p-group, metabelian, nilpotent (class 4), monomial
Aliases: D4○Q32, Q8○D16, D4.14D8, Q8.14D8, C16.5C23, C8.18C24, D8.7C23, SD32.C22, D16.4C22, Q32.4C22, Q16.7C23, M4(2).23D4, M5(2).14C22, Q8○D8⋊6C2, D4○C16⋊5C2, C4○D16⋊6C2, C4.51(C2×D8), C8.17(C2×D4), (C2×Q32)⋊13C2, C4○D4.37D4, Q32⋊C2⋊6C2, C22.8(C2×D8), C4.24(C22×D4), C2.33(C22×D8), (C2×C16).35C22, (C2×C8).296C23, C4○D8.11C22, C8○D4.14C22, (C2×Q16).97C22, (C2×C4).186(C2×D4), SmallGroup(128,2149)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Subgroups: 352 in 175 conjugacy classes, 90 normal (11 characteristic)
C1, C2, C2 [×5], C4, C4 [×3], C4 [×6], C22 [×3], C22 [×2], C8, C8 [×3], C2×C4 [×3], C2×C4 [×12], D4 [×3], D4 [×8], Q8, Q8 [×12], C16, C16 [×3], C2×C8 [×3], M4(2) [×3], D8 [×2], SD16 [×6], Q16 [×6], Q16 [×6], C2×Q8 [×8], C4○D4, C4○D4 [×12], C2×C16 [×3], M5(2) [×3], D16, SD32 [×6], Q32 [×9], C8○D4, C2×Q16 [×6], C4○D8 [×6], C8.C22 [×6], 2- (1+4) [×2], D4○C16, C2×Q32 [×3], C4○D16 [×3], Q32⋊C2 [×6], Q8○D8 [×2], D4○Q32
Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], D8 [×4], C2×D4 [×6], C24, C2×D8 [×6], C22×D4, C22×D8, D4○Q32
Generators and relations
G = < a,b,c,d | a4=b2=1, c8=d2=a2, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=a2c7 >
►On 64 points
(1 51 9 59)(2 52 10 60)(3 53 11 61)(4 54 12 62)(5 55 13 63)(6 56 14 64)(7 57 15 49)(8 58 16 50)(17 47 25 39)(18 48 26 40)(19 33 27 41)(20 34 28 42)(21 35 29 43)(22 36 30 44)(23 37 31 45)(24 38 32 46)
(1 25)(2 26)(3 27)(4 28)(5 29)(6 30)(7 31)(8 32)(9 17)(10 18)(11 19)(12 20)(13 21)(14 22)(15 23)(16 24)(33 53)(34 54)(35 55)(36 56)(37 57)(38 58)(39 59)(40 60)(41 61)(42 62)(43 63)(44 64)(45 49)(46 50)(47 51)(48 52)
(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 52 9 60)(2 51 10 59)(3 50 11 58)(4 49 12 57)(5 64 13 56)(6 63 14 55)(7 62 15 54)(8 61 16 53)(17 40 25 48)(18 39 26 47)(19 38 27 46)(20 37 28 45)(21 36 29 44)(22 35 30 43)(23 34 31 42)(24 33 32 41)
G:=sub<Sym(64)| (1,51,9,59)(2,52,10,60)(3,53,11,61)(4,54,12,62)(5,55,13,63)(6,56,14,64)(7,57,15,49)(8,58,16,50)(17,47,25,39)(18,48,26,40)(19,33,27,41)(20,34,28,42)(21,35,29,43)(22,36,30,44)(23,37,31,45)(24,38,32,46), (1,25)(2,26)(3,27)(4,28)(5,29)(6,30)(7,31)(8,32)(9,17)(10,18)(11,19)(12,20)(13,21)(14,22)(15,23)(16,24)(33,53)(34,54)(35,55)(36,56)(37,57)(38,58)(39,59)(40,60)(41,61)(42,62)(43,63)(44,64)(45,49)(46,50)(47,51)(48,52), (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), (1,52,9,60)(2,51,10,59)(3,50,11,58)(4,49,12,57)(5,64,13,56)(6,63,14,55)(7,62,15,54)(8,61,16,53)(17,40,25,48)(18,39,26,47)(19,38,27,46)(20,37,28,45)(21,36,29,44)(22,35,30,43)(23,34,31,42)(24,33,32,41)>;
G:=Group( (1,51,9,59)(2,52,10,60)(3,53,11,61)(4,54,12,62)(5,55,13,63)(6,56,14,64)(7,57,15,49)(8,58,16,50)(17,47,25,39)(18,48,26,40)(19,33,27,41)(20,34,28,42)(21,35,29,43)(22,36,30,44)(23,37,31,45)(24,38,32,46), (1,25)(2,26)(3,27)(4,28)(5,29)(6,30)(7,31)(8,32)(9,17)(10,18)(11,19)(12,20)(13,21)(14,22)(15,23)(16,24)(33,53)(34,54)(35,55)(36,56)(37,57)(38,58)(39,59)(40,60)(41,61)(42,62)(43,63)(44,64)(45,49)(46,50)(47,51)(48,52), (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), (1,52,9,60)(2,51,10,59)(3,50,11,58)(4,49,12,57)(5,64,13,56)(6,63,14,55)(7,62,15,54)(8,61,16,53)(17,40,25,48)(18,39,26,47)(19,38,27,46)(20,37,28,45)(21,36,29,44)(22,35,30,43)(23,34,31,42)(24,33,32,41) );
G=PermutationGroup([(1,51,9,59),(2,52,10,60),(3,53,11,61),(4,54,12,62),(5,55,13,63),(6,56,14,64),(7,57,15,49),(8,58,16,50),(17,47,25,39),(18,48,26,40),(19,33,27,41),(20,34,28,42),(21,35,29,43),(22,36,30,44),(23,37,31,45),(24,38,32,46)], [(1,25),(2,26),(3,27),(4,28),(5,29),(6,30),(7,31),(8,32),(9,17),(10,18),(11,19),(12,20),(13,21),(14,22),(15,23),(16,24),(33,53),(34,54),(35,55),(36,56),(37,57),(38,58),(39,59),(40,60),(41,61),(42,62),(43,63),(44,64),(45,49),(46,50),(47,51),(48,52)], [(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)], [(1,52,9,60),(2,51,10,59),(3,50,11,58),(4,49,12,57),(5,64,13,56),(6,63,14,55),(7,62,15,54),(8,61,16,53),(17,40,25,48),(18,39,26,47),(19,38,27,46),(20,37,28,45),(21,36,29,44),(22,35,30,43),(23,34,31,42),(24,33,32,41)])
Matrix representation ►G ⊆ GL4(𝔽17) generated by
| 7 | 0 | 16 | 2 |
| 0 | 7 | 16 | 1 |
| 1 | 15 | 10 | 0 |
| 1 | 16 | 0 | 10 |
| 1 | 15 | 10 | 0 |
| 1 | 16 | 0 | 10 |
| 7 | 0 | 16 | 2 |
| 0 | 7 | 16 | 1 |
| 15 | 9 | 0 | 0 |
| 4 | 7 | 0 | 0 |
| 0 | 0 | 15 | 9 |
| 0 | 0 | 4 | 7 |
| 15 | 5 | 2 | 5 |
| 9 | 2 | 13 | 15 |
| 2 | 5 | 15 | 5 |
| 13 | 15 | 9 | 2 |
G:=sub<GL(4,GF(17))| [7,0,1,1,0,7,15,16,16,16,10,0,2,1,0,10],[1,1,7,0,15,16,0,7,10,0,16,16,0,10,2,1],[15,4,0,0,9,7,0,0,0,0,15,4,0,0,9,7],[15,9,2,13,5,2,5,15,2,13,15,9,5,15,5,2] >;
32 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 4A | 4B | 4C | 4D | 4E | ··· | 4J | 8A | 8B | 8C | 8D | 8E | 16A | 16B | 16C | 16D | 16E | ··· | 16J |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 8 | 8 | 8 | 8 | 8 | 16 | 16 | 16 | 16 | 16 | ··· | 16 |
| size | 1 | 1 | 2 | 2 | 2 | 8 | 8 | 2 | 2 | 2 | 2 | 8 | ··· | 8 | 2 | 2 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | ··· | 4 |
32 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | - |
| image | C1 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D8 | D8 | D4○Q32 |
| kernel | D4○Q32 | D4○C16 | C2×Q32 | C4○D16 | Q32⋊C2 | Q8○D8 | M4(2) | C4○D4 | D4 | Q8 | C1 |
| # reps | 1 | 1 | 3 | 3 | 6 | 2 | 3 | 1 | 6 | 2 | 4 |
In GAP, Magma, Sage, TeX
D_4\circ Q_{32} % in TeX
G:=Group("D4oQ32"); // GroupNames label
G:=SmallGroup(128,2149);
// by ID
G=gap.SmallGroup(128,2149);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,-2,-2,448,253,456,521,1684,851,242,4037,2028,124]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^2=1,c^8=d^2=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^7>;
// generators/relations