p-group, metabelian, nilpotent (class 3), monomial
Aliases: D4⋊8SD16, C42.463C23, C4.432+ (1+4), (D4×Q8)⋊7C2, (C8×D4)⋊32C2, C8⋊8D4⋊37C2, C4⋊C4.262D4, Q8⋊5(C4○D4), D4○2(Q8⋊C4), Q8⋊Q8⋊41C2, Q8⋊D4⋊34C2, (C4×SD16)⋊39C2, D4⋊6D4.5C2, (C2×D4).352D4, C2.44(Q8○D8), C4.45(C2×SD16), D4.D4⋊43C2, C4⋊C4.400C23, C4⋊C8.343C22, (C2×C8).348C23, (C4×C8).272C22, (C2×C4).490C24, C22⋊C4.102D4, C4.SD16⋊29C2, C23.471(C2×D4), C4⋊Q8.141C22, C22.6(C2×SD16), C4.Q8.99C22, (C4×D4).331C22, (C2×D4).222C23, C4⋊D4.72C22, (C2×Q8).207C23, (C4×Q8).147C22, C2.126(D4⋊5D4), C2.29(C22×SD16), C22⋊Q8.70C22, C23.47D4⋊32C2, C22⋊C8.223C22, (C22×C8).354C22, C22.750(C22×D4), D4⋊C4.150C22, (C22×C4).1134C23, Q8⋊C4.112C22, (C2×SD16).155C22, (C22×Q8).337C22, (C2×D4)○(Q8⋊C4), C4.215(C2×C4○D4), (C2×C4).167(C2×D4), (C2×Q8⋊C4)⋊41C2, (C2×C4⋊C4).660C22, SmallGroup(128,2030)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Subgroups: 408 in 210 conjugacy classes, 96 normal (38 characteristic)
C1, C2 [×3], C2 [×5], C4 [×2], C4 [×2], C4 [×11], C22, C22 [×4], C22 [×7], C8 [×4], C2×C4 [×3], C2×C4 [×2], C2×C4 [×21], D4 [×4], D4 [×6], Q8 [×2], Q8 [×11], C23 [×2], C23, C42, C42, C22⋊C4 [×2], C22⋊C4 [×5], C4⋊C4 [×3], C4⋊C4 [×2], C4⋊C4 [×8], C2×C8 [×2], C2×C8 [×2], C2×C8 [×2], SD16 [×4], C22×C4 [×2], C22×C4 [×5], C2×D4 [×2], C2×D4 [×2], C2×Q8, C2×Q8 [×2], C2×Q8 [×9], C4○D4 [×4], C4×C8, C22⋊C8 [×2], D4⋊C4, Q8⋊C4, Q8⋊C4 [×8], C4⋊C8, C4.Q8, C4.Q8 [×2], C2×C4⋊C4 [×2], C4×D4 [×2], C4×D4, C4×Q8, C4⋊D4 [×2], C22⋊Q8 [×2], C22⋊Q8 [×3], C22.D4 [×2], C4⋊Q8 [×2], C4⋊Q8, C22×C8 [×2], C2×SD16, C2×SD16 [×2], C22×Q8 [×2], C2×C4○D4, C2×Q8⋊C4 [×2], C8×D4, C4×SD16, Q8⋊D4 [×2], D4.D4, C8⋊8D4 [×2], Q8⋊Q8, C23.47D4 [×2], C4.SD16, D4⋊6D4, D4×Q8, D4⋊8SD16
Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], SD16 [×4], C2×D4 [×6], C4○D4 [×2], C24, C2×SD16 [×6], C22×D4, C2×C4○D4, 2+ (1+4), D4⋊5D4, C22×SD16, Q8○D8, D4⋊8SD16
Generators and relations
G = < a,b,c,d | a4=b2=c8=d2=1, bab=a-1, ac=ca, ad=da, cbc-1=dbd=a2b, dcd=c3 >
►On 64 points
(1 22 27 42)(2 23 28 43)(3 24 29 44)(4 17 30 45)(5 18 31 46)(6 19 32 47)(7 20 25 48)(8 21 26 41)(9 62 34 53)(10 63 35 54)(11 64 36 55)(12 57 37 56)(13 58 38 49)(14 59 39 50)(15 60 40 51)(16 61 33 52)
(1 63)(2 55)(3 57)(4 49)(5 59)(6 51)(7 61)(8 53)(9 41)(10 22)(11 43)(12 24)(13 45)(14 18)(15 47)(16 20)(17 38)(19 40)(21 34)(23 36)(25 52)(26 62)(27 54)(28 64)(29 56)(30 58)(31 50)(32 60)(33 48)(35 42)(37 44)(39 46)
(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 5)(2 8)(4 6)(9 36)(10 39)(11 34)(12 37)(13 40)(14 35)(15 38)(16 33)(17 19)(18 22)(21 23)(26 28)(27 31)(30 32)(41 43)(42 46)(45 47)(49 60)(50 63)(51 58)(52 61)(53 64)(54 59)(55 62)(56 57)
G:=sub<Sym(64)| (1,22,27,42)(2,23,28,43)(3,24,29,44)(4,17,30,45)(5,18,31,46)(6,19,32,47)(7,20,25,48)(8,21,26,41)(9,62,34,53)(10,63,35,54)(11,64,36,55)(12,57,37,56)(13,58,38,49)(14,59,39,50)(15,60,40,51)(16,61,33,52), (1,63)(2,55)(3,57)(4,49)(5,59)(6,51)(7,61)(8,53)(9,41)(10,22)(11,43)(12,24)(13,45)(14,18)(15,47)(16,20)(17,38)(19,40)(21,34)(23,36)(25,52)(26,62)(27,54)(28,64)(29,56)(30,58)(31,50)(32,60)(33,48)(35,42)(37,44)(39,46), (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), (1,5)(2,8)(4,6)(9,36)(10,39)(11,34)(12,37)(13,40)(14,35)(15,38)(16,33)(17,19)(18,22)(21,23)(26,28)(27,31)(30,32)(41,43)(42,46)(45,47)(49,60)(50,63)(51,58)(52,61)(53,64)(54,59)(55,62)(56,57)>;
G:=Group( (1,22,27,42)(2,23,28,43)(3,24,29,44)(4,17,30,45)(5,18,31,46)(6,19,32,47)(7,20,25,48)(8,21,26,41)(9,62,34,53)(10,63,35,54)(11,64,36,55)(12,57,37,56)(13,58,38,49)(14,59,39,50)(15,60,40,51)(16,61,33,52), (1,63)(2,55)(3,57)(4,49)(5,59)(6,51)(7,61)(8,53)(9,41)(10,22)(11,43)(12,24)(13,45)(14,18)(15,47)(16,20)(17,38)(19,40)(21,34)(23,36)(25,52)(26,62)(27,54)(28,64)(29,56)(30,58)(31,50)(32,60)(33,48)(35,42)(37,44)(39,46), (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), (1,5)(2,8)(4,6)(9,36)(10,39)(11,34)(12,37)(13,40)(14,35)(15,38)(16,33)(17,19)(18,22)(21,23)(26,28)(27,31)(30,32)(41,43)(42,46)(45,47)(49,60)(50,63)(51,58)(52,61)(53,64)(54,59)(55,62)(56,57) );
G=PermutationGroup([(1,22,27,42),(2,23,28,43),(3,24,29,44),(4,17,30,45),(5,18,31,46),(6,19,32,47),(7,20,25,48),(8,21,26,41),(9,62,34,53),(10,63,35,54),(11,64,36,55),(12,57,37,56),(13,58,38,49),(14,59,39,50),(15,60,40,51),(16,61,33,52)], [(1,63),(2,55),(3,57),(4,49),(5,59),(6,51),(7,61),(8,53),(9,41),(10,22),(11,43),(12,24),(13,45),(14,18),(15,47),(16,20),(17,38),(19,40),(21,34),(23,36),(25,52),(26,62),(27,54),(28,64),(29,56),(30,58),(31,50),(32,60),(33,48),(35,42),(37,44),(39,46)], [(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)], [(1,5),(2,8),(4,6),(9,36),(10,39),(11,34),(12,37),(13,40),(14,35),(15,38),(16,33),(17,19),(18,22),(21,23),(26,28),(27,31),(30,32),(41,43),(42,46),(45,47),(49,60),(50,63),(51,58),(52,61),(53,64),(54,59),(55,62),(56,57)])
Matrix representation ►G ⊆ GL4(𝔽17) generated by
| 13 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 |
| 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 16 |
| 0 | 4 | 0 | 0 |
| 13 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 16 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 7 | 10 |
| 0 | 0 | 12 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 16 | 0 |
| 0 | 0 | 16 | 1 |
G:=sub<GL(4,GF(17))| [13,0,0,0,0,4,0,0,0,0,16,0,0,0,0,16],[0,13,0,0,4,0,0,0,0,0,1,0,0,0,0,1],[16,0,0,0,0,1,0,0,0,0,7,12,0,0,10,0],[1,0,0,0,0,16,0,0,0,0,16,16,0,0,0,1] >;
35 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 4A | 4B | 4C | 4D | 4E | ··· | 4K | 4L | ··· | 4P | 8A | 8B | 8C | 8D | 8E | ··· | 8J |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | ··· | 4 | 8 | 8 | 8 | 8 | 8 | ··· | 8 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 8 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 8 | ··· | 8 | 2 | 2 | 2 | 2 | 4 | ··· | 4 |
35 irreducible representations
| dim | 1 | 1 | 1 | 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 | C2 | C2 | C2 | D4 | D4 | D4 | SD16 | C4○D4 | 2+ (1+4) | Q8○D8 |
| kernel | D4⋊8SD16 | C2×Q8⋊C4 | C8×D4 | C4×SD16 | Q8⋊D4 | D4.D4 | C8⋊8D4 | Q8⋊Q8 | C23.47D4 | C4.SD16 | D4⋊6D4 | D4×Q8 | C22⋊C4 | C4⋊C4 | C2×D4 | D4 | Q8 | C4 | C2 |
| # reps | 1 | 2 | 1 | 1 | 2 | 1 | 2 | 1 | 2 | 1 | 1 | 1 | 2 | 1 | 1 | 8 | 4 | 1 | 2 |
In GAP, Magma, Sage, TeX
D_4\rtimes_8SD_{16} % in TeX
G:=Group("D4:8SD16"); // GroupNames label
G:=SmallGroup(128,2030);
// by ID
G=gap.SmallGroup(128,2030);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,112,253,758,352,346,4037,1027,124]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^2=c^8=d^2=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d=a^2*b,d*c*d=c^3>;
// generators/relations