4), C4○(C8×D4), C8○2(C4×D4), C23⋊5(C2×C8), (C23×C8)⋊7C2, C4⋊1(C22×C8), C4⋊C8⋊94C22, (C4×C8)⋊69C22, (C4×D4).42C4, C4.182(C4×D4), C2.4(C23×C8), C24.99(C2×C4), C22⋊1(C22×C8), C22⋊C8⋊82C22, C42.281(C2×C4), (C2×C4).645C24, (C2×C8).476C23, (C22×C8)⋊64C22, (C22×D4).46C4, C4.191(C22×D4), C22.112(C4×D4), (C4×D4).359C22, C22.40(C8○D4), C22.40(C23×C4), (C23×C4).659C22, C23.140(C22×C4), (C22×C4).1274C23, (C2×C42).1108C22, C8○2(C2×C4⋊C8), C8○2(C2×C4⋊C4), C4⋊C4○3(C2×C8), (C2×C4×C8)⋊18C2, (C2×C8)○(C8×D4), (C2×C4)○(C8×D4), C2.4(C2×C4×D4), (C2×C4)⋊8(C2×C8), (C2×C8)○2(C4×D4), C4⋊C4○(C22×C8), (C2×C4⋊C8)⋊55C2, (C2×C8)○3(C4⋊C8), C22⋊C4○3(C2×C8), C8○2(C2×C22⋊C8), C4⋊C8○2(C22×C8), C8○2(C2×C22⋊C4), (C2×C4×D4).92C2, C2.3(C2×C8○D4), (C2×C8)○(C22×D4), (C4×D4)○(C22×C8), (C2×D4)○(C22×C8), (C2×C4⋊C4).84C4, C22⋊C4○(C22×C8), C4⋊C4.246(C2×C4), (C2×C8)○3(C22⋊C8), (C2×C22⋊C8)⋊49C2, C4.296(C2×C4○D4), C22⋊C8○2(C22×C8), (C22×C8)○(C22×D4), (C2×D4).248(C2×C4), (C2×C4).1570(C2×D4), (C2×C22⋊C4).55C4, C22⋊C4.90(C2×C4), (C2×C4).955(C4○D4), (C2×C4).292(C22×C4), (C22×C4).383(C2×C4), (C2×C8)○(C2×C4×D4), (C2×C8)○2(C2×C4⋊C8), (C2×C8)○2(C2×C4⋊C4), (C22×C8)○(C2×C4×D4), (C2×C4⋊C4)○(C22×C8), (C22×C8)○(C2×C4⋊C8), (C2×C8)○2(C2×C22⋊C4), (C2×C8)○2(C2×C22⋊C8), (C2×C22⋊C4)○(C22×C8), (C22×C8)○(C2×C22⋊C8), SmallGroup(128,1658)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for D4×C2×C8
G = < a,b,c,d | a2=b8=c4=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >
Subgroups: 420 in 308 conjugacy classes, 196 normal (22 characteristic)
C1, C2, C2, C2, C4, C4, C4, C22, C22, C22, C8, C8, C2×C4, C2×C4, C2×C4, D4, C23, C23, C23, C42, C22⋊C4, C4⋊C4, C2×C8, C2×C8, C22×C4, C22×C4, C22×C4, C2×D4, C24, C4×C8, C22⋊C8, C4⋊C8, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C4×D4, C22×C8, C22×C8, C22×C8, C23×C4, C22×D4, C2×C4×C8, C2×C22⋊C8, C2×C4⋊C8, C8×D4, C2×C4×D4, C23×C8, D4×C2×C8
Quotients: C1, C2, C4, C22, C8, C2×C4, D4, C23, C2×C8, C22×C4, C2×D4, C4○D4, C24, C4×D4, C22×C8, C8○D4, C23×C4, C22×D4, C2×C4○D4, C8×D4, C2×C4×D4, C23×C8, C2×C8○D4, D4×C2×C8
►On 64 points
(1 14)(2 15)(3 16)(4 9)(5 10)(6 11)(7 12)(8 13)(17 41)(18 42)(19 43)(20 44)(21 45)(22 46)(23 47)(24 48)(25 40)(26 33)(27 34)(28 35)(29 36)(30 37)(31 38)(32 39)(49 62)(50 63)(51 64)(52 57)(53 58)(54 59)(55 60)(56 61)
(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 43 54)(2 32 44 55)(3 25 45 56)(4 26 46 49)(5 27 47 50)(6 28 48 51)(7 29 41 52)(8 30 42 53)(9 33 22 62)(10 34 23 63)(11 35 24 64)(12 36 17 57)(13 37 18 58)(14 38 19 59)(15 39 20 60)(16 40 21 61)
(1 19)(2 20)(3 21)(4 22)(5 23)(6 24)(7 17)(8 18)(9 46)(10 47)(11 48)(12 41)(13 42)(14 43)(15 44)(16 45)(25 40)(26 33)(27 34)(28 35)(29 36)(30 37)(31 38)(32 39)(49 62)(50 63)(51 64)(52 57)(53 58)(54 59)(55 60)(56 61)
G:=sub<Sym(64)| (1,14)(2,15)(3,16)(4,9)(5,10)(6,11)(7,12)(8,13)(17,41)(18,42)(19,43)(20,44)(21,45)(22,46)(23,47)(24,48)(25,40)(26,33)(27,34)(28,35)(29,36)(30,37)(31,38)(32,39)(49,62)(50,63)(51,64)(52,57)(53,58)(54,59)(55,60)(56,61), (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), (1,31,43,54)(2,32,44,55)(3,25,45,56)(4,26,46,49)(5,27,47,50)(6,28,48,51)(7,29,41,52)(8,30,42,53)(9,33,22,62)(10,34,23,63)(11,35,24,64)(12,36,17,57)(13,37,18,58)(14,38,19,59)(15,39,20,60)(16,40,21,61), (1,19)(2,20)(3,21)(4,22)(5,23)(6,24)(7,17)(8,18)(9,46)(10,47)(11,48)(12,41)(13,42)(14,43)(15,44)(16,45)(25,40)(26,33)(27,34)(28,35)(29,36)(30,37)(31,38)(32,39)(49,62)(50,63)(51,64)(52,57)(53,58)(54,59)(55,60)(56,61)>;
G:=Group( (1,14)(2,15)(3,16)(4,9)(5,10)(6,11)(7,12)(8,13)(17,41)(18,42)(19,43)(20,44)(21,45)(22,46)(23,47)(24,48)(25,40)(26,33)(27,34)(28,35)(29,36)(30,37)(31,38)(32,39)(49,62)(50,63)(51,64)(52,57)(53,58)(54,59)(55,60)(56,61), (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), (1,31,43,54)(2,32,44,55)(3,25,45,56)(4,26,46,49)(5,27,47,50)(6,28,48,51)(7,29,41,52)(8,30,42,53)(9,33,22,62)(10,34,23,63)(11,35,24,64)(12,36,17,57)(13,37,18,58)(14,38,19,59)(15,39,20,60)(16,40,21,61), (1,19)(2,20)(3,21)(4,22)(5,23)(6,24)(7,17)(8,18)(9,46)(10,47)(11,48)(12,41)(13,42)(14,43)(15,44)(16,45)(25,40)(26,33)(27,34)(28,35)(29,36)(30,37)(31,38)(32,39)(49,62)(50,63)(51,64)(52,57)(53,58)(54,59)(55,60)(56,61) );
G=PermutationGroup([[(1,14),(2,15),(3,16),(4,9),(5,10),(6,11),(7,12),(8,13),(17,41),(18,42),(19,43),(20,44),(21,45),(22,46),(23,47),(24,48),(25,40),(26,33),(27,34),(28,35),(29,36),(30,37),(31,38),(32,39),(49,62),(50,63),(51,64),(52,57),(53,58),(54,59),(55,60),(56,61)], [(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)], [(1,31,43,54),(2,32,44,55),(3,25,45,56),(4,26,46,49),(5,27,47,50),(6,28,48,51),(7,29,41,52),(8,30,42,53),(9,33,22,62),(10,34,23,63),(11,35,24,64),(12,36,17,57),(13,37,18,58),(14,38,19,59),(15,39,20,60),(16,40,21,61)], [(1,19),(2,20),(3,21),(4,22),(5,23),(6,24),(7,17),(8,18),(9,46),(10,47),(11,48),(12,41),(13,42),(14,43),(15,44),(16,45),(25,40),(26,33),(27,34),(28,35),(29,36),(30,37),(31,38),(32,39),(49,62),(50,63),(51,64),(52,57),(53,58),(54,59),(55,60),(56,61)]])
80 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | ··· | 2O | 4A | ··· | 4H | 4I | ··· | 4X | 8A | ··· | 8P | 8Q | ··· | 8AN |
| order | 1 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 8 | ··· | 8 | 8 | ··· | 8 |
| size | 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 | 1 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | |||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | C4 | C8 | D4 | C4○D4 | C8○D4 |
| kernel | D4×C2×C8 | C2×C4×C8 | C2×C22⋊C8 | C2×C4⋊C8 | C8×D4 | C2×C4×D4 | C23×C8 | C2×C22⋊C4 | C2×C4⋊C4 | C4×D4 | C22×D4 | C2×D4 | C2×C8 | C2×C4 | C22 |
| # reps | 1 | 1 | 2 | 1 | 8 | 1 | 2 | 4 | 2 | 8 | 2 | 32 | 4 | 4 | 8 |
Matrix representation of D4×C2×C8 ►in GL4(𝔽17) generated by
| 16 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 15 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 15 | 0 |
| 0 | 0 | 0 | 15 |
| 16 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 16 | 1 |
| 0 | 0 | 15 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 16 | 1 |
| 0 | 0 | 0 | 1 |
G:=sub<GL(4,GF(17))| [16,0,0,0,0,16,0,0,0,0,1,0,0,0,0,1],[15,0,0,0,0,16,0,0,0,0,15,0,0,0,0,15],[16,0,0,0,0,16,0,0,0,0,16,15,0,0,1,1],[1,0,0,0,0,1,0,0,0,0,16,0,0,0,1,1] >;
D4×C2×C8 in GAP, Magma, Sage, TeX
D_4\times C_2\times C_8
% in TeX
G:=Group("D4xC2xC8"); // GroupNames label
G:=SmallGroup(128,1658);
// by ID
G=gap.SmallGroup(128,1658);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,224,253,184,124]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^8=c^4=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations