p-group, metabelian, nilpotent (class 3), monomial
Aliases: D8⋊13D4, C42.450C23, C4.1372+ (1+4), C4⋊C4○2D8, (C4×D8)⋊10C2, (C8×D4)⋊12C2, C4⋊2(C4○D8), C2.67(D42), C8.80(C2×D4), D4⋊6D4⋊5C2, C4⋊C4.405D4, D4.29(C2×D4), C4⋊Q16⋊11C2, (C2×D4).229D4, D4.7D4⋊6C2, (C4×C8).80C22, C22⋊C4.94D4, C2.41(Q8○D8), C4.97(C22×D4), D4.D4⋊42C2, C8.18D4⋊21C2, C4⋊C8.341C22, C4⋊C4.222C23, (C2×C8).569C23, (C2×C4).481C24, C23.103(C2×D4), C4⋊Q8.137C22, (C4×D4).151C22, (C2×D4).419C23, (C2×D8).175C22, (C2×Q8).202C23, (C2×Q16).35C22, C2.D8.187C22, C22⋊Q8.65C22, C22⋊C8.198C22, (C22×C8).192C22, C22.741(C22×D4), D4⋊C4.166C22, (C22×C4).1125C23, Q8⋊C4.111C22, (C2×SD16).152C22, C4⋊C4○(C2×D8), (C2×C4○D8)⋊11C2, C2.54(C2×C4○D8), (C2×C4).919(C2×D4), (C2×C4○D4).192C22, SmallGroup(128,2015)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Subgroups: 472 in 242 conjugacy classes, 96 normal (24 characteristic)
C1, C2 [×3], C2 [×6], C4 [×2], C4 [×2], C4 [×9], C22, C22 [×14], C8 [×2], C8 [×3], C2×C4 [×3], C2×C4 [×2], C2×C4 [×22], D4 [×4], D4 [×12], Q8 [×8], C23 [×2], C23 [×2], C42, C22⋊C4 [×2], C22⋊C4 [×6], C4⋊C4, C4⋊C4 [×2], C4⋊C4 [×8], C2×C8 [×2], C2×C8 [×2], C2×C8 [×4], D8 [×4], SD16 [×8], Q16 [×6], C22×C4 [×2], C22×C4 [×6], C2×D4, C2×D4 [×2], C2×D4 [×4], C2×Q8 [×4], C4○D4 [×12], C4×C8, C22⋊C8 [×2], D4⋊C4 [×2], Q8⋊C4 [×4], C4⋊C8, C2.D8, C2×C4⋊C4 [×2], C4×D4, C4×D4 [×2], C4⋊D4 [×2], C22⋊Q8 [×4], C22.D4 [×4], C4⋊Q8 [×2], C22×C8 [×2], C2×D8, C2×SD16 [×4], C2×Q16 [×4], C4○D8 [×8], C2×C4○D4 [×4], C8×D4, C4×D8, D4.7D4 [×4], D4.D4 [×2], C8.18D4 [×2], C4⋊Q16, D4⋊6D4 [×2], C2×C4○D8 [×2], D8⋊13D4
Quotients:
C1, C2 [×15], C22 [×35], D4 [×8], C23 [×15], C2×D4 [×12], C24, C4○D8 [×2], C22×D4 [×2], 2+ (1+4), D42, C2×C4○D8, Q8○D8, D8⋊13D4
Generators and relations
G = < a,b,c,d | a8=b2=c4=d2=1, bab=a-1, ac=ca, ad=da, bc=cb, dbd=a4b, dcd=c-1 >
►On 64 points
(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 26)(2 25)(3 32)(4 31)(5 30)(6 29)(7 28)(8 27)(9 37)(10 36)(11 35)(12 34)(13 33)(14 40)(15 39)(16 38)(17 42)(18 41)(19 48)(20 47)(21 46)(22 45)(23 44)(24 43)(49 57)(50 64)(51 63)(52 62)(53 61)(54 60)(55 59)(56 58)
(1 19 27 41)(2 20 28 42)(3 21 29 43)(4 22 30 44)(5 23 31 45)(6 24 32 46)(7 17 25 47)(8 18 26 48)(9 54 34 57)(10 55 35 58)(11 56 36 59)(12 49 37 60)(13 50 38 61)(14 51 39 62)(15 52 40 63)(16 53 33 64)
(1 54)(2 55)(3 56)(4 49)(5 50)(6 51)(7 52)(8 53)(9 19)(10 20)(11 21)(12 22)(13 23)(14 24)(15 17)(16 18)(25 63)(26 64)(27 57)(28 58)(29 59)(30 60)(31 61)(32 62)(33 48)(34 41)(35 42)(36 43)(37 44)(38 45)(39 46)(40 47)
G:=sub<Sym(64)| (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), (1,26)(2,25)(3,32)(4,31)(5,30)(6,29)(7,28)(8,27)(9,37)(10,36)(11,35)(12,34)(13,33)(14,40)(15,39)(16,38)(17,42)(18,41)(19,48)(20,47)(21,46)(22,45)(23,44)(24,43)(49,57)(50,64)(51,63)(52,62)(53,61)(54,60)(55,59)(56,58), (1,19,27,41)(2,20,28,42)(3,21,29,43)(4,22,30,44)(5,23,31,45)(6,24,32,46)(7,17,25,47)(8,18,26,48)(9,54,34,57)(10,55,35,58)(11,56,36,59)(12,49,37,60)(13,50,38,61)(14,51,39,62)(15,52,40,63)(16,53,33,64), (1,54)(2,55)(3,56)(4,49)(5,50)(6,51)(7,52)(8,53)(9,19)(10,20)(11,21)(12,22)(13,23)(14,24)(15,17)(16,18)(25,63)(26,64)(27,57)(28,58)(29,59)(30,60)(31,61)(32,62)(33,48)(34,41)(35,42)(36,43)(37,44)(38,45)(39,46)(40,47)>;
G:=Group( (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), (1,26)(2,25)(3,32)(4,31)(5,30)(6,29)(7,28)(8,27)(9,37)(10,36)(11,35)(12,34)(13,33)(14,40)(15,39)(16,38)(17,42)(18,41)(19,48)(20,47)(21,46)(22,45)(23,44)(24,43)(49,57)(50,64)(51,63)(52,62)(53,61)(54,60)(55,59)(56,58), (1,19,27,41)(2,20,28,42)(3,21,29,43)(4,22,30,44)(5,23,31,45)(6,24,32,46)(7,17,25,47)(8,18,26,48)(9,54,34,57)(10,55,35,58)(11,56,36,59)(12,49,37,60)(13,50,38,61)(14,51,39,62)(15,52,40,63)(16,53,33,64), (1,54)(2,55)(3,56)(4,49)(5,50)(6,51)(7,52)(8,53)(9,19)(10,20)(11,21)(12,22)(13,23)(14,24)(15,17)(16,18)(25,63)(26,64)(27,57)(28,58)(29,59)(30,60)(31,61)(32,62)(33,48)(34,41)(35,42)(36,43)(37,44)(38,45)(39,46)(40,47) );
G=PermutationGroup([(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)], [(1,26),(2,25),(3,32),(4,31),(5,30),(6,29),(7,28),(8,27),(9,37),(10,36),(11,35),(12,34),(13,33),(14,40),(15,39),(16,38),(17,42),(18,41),(19,48),(20,47),(21,46),(22,45),(23,44),(24,43),(49,57),(50,64),(51,63),(52,62),(53,61),(54,60),(55,59),(56,58)], [(1,19,27,41),(2,20,28,42),(3,21,29,43),(4,22,30,44),(5,23,31,45),(6,24,32,46),(7,17,25,47),(8,18,26,48),(9,54,34,57),(10,55,35,58),(11,56,36,59),(12,49,37,60),(13,50,38,61),(14,51,39,62),(15,52,40,63),(16,53,33,64)], [(1,54),(2,55),(3,56),(4,49),(5,50),(6,51),(7,52),(8,53),(9,19),(10,20),(11,21),(12,22),(13,23),(14,24),(15,17),(16,18),(25,63),(26,64),(27,57),(28,58),(29,59),(30,60),(31,61),(32,62),(33,48),(34,41),(35,42),(36,43),(37,44),(38,45),(39,46),(40,47)])
Matrix representation ►G ⊆ GL4(𝔽17) generated by
| 16 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 0 | 6 |
| 0 | 0 | 14 | 6 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 6 |
| 0 | 0 | 3 | 0 |
| 5 | 3 | 0 | 0 |
| 14 | 12 | 0 | 0 |
| 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 16 |
| 14 | 12 | 0 | 0 |
| 5 | 3 | 0 | 0 |
| 0 | 0 | 13 | 8 |
| 0 | 0 | 13 | 4 |
G:=sub<GL(4,GF(17))| [16,0,0,0,0,16,0,0,0,0,0,14,0,0,6,6],[1,0,0,0,0,1,0,0,0,0,0,3,0,0,6,0],[5,14,0,0,3,12,0,0,0,0,16,0,0,0,0,16],[14,5,0,0,12,3,0,0,0,0,13,13,0,0,8,4] >;
35 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | ··· | 2I | 4A | ··· | 4H | 4I | 4J | ··· | 4O | 8A | 8B | 8C | 8D | 8E | ··· | 8J |
| order | 1 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | 4 | ··· | 4 | 8 | 8 | 8 | 8 | 8 | ··· | 8 |
| size | 1 | 1 | 1 | 1 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | 8 | ··· | 8 | 2 | 2 | 2 | 2 | 4 | ··· | 4 |
35 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D4 | D4 | C4○D8 | 2+ (1+4) | Q8○D8 |
| kernel | D8⋊13D4 | C8×D4 | C4×D8 | D4.7D4 | D4.D4 | C8.18D4 | C4⋊Q16 | D4⋊6D4 | C2×C4○D8 | C22⋊C4 | C4⋊C4 | D8 | C2×D4 | C4 | C4 | C2 |
| # reps | 1 | 1 | 1 | 4 | 2 | 2 | 1 | 2 | 2 | 2 | 1 | 4 | 1 | 8 | 1 | 2 |
In GAP, Magma, Sage, TeX
D_8\rtimes_{13}D_4 % in TeX
G:=Group("D8:13D4"); // GroupNames label
G:=SmallGroup(128,2015);
// by ID
G=gap.SmallGroup(128,2015);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,448,253,758,352,2019,346,2804,1411,375,172]);
// Polycyclic
G:=Group<a,b,c,d|a^8=b^2=c^4=d^2=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d=a^4*b,d*c*d=c^-1>;
// generators/relations