p-group, metabelian, nilpotent (class 2), monomial
Aliases: D4.5C42, Q8.5C42, C42.590C23, C8○D4⋊8C4, D4○(C8⋊C4), Q8○(C8⋊C4), (C4×D4).21C4, (C4×Q8).20C4, C4.61(C23×C4), C8.49(C22×C4), C4.13(C2×C42), M4(2)⋊25(C2×C4), (C4×M4(2))⋊28C2, M4(2)○(C8⋊C4), (C4×C8).324C22, (C2×C8).612C23, (C2×C4).625C24, C42.199(C2×C4), C22.3(C2×C42), C8○2M4(2)⋊27C2, C2.2(Q8○M4(2)), C22.36(C23×C4), C2.17(C22×C42), C8⋊C4.173C22, C23.137(C22×C4), (C22×C8).422C22, (C2×C42).750C22, (C22×C4).1490C23, C42⋊C2.348C22, (C2×M4(2)).382C22, (C2×C8)⋊26(C2×C4), C4○D4○(C8⋊C4), (C2×Q8)○(C8⋊C4), (C4×C4○D4).9C2, (C2×C8⋊C4)⋊29C2, C4⋊C4.245(C2×C4), (C2×C8○D4).20C2, C4○D4.37(C2×C4), C8⋊C4○(C2×M4(2)), (C2×D4).246(C2×C4), C22⋊C4.88(C2×C4), (C2×Q8).222(C2×C4), C8⋊C4○(C42⋊C2), (C22×C4).135(C2×C4), (C2×C4).455(C22×C4), (C2×C4○D4).338C22, C8⋊C4○(C2×C4○D4), SmallGroup(128,1607)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Subgroups: 316 in 278 conjugacy classes, 252 normal (10 characteristic)
C1, C2, C2 [×2], C2 [×6], C4 [×8], C4 [×8], C22, C22 [×6], C22 [×6], C8 [×16], C2×C4, C2×C4 [×23], C2×C4 [×6], 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 [×6], C8⋊C4, C8⋊C4 [×9], 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×C8⋊C4 [×3], C4×M4(2) [×3], C8○2M4(2) [×6], C4×C4○D4, C2×C8○D4 [×2], D4.5C42
Quotients:
C1, C2 [×15], C4 [×24], C22 [×35], C2×C4 [×84], C23 [×15], C42 [×16], C22×C4 [×42], C24, C2×C42 [×12], C23×C4 [×3], C22×C42, Q8○M4(2) [×2], D4.5C42
Generators and relations
G = < a,b,c,d | a4=b2=d4=1, c4=a2, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=a2c >
►On 64 points
(1 7 5 3)(2 8 6 4)(9 11 13 15)(10 12 14 16)(17 19 21 23)(18 20 22 24)(25 27 29 31)(26 28 30 32)(33 35 37 39)(34 36 38 40)(41 47 45 43)(42 48 46 44)(49 55 53 51)(50 56 54 52)(57 63 61 59)(58 64 62 60)
(1 14)(2 15)(3 16)(4 9)(5 10)(6 11)(7 12)(8 13)(17 44)(18 45)(19 46)(20 47)(21 48)(22 41)(23 42)(24 43)(25 51)(26 52)(27 53)(28 54)(29 55)(30 56)(31 49)(32 50)(33 61)(34 62)(35 63)(36 64)(37 57)(38 58)(39 59)(40 60)
(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 63 55 46)(2 60 56 43)(3 57 49 48)(4 62 50 45)(5 59 51 42)(6 64 52 47)(7 61 53 44)(8 58 54 41)(9 34 32 18)(10 39 25 23)(11 36 26 20)(12 33 27 17)(13 38 28 22)(14 35 29 19)(15 40 30 24)(16 37 31 21)
G:=sub<Sym(64)| (1,7,5,3)(2,8,6,4)(9,11,13,15)(10,12,14,16)(17,19,21,23)(18,20,22,24)(25,27,29,31)(26,28,30,32)(33,35,37,39)(34,36,38,40)(41,47,45,43)(42,48,46,44)(49,55,53,51)(50,56,54,52)(57,63,61,59)(58,64,62,60), (1,14)(2,15)(3,16)(4,9)(5,10)(6,11)(7,12)(8,13)(17,44)(18,45)(19,46)(20,47)(21,48)(22,41)(23,42)(24,43)(25,51)(26,52)(27,53)(28,54)(29,55)(30,56)(31,49)(32,50)(33,61)(34,62)(35,63)(36,64)(37,57)(38,58)(39,59)(40,60), (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), (1,63,55,46)(2,60,56,43)(3,57,49,48)(4,62,50,45)(5,59,51,42)(6,64,52,47)(7,61,53,44)(8,58,54,41)(9,34,32,18)(10,39,25,23)(11,36,26,20)(12,33,27,17)(13,38,28,22)(14,35,29,19)(15,40,30,24)(16,37,31,21)>;
G:=Group( (1,7,5,3)(2,8,6,4)(9,11,13,15)(10,12,14,16)(17,19,21,23)(18,20,22,24)(25,27,29,31)(26,28,30,32)(33,35,37,39)(34,36,38,40)(41,47,45,43)(42,48,46,44)(49,55,53,51)(50,56,54,52)(57,63,61,59)(58,64,62,60), (1,14)(2,15)(3,16)(4,9)(5,10)(6,11)(7,12)(8,13)(17,44)(18,45)(19,46)(20,47)(21,48)(22,41)(23,42)(24,43)(25,51)(26,52)(27,53)(28,54)(29,55)(30,56)(31,49)(32,50)(33,61)(34,62)(35,63)(36,64)(37,57)(38,58)(39,59)(40,60), (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), (1,63,55,46)(2,60,56,43)(3,57,49,48)(4,62,50,45)(5,59,51,42)(6,64,52,47)(7,61,53,44)(8,58,54,41)(9,34,32,18)(10,39,25,23)(11,36,26,20)(12,33,27,17)(13,38,28,22)(14,35,29,19)(15,40,30,24)(16,37,31,21) );
G=PermutationGroup([(1,7,5,3),(2,8,6,4),(9,11,13,15),(10,12,14,16),(17,19,21,23),(18,20,22,24),(25,27,29,31),(26,28,30,32),(33,35,37,39),(34,36,38,40),(41,47,45,43),(42,48,46,44),(49,55,53,51),(50,56,54,52),(57,63,61,59),(58,64,62,60)], [(1,14),(2,15),(3,16),(4,9),(5,10),(6,11),(7,12),(8,13),(17,44),(18,45),(19,46),(20,47),(21,48),(22,41),(23,42),(24,43),(25,51),(26,52),(27,53),(28,54),(29,55),(30,56),(31,49),(32,50),(33,61),(34,62),(35,63),(36,64),(37,57),(38,58),(39,59),(40,60)], [(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)], [(1,63,55,46),(2,60,56,43),(3,57,49,48),(4,62,50,45),(5,59,51,42),(6,64,52,47),(7,61,53,44),(8,58,54,41),(9,34,32,18),(10,39,25,23),(11,36,26,20),(12,33,27,17),(13,38,28,22),(14,35,29,19),(15,40,30,24),(16,37,31,21)])
Matrix representation ►G ⊆ GL5(𝔽17)
| 16 | 0 | 0 | 0 | 0 |
| 0 | 13 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 0 | 13 | 0 |
| 0 | 0 | 0 | 0 | 4 |
| 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 |
| 0 | 13 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 |
| 0 | 0 | 0 | 13 | 0 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 9 | 0 | 5 | 0 |
| 0 | 0 | 9 | 0 | 5 |
| 0 | 5 | 0 | 8 | 0 |
| 0 | 0 | 5 | 0 | 8 |
| 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 |
G:=sub<GL(5,GF(17))| [16,0,0,0,0,0,13,0,0,0,0,0,4,0,0,0,0,0,13,0,0,0,0,0,4],[16,0,0,0,0,0,0,13,0,0,0,4,0,0,0,0,0,0,0,13,0,0,0,4,0],[1,0,0,0,0,0,9,0,5,0,0,0,9,0,5,0,5,0,8,0,0,0,5,0,8],[4,0,0,0,0,0,0,0,16,0,0,0,0,0,16,0,1,0,0,0,0,0,1,0,0] >;
68 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | ··· | 2I | 4A | 4B | 4C | 4D | 4E | ··· | 4Z | 8A | ··· | 8AF |
| order | 1 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 8 | ··· | 8 |
| size | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 2 | ··· | 2 |
68 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 4 |
| type | + | + | + | + | + | + | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | Q8○M4(2) |
| kernel | D4.5C42 | C2×C8⋊C4 | C4×M4(2) | C8○2M4(2) | C4×C4○D4 | C2×C8○D4 | C4×D4 | C4×Q8 | C8○D4 | C2 |
| # reps | 1 | 3 | 3 | 6 | 1 | 2 | 12 | 4 | 32 | 4 |
In GAP, Magma, Sage, TeX
D_4._5C_4^2
% in TeX
G:=Group("D4.5C4^2"); // GroupNames label
G:=SmallGroup(128,1607);
// by ID
G=gap.SmallGroup(128,1607);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,112,925,232,521,172]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^2=d^4=1,c^4=a^2,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=a^2*c>;
// generators/relations