p-group, metabelian, nilpotent (class 3), monomial
Aliases: Q8⋊8SD16, C42.503C23, C4.242- (1+4), (Q82)⋊3C2, (C8×Q8)⋊22C2, C4⋊C4.276D4, Q8⋊8(C4○D4), Q8○2(Q8⋊C4), D4⋊2Q8⋊44C2, C4⋊SD16⋊44C2, (C4×SD16)⋊47C2, C2.62(D4○D8), C4.4D8⋊32C2, Q8⋊6D4.8C2, (C2×Q8).268D4, C4.48(C2×SD16), C4⋊C4.430C23, C4⋊C8.350C22, (C4×C8).280C22, (C2×C8).369C23, (C2×C4).554C24, C4⋊Q8.183C22, C2.62(Q8⋊5D4), (C2×D4).267C23, (C4×D4).194C22, C4⋊1D4.96C22, (C2×Q8).402C23, (C4×Q8).308C22, C2.32(C22×SD16), C4.Q8.175C22, C22.814(C22×D4), D4⋊C4.127C22, Q8⋊C4.216C22, (C2×SD16).170C22, C4.255(C2×C4○D4), (C2×C4).1100(C2×D4), SmallGroup(128,2094)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Subgroups: 408 in 196 conjugacy classes, 96 normal (18 characteristic)
C1, C2 [×3], C2 [×3], C4 [×2], C4 [×6], C4 [×9], C22, C22 [×9], C8 [×4], C2×C4, C2×C4 [×6], C2×C4 [×11], D4 [×15], Q8 [×6], Q8 [×4], C23 [×3], C42 [×3], C42 [×3], C22⋊C4 [×3], C4⋊C4, C4⋊C4 [×6], C4⋊C4 [×6], C2×C8, C2×C8 [×3], SD16 [×6], C22×C4 [×3], C2×D4 [×3], C2×D4 [×6], C2×Q8 [×2], C2×Q8 [×3], C4○D4 [×4], C4×C8 [×3], D4⋊C4 [×9], Q8⋊C4, C4⋊C8 [×3], C4.Q8 [×3], C4×D4 [×3], C4×Q8, C4×Q8 [×3], C4×Q8, C4⋊D4 [×3], C4⋊1D4 [×3], C4⋊Q8 [×3], C4⋊Q8 [×3], C2×SD16 [×3], C2×C4○D4, C4×SD16 [×3], C8×Q8, C4⋊SD16 [×3], D4⋊2Q8 [×3], C4.4D8 [×3], Q8⋊6D4, Q82, Q8⋊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), Q8⋊5D4, C22×SD16, D4○D8, Q8⋊8SD16
Generators and relations
G = < a,b,c,d | a4=c8=d2=1, b2=a2, bab-1=a-1, ac=ca, ad=da, cbc-1=dbd=a2b, dcd=c3 >
►On 64 points
(1 51 59 27)(2 52 60 28)(3 53 61 29)(4 54 62 30)(5 55 63 31)(6 56 64 32)(7 49 57 25)(8 50 58 26)(9 20 39 46)(10 21 40 47)(11 22 33 48)(12 23 34 41)(13 24 35 42)(14 17 36 43)(15 18 37 44)(16 19 38 45)
(1 46 59 20)(2 21 60 47)(3 48 61 22)(4 23 62 41)(5 42 63 24)(6 17 64 43)(7 44 57 18)(8 19 58 45)(9 51 39 27)(10 28 40 52)(11 53 33 29)(12 30 34 54)(13 55 35 31)(14 32 36 56)(15 49 37 25)(16 26 38 50)
(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 39)(2 34)(3 37)(4 40)(5 35)(6 38)(7 33)(8 36)(9 59)(10 62)(11 57)(12 60)(13 63)(14 58)(15 61)(16 64)(17 26)(18 29)(19 32)(20 27)(21 30)(22 25)(23 28)(24 31)(41 52)(42 55)(43 50)(44 53)(45 56)(46 51)(47 54)(48 49)
G:=sub<Sym(64)| (1,51,59,27)(2,52,60,28)(3,53,61,29)(4,54,62,30)(5,55,63,31)(6,56,64,32)(7,49,57,25)(8,50,58,26)(9,20,39,46)(10,21,40,47)(11,22,33,48)(12,23,34,41)(13,24,35,42)(14,17,36,43)(15,18,37,44)(16,19,38,45), (1,46,59,20)(2,21,60,47)(3,48,61,22)(4,23,62,41)(5,42,63,24)(6,17,64,43)(7,44,57,18)(8,19,58,45)(9,51,39,27)(10,28,40,52)(11,53,33,29)(12,30,34,54)(13,55,35,31)(14,32,36,56)(15,49,37,25)(16,26,38,50), (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), (1,39)(2,34)(3,37)(4,40)(5,35)(6,38)(7,33)(8,36)(9,59)(10,62)(11,57)(12,60)(13,63)(14,58)(15,61)(16,64)(17,26)(18,29)(19,32)(20,27)(21,30)(22,25)(23,28)(24,31)(41,52)(42,55)(43,50)(44,53)(45,56)(46,51)(47,54)(48,49)>;
G:=Group( (1,51,59,27)(2,52,60,28)(3,53,61,29)(4,54,62,30)(5,55,63,31)(6,56,64,32)(7,49,57,25)(8,50,58,26)(9,20,39,46)(10,21,40,47)(11,22,33,48)(12,23,34,41)(13,24,35,42)(14,17,36,43)(15,18,37,44)(16,19,38,45), (1,46,59,20)(2,21,60,47)(3,48,61,22)(4,23,62,41)(5,42,63,24)(6,17,64,43)(7,44,57,18)(8,19,58,45)(9,51,39,27)(10,28,40,52)(11,53,33,29)(12,30,34,54)(13,55,35,31)(14,32,36,56)(15,49,37,25)(16,26,38,50), (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), (1,39)(2,34)(3,37)(4,40)(5,35)(6,38)(7,33)(8,36)(9,59)(10,62)(11,57)(12,60)(13,63)(14,58)(15,61)(16,64)(17,26)(18,29)(19,32)(20,27)(21,30)(22,25)(23,28)(24,31)(41,52)(42,55)(43,50)(44,53)(45,56)(46,51)(47,54)(48,49) );
G=PermutationGroup([(1,51,59,27),(2,52,60,28),(3,53,61,29),(4,54,62,30),(5,55,63,31),(6,56,64,32),(7,49,57,25),(8,50,58,26),(9,20,39,46),(10,21,40,47),(11,22,33,48),(12,23,34,41),(13,24,35,42),(14,17,36,43),(15,18,37,44),(16,19,38,45)], [(1,46,59,20),(2,21,60,47),(3,48,61,22),(4,23,62,41),(5,42,63,24),(6,17,64,43),(7,44,57,18),(8,19,58,45),(9,51,39,27),(10,28,40,52),(11,53,33,29),(12,30,34,54),(13,55,35,31),(14,32,36,56),(15,49,37,25),(16,26,38,50)], [(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)], [(1,39),(2,34),(3,37),(4,40),(5,35),(6,38),(7,33),(8,36),(9,59),(10,62),(11,57),(12,60),(13,63),(14,58),(15,61),(16,64),(17,26),(18,29),(19,32),(20,27),(21,30),(22,25),(23,28),(24,31),(41,52),(42,55),(43,50),(44,53),(45,56),(46,51),(47,54),(48,49)])
Matrix representation ►G ⊆ GL4(𝔽17) generated by
| 16 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 16 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 13 |
| 12 | 5 | 0 | 0 |
| 12 | 12 | 0 | 0 |
| 0 | 0 | 0 | 4 |
| 0 | 0 | 13 | 0 |
| 0 | 16 | 0 | 0 |
| 16 | 0 | 0 | 0 |
| 0 | 0 | 0 | 13 |
| 0 | 0 | 4 | 0 |
G:=sub<GL(4,GF(17))| [16,0,0,0,0,16,0,0,0,0,0,16,0,0,1,0],[1,0,0,0,0,1,0,0,0,0,4,0,0,0,0,13],[12,12,0,0,5,12,0,0,0,0,0,13,0,0,4,0],[0,16,0,0,16,0,0,0,0,0,0,4,0,0,13,0] >;
35 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 4A | ··· | 4H | 4I | ··· | 4O | 4P | 4Q | 4R | 8A | 8B | 8C | 8D | 8E | ··· | 8J |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 8 | ··· | 8 |
| size | 1 | 1 | 1 | 1 | 8 | 8 | 8 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | 8 | 8 | 2 | 2 | 2 | 2 | 4 | ··· | 4 |
35 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | - | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | SD16 | C4○D4 | 2- (1+4) | D4○D8 |
| kernel | Q8⋊8SD16 | C4×SD16 | C8×Q8 | C4⋊SD16 | D4⋊2Q8 | C4.4D8 | Q8⋊6D4 | Q82 | C4⋊C4 | C2×Q8 | Q8 | Q8 | C4 | C2 |
| # reps | 1 | 3 | 1 | 3 | 3 | 3 | 1 | 1 | 3 | 1 | 8 | 4 | 1 | 2 |
In GAP, Magma, Sage, TeX
Q_8\rtimes_8SD_{16} % in TeX
G:=Group("Q8:8SD16"); // GroupNames label
G:=SmallGroup(128,2094);
// by ID
G=gap.SmallGroup(128,2094);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,120,758,352,346,80,4037,1027,124]);
// Polycyclic
G:=Group<a,b,c,d|a^4=c^8=d^2=1,b^2=a^2,b*a*b^-1=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