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G = Q89SD16order 128 = 27

3rd semidirect product of Q8 and SD16 acting through Inn(Q8)

p-group, metabelian, nilpotent (class 3), monomial

Aliases: Q89SD16, C42.526C23, C4.1432+ 1+4, (C8×Q8)⋊24C2, C818(C4○D4), C85D428C2, C4⋊C4.284D4, Q82(C4.Q8), Q83Q812C2, C4⋊SD1645C2, (C4×SD16)⋊49C2, (C2×Q8).274D4, C4.50(C2×SD16), D4.D446C2, C4⋊C4.443C23, C4⋊C8.352C22, (C2×C4).584C24, (C2×C8).378C23, (C4×C8).283C22, Q86D4.10C2, C4⋊Q8.212C22, C2.38(Q86D4), (C2×D4).279C23, (C4×D4).218C22, (C4×Q8).315C22, (C2×Q8).263C23, C2.34(C22×SD16), C4.Q8.186C22, C2.111(D4○SD16), C41D4.105C22, C22.844(C22×D4), D4⋊C4.191C22, Q8⋊C4.189C22, (C2×SD16).102C22, C4.162(C2×C4○D4), (C2×C4).1106(C2×D4), SmallGroup(128,2124)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — Q89SD16
C1C2C4C2×C4C42C4×D4Q86D4 — Q89SD16
C1C2C2×C4 — Q89SD16
C1C22C4×Q8 — Q89SD16
C1C2C2C2×C4 — Q89SD16

Generators and relations for Q89SD16
 G = < a,b,c,d | a4=c8=d2=1, b2=a2, bab-1=cac-1=dad=a-1, cbc-1=a2b, bd=db, dcd=c3 >

Subgroups: 416 in 202 conjugacy classes, 96 normal (18 characteristic)
C1, C2 [×3], C2 [×3], C4 [×2], C4 [×6], C4 [×8], C22, C22 [×9], C8 [×2], C8 [×3], C2×C4, C2×C4 [×6], C2×C4 [×11], D4 [×15], Q8 [×4], Q8 [×6], C23 [×3], C42 [×3], C42 [×3], C22⋊C4 [×3], C4⋊C4 [×2], C4⋊C4 [×3], C4⋊C4 [×9], C2×C8, C2×C8 [×3], SD16 [×12], C22×C4 [×3], C2×D4 [×3], C2×D4 [×6], C2×Q8, C2×Q8 [×3], C4○D4 [×4], C4×C8 [×3], D4⋊C4 [×3], Q8⋊C4 [×3], C4⋊C8 [×3], C4.Q8, C4×D4 [×3], C4×Q8, C4×Q8 [×3], C4×Q8, C4⋊D4 [×3], C42.C2 [×3], C41D4 [×3], C4⋊Q8 [×3], C2×SD16 [×9], C2×C4○D4, C4×SD16 [×3], C8×Q8, C4⋊SD16 [×3], D4.D4 [×3], C85D4 [×3], Q86D4, Q83Q8, Q89SD16
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, Q86D4, C22×SD16, D4○SD16, Q89SD16

Smallest permutation representation of Q89SD16
On 64 points
Generators in S64
(1 30 59 51)(2 52 60 31)(3 32 61 53)(4 54 62 25)(5 26 63 55)(6 56 64 27)(7 28 57 49)(8 50 58 29)(9 20 40 45)(10 46 33 21)(11 22 34 47)(12 48 35 23)(13 24 36 41)(14 42 37 17)(15 18 38 43)(16 44 39 19)
(1 34 59 11)(2 12 60 35)(3 36 61 13)(4 14 62 37)(5 38 63 15)(6 16 64 39)(7 40 57 9)(8 10 58 33)(17 25 42 54)(18 55 43 26)(19 27 44 56)(20 49 45 28)(21 29 46 50)(22 51 47 30)(23 31 48 52)(24 53 41 32)
(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 43)(2 46)(3 41)(4 44)(5 47)(6 42)(7 45)(8 48)(9 49)(10 52)(11 55)(12 50)(13 53)(14 56)(15 51)(16 54)(17 64)(18 59)(19 62)(20 57)(21 60)(22 63)(23 58)(24 61)(25 39)(26 34)(27 37)(28 40)(29 35)(30 38)(31 33)(32 36)

G:=sub<Sym(64)| (1,30,59,51)(2,52,60,31)(3,32,61,53)(4,54,62,25)(5,26,63,55)(6,56,64,27)(7,28,57,49)(8,50,58,29)(9,20,40,45)(10,46,33,21)(11,22,34,47)(12,48,35,23)(13,24,36,41)(14,42,37,17)(15,18,38,43)(16,44,39,19), (1,34,59,11)(2,12,60,35)(3,36,61,13)(4,14,62,37)(5,38,63,15)(6,16,64,39)(7,40,57,9)(8,10,58,33)(17,25,42,54)(18,55,43,26)(19,27,44,56)(20,49,45,28)(21,29,46,50)(22,51,47,30)(23,31,48,52)(24,53,41,32), (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), (1,43)(2,46)(3,41)(4,44)(5,47)(6,42)(7,45)(8,48)(9,49)(10,52)(11,55)(12,50)(13,53)(14,56)(15,51)(16,54)(17,64)(18,59)(19,62)(20,57)(21,60)(22,63)(23,58)(24,61)(25,39)(26,34)(27,37)(28,40)(29,35)(30,38)(31,33)(32,36)>;

G:=Group( (1,30,59,51)(2,52,60,31)(3,32,61,53)(4,54,62,25)(5,26,63,55)(6,56,64,27)(7,28,57,49)(8,50,58,29)(9,20,40,45)(10,46,33,21)(11,22,34,47)(12,48,35,23)(13,24,36,41)(14,42,37,17)(15,18,38,43)(16,44,39,19), (1,34,59,11)(2,12,60,35)(3,36,61,13)(4,14,62,37)(5,38,63,15)(6,16,64,39)(7,40,57,9)(8,10,58,33)(17,25,42,54)(18,55,43,26)(19,27,44,56)(20,49,45,28)(21,29,46,50)(22,51,47,30)(23,31,48,52)(24,53,41,32), (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), (1,43)(2,46)(3,41)(4,44)(5,47)(6,42)(7,45)(8,48)(9,49)(10,52)(11,55)(12,50)(13,53)(14,56)(15,51)(16,54)(17,64)(18,59)(19,62)(20,57)(21,60)(22,63)(23,58)(24,61)(25,39)(26,34)(27,37)(28,40)(29,35)(30,38)(31,33)(32,36) );

G=PermutationGroup([(1,30,59,51),(2,52,60,31),(3,32,61,53),(4,54,62,25),(5,26,63,55),(6,56,64,27),(7,28,57,49),(8,50,58,29),(9,20,40,45),(10,46,33,21),(11,22,34,47),(12,48,35,23),(13,24,36,41),(14,42,37,17),(15,18,38,43),(16,44,39,19)], [(1,34,59,11),(2,12,60,35),(3,36,61,13),(4,14,62,37),(5,38,63,15),(6,16,64,39),(7,40,57,9),(8,10,58,33),(17,25,42,54),(18,55,43,26),(19,27,44,56),(20,49,45,28),(21,29,46,50),(22,51,47,30),(23,31,48,52),(24,53,41,32)], [(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)], [(1,43),(2,46),(3,41),(4,44),(5,47),(6,42),(7,45),(8,48),(9,49),(10,52),(11,55),(12,50),(13,53),(14,56),(15,51),(16,54),(17,64),(18,59),(19,62),(20,57),(21,60),(22,63),(23,58),(24,61),(25,39),(26,34),(27,37),(28,40),(29,35),(30,38),(31,33),(32,36)])

35 conjugacy classes

class 1 2A2B2C2D2E2F4A···4H4I···4O4P4Q4R8A8B8C8D8E···8J
order12222224···44···444488888···8
size11118882···24···488822224···4

35 irreducible representations

dim11111111222244
type+++++++++++
imageC1C2C2C2C2C2C2C2D4D4C4○D4SD162+ 1+4D4○SD16
kernelQ89SD16C4×SD16C8×Q8C4⋊SD16D4.D4C85D4Q86D4Q83Q8C4⋊C4C2×Q8C8Q8C4C2
# reps13133311314812

Matrix representation of Q89SD16 in GL4(𝔽17) generated by

4000
01300
00160
00016
,
0100
16000
0010
0001
,
0400
4000
00010
001210
,
01300
4000
0010
00116
G:=sub<GL(4,GF(17))| [4,0,0,0,0,13,0,0,0,0,16,0,0,0,0,16],[0,16,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[0,4,0,0,4,0,0,0,0,0,0,12,0,0,10,10],[0,4,0,0,13,0,0,0,0,0,1,1,0,0,0,16] >;

Q89SD16 in GAP, Magma, Sage, TeX

Q_8\rtimes_9{\rm SD}_{16}
% in TeX

G:=Group("Q8:9SD16");
// GroupNames label

G:=SmallGroup(128,2124);
// by ID

G=gap.SmallGroup(128,2124);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,448,253,120,758,100,346,80,4037,1027,124]);
// Polycyclic

G:=Group<a,b,c,d|a^4=c^8=d^2=1,b^2=a^2,b*a*b^-1=c*a*c^-1=d*a*d=a^-1,c*b*c^-1=a^2*b,b*d=d*b,d*c*d=c^3>;
// generators/relations

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