direct product, p-group, metabelian, nilpotent (class 2), monomial
Aliases: C4×C8○D4, D4.4C42, Q8.4C42, C42.589C23, C8○(C4×D4), D4○(C4×C8), C8○(C4×Q8), Q8○(C4×C8), C8○2(C8○D4), M4(2)○(C4×C8), C8○(C4×M4(2)), (C4×D4).41C4, (C4×Q8).38C4, C8.48(C22×C4), C4.60(C23×C4), C4.12(C2×C42), (C4×M4(2))⋊44C2, M4(2)⋊24(C2×C4), (C2×C8).638C23, C42.280(C2×C4), (C2×C4).624C24, (C4×C8).435C22, C22.2(C2×C42), C8○2(C8○2M4(2)), C4○2(C8○2M4(2)), C8○2M4(2)⋊39C2, C22.35(C23×C4), C2.16(C22×C42), C8⋊C4.172C22, C42○(C8○2M4(2)), C23.136(C22×C4), (C22×C8).584C22, (C22×C4).1489C23, (C2×C42).1101C22, C42⋊C2.347C22, (C2×M4(2)).381C22, (C2×C4×C8)⋊41C2, C8○(C2×C8○D4), C4○D4○(C4×C8), C8○(C4×C4○D4), (C4×C8)○(C4×D4), (C4×C8)○(C4×Q8), (C2×Q8)○(C4×C8), (C2×C8)⋊33(C2×C4), (C2×C8)○(C8○D4), C2.2(C2×C8○D4), C42○(C2×C8○D4), (C4×C8)○(C4×M4(2)), (C4×C8)○2(C8⋊C4), C4⋊C4.244(C2×C4), (C2×C8○D4).23C2, (C4×C4○D4).32C2, C4○D4.36(C2×C4), (C4×C8)○2(C2×M4(2)), (C2×C8)○2(C4×M4(2)), (C4×C8)○(C42⋊C2), (C2×D4).245(C2×C4), C22⋊C4.87(C2×C4), (C2×Q8).221(C2×C4), (C4×C8)○(C8○2M4(2)), (C2×C4).454(C22×C4), (C22×C4).382(C2×C4), (C2×C4○D4).337C22, (C4×C8)○(C2×C8○D4), (C4×C8)○(C2×C4○D4), (C2×C8)○(C4×C4○D4), SmallGroup(128,1606)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Subgroups: 316 in 286 conjugacy classes, 256 normal (10 characteristic)
C1, C2, C2 [×2], C2 [×6], C4 [×12], C4 [×6], C22, C22 [×6], C22 [×6], C8 [×16], C2×C4, C2×C4 [×23], C2×C4 [×12], D4 [×12], Q8 [×4], C23 [×3], C42, C42 [×9], C22⋊C4 [×6], C4⋊C4 [×6], C2×C8 [×32], M4(2) [×24], C22×C4 [×9], C2×D4 [×3], C2×Q8, C4○D4 [×8], C4×C8, C4×C8 [×9], C8⋊C4 [×6], C2×C42 [×3], C42⋊C2 [×3], C4×D4 [×6], C4×Q8 [×2], C22×C8 [×6], C2×M4(2) [×6], C8○D4 [×16], C2×C4○D4, C2×C4×C8 [×3], C4×M4(2) [×3], C8○2M4(2) [×6], C4×C4○D4, C2×C8○D4 [×2], C4×C8○D4
Quotients:
C1, C2 [×15], C4 [×24], C22 [×35], C2×C4 [×84], C23 [×15], C42 [×16], C22×C4 [×42], C24, C2×C42 [×12], C8○D4 [×4], C23×C4 [×3], C22×C42, C2×C8○D4 [×2], C4×C8○D4
Generators and relations
G = < a,b,c,d | a4=b8=d2=1, c2=b4, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=b4c >
►On 64 points
(1 31 63 47)(2 32 64 48)(3 25 57 41)(4 26 58 42)(5 27 59 43)(6 28 60 44)(7 29 61 45)(8 30 62 46)(9 18 38 49)(10 19 39 50)(11 20 40 51)(12 21 33 52)(13 22 34 53)(14 23 35 54)(15 24 36 55)(16 17 37 56)
(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 61 5 57)(2 62 6 58)(3 63 7 59)(4 64 8 60)(9 40 13 36)(10 33 14 37)(11 34 15 38)(12 35 16 39)(17 50 21 54)(18 51 22 55)(19 52 23 56)(20 53 24 49)(25 47 29 43)(26 48 30 44)(27 41 31 45)(28 42 32 46)
(1 20)(2 21)(3 22)(4 23)(5 24)(6 17)(7 18)(8 19)(9 45)(10 46)(11 47)(12 48)(13 41)(14 42)(15 43)(16 44)(25 34)(26 35)(27 36)(28 37)(29 38)(30 39)(31 40)(32 33)(49 61)(50 62)(51 63)(52 64)(53 57)(54 58)(55 59)(56 60)
G:=sub<Sym(64)| (1,31,63,47)(2,32,64,48)(3,25,57,41)(4,26,58,42)(5,27,59,43)(6,28,60,44)(7,29,61,45)(8,30,62,46)(9,18,38,49)(10,19,39,50)(11,20,40,51)(12,21,33,52)(13,22,34,53)(14,23,35,54)(15,24,36,55)(16,17,37,56), (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), (1,61,5,57)(2,62,6,58)(3,63,7,59)(4,64,8,60)(9,40,13,36)(10,33,14,37)(11,34,15,38)(12,35,16,39)(17,50,21,54)(18,51,22,55)(19,52,23,56)(20,53,24,49)(25,47,29,43)(26,48,30,44)(27,41,31,45)(28,42,32,46), (1,20)(2,21)(3,22)(4,23)(5,24)(6,17)(7,18)(8,19)(9,45)(10,46)(11,47)(12,48)(13,41)(14,42)(15,43)(16,44)(25,34)(26,35)(27,36)(28,37)(29,38)(30,39)(31,40)(32,33)(49,61)(50,62)(51,63)(52,64)(53,57)(54,58)(55,59)(56,60)>;
G:=Group( (1,31,63,47)(2,32,64,48)(3,25,57,41)(4,26,58,42)(5,27,59,43)(6,28,60,44)(7,29,61,45)(8,30,62,46)(9,18,38,49)(10,19,39,50)(11,20,40,51)(12,21,33,52)(13,22,34,53)(14,23,35,54)(15,24,36,55)(16,17,37,56), (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), (1,61,5,57)(2,62,6,58)(3,63,7,59)(4,64,8,60)(9,40,13,36)(10,33,14,37)(11,34,15,38)(12,35,16,39)(17,50,21,54)(18,51,22,55)(19,52,23,56)(20,53,24,49)(25,47,29,43)(26,48,30,44)(27,41,31,45)(28,42,32,46), (1,20)(2,21)(3,22)(4,23)(5,24)(6,17)(7,18)(8,19)(9,45)(10,46)(11,47)(12,48)(13,41)(14,42)(15,43)(16,44)(25,34)(26,35)(27,36)(28,37)(29,38)(30,39)(31,40)(32,33)(49,61)(50,62)(51,63)(52,64)(53,57)(54,58)(55,59)(56,60) );
G=PermutationGroup([(1,31,63,47),(2,32,64,48),(3,25,57,41),(4,26,58,42),(5,27,59,43),(6,28,60,44),(7,29,61,45),(8,30,62,46),(9,18,38,49),(10,19,39,50),(11,20,40,51),(12,21,33,52),(13,22,34,53),(14,23,35,54),(15,24,36,55),(16,17,37,56)], [(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)], [(1,61,5,57),(2,62,6,58),(3,63,7,59),(4,64,8,60),(9,40,13,36),(10,33,14,37),(11,34,15,38),(12,35,16,39),(17,50,21,54),(18,51,22,55),(19,52,23,56),(20,53,24,49),(25,47,29,43),(26,48,30,44),(27,41,31,45),(28,42,32,46)], [(1,20),(2,21),(3,22),(4,23),(5,24),(6,17),(7,18),(8,19),(9,45),(10,46),(11,47),(12,48),(13,41),(14,42),(15,43),(16,44),(25,34),(26,35),(27,36),(28,37),(29,38),(30,39),(31,40),(32,33),(49,61),(50,62),(51,63),(52,64),(53,57),(54,58),(55,59),(56,60)])
Matrix representation ►G ⊆ GL3(𝔽17) generated by
| 13 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 0 | 1 |
| 1 | 0 | 0 |
| 0 | 2 | 0 |
| 0 | 0 | 2 |
| 1 | 0 | 0 |
| 0 | 13 | 0 |
| 0 | 0 | 4 |
| 1 | 0 | 0 |
| 0 | 0 | 4 |
| 0 | 13 | 0 |
G:=sub<GL(3,GF(17))| [13,0,0,0,1,0,0,0,1],[1,0,0,0,2,0,0,0,2],[1,0,0,0,13,0,0,0,4],[1,0,0,0,0,13,0,4,0] >;
80 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | ··· | 2I | 4A | ··· | 4L | 4M | ··· | 4AD | 8A | ··· | 8P | 8Q | ··· | 8AN |
| order | 1 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 8 | ··· | 8 | 8 | ··· | 8 |
| size | 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 | 2 |
| type | + | + | + | + | + | + | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | C8○D4 |
| kernel | C4×C8○D4 | C2×C4×C8 | C4×M4(2) | C8○2M4(2) | C4×C4○D4 | C2×C8○D4 | C4×D4 | C4×Q8 | C8○D4 | C4 |
| # reps | 1 | 3 | 3 | 6 | 1 | 2 | 12 | 4 | 32 | 16 |
In GAP, Magma, Sage, TeX
C_4\times C_8\circ D_4
% in TeX
G:=Group("C4xC8oD4"); // GroupNames label
G:=SmallGroup(128,1606);
// by ID
G=gap.SmallGroup(128,1606);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,112,232,521,172]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^8=d^2=1,c^2=b^4,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=b^4*c>;
// generators/relations