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G = Q8×SD16order 128 = 27

Direct product of Q8 and SD16

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

Aliases: Q8×SD16, C42.520C23, C4.942- 1+4, Q826C2, C89(C2×Q8), Q84(C2×Q8), (C8×Q8)⋊23C2, (D4×Q8).9C2, D4.8(C2×Q8), C2.37(D4×Q8), C4⋊C4.281D4, C83Q827C2, Q8⋊Q847C2, (C2×Q8).271D4, C4.49(C2×SD16), C4.37(C22×Q8), C4⋊C4.268C23, C4⋊C8.351C22, (C2×C8).373C23, (C2×C4).571C24, (C4×C8).282C22, D42Q8.13C2, (C4×SD16).14C2, C4⋊Q8.200C22, (C2×D4).434C23, (C4×D4).209C22, (C2×Q8).407C23, (C4×Q8).201C22, C2.33(C22×SD16), C4.Q8.114C22, C2.106(D4○SD16), C22.831(C22×D4), D4⋊C4.189C22, Q8⋊C4.185C22, (C2×SD16).179C22, (C2×C4).1103(C2×D4), SmallGroup(128,2111)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — Q8×SD16
C1C2C4C2×C4C42C4×D4D4×Q8 — Q8×SD16
C1C2C2×C4 — Q8×SD16
C1C22C4×Q8 — Q8×SD16
C1C2C2C2×C4 — Q8×SD16

Generators and relations for Q8×SD16
 G = < a,b,c,d | a4=c8=d2=1, b2=a2, bab-1=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd=c3 >

Subgroups: 336 in 185 conjugacy classes, 104 normal (18 characteristic)
C1, C2 [×3], C2 [×2], C4 [×2], C4 [×6], C4 [×11], C22, C22 [×4], C8 [×2], C8 [×3], C2×C4, C2×C4 [×6], C2×C4 [×13], D4 [×2], D4, Q8 [×6], Q8 [×10], C23, C42 [×3], C42 [×3], C22⋊C4 [×3], C4⋊C4 [×9], C4⋊C4 [×9], C2×C8, C2×C8 [×3], SD16 [×4], C22×C4 [×3], C2×D4, C2×Q8 [×2], C2×Q8 [×10], C4×C8 [×3], D4⋊C4 [×3], Q8⋊C4 [×3], C4⋊C8 [×3], C4.Q8 [×9], C4×D4 [×3], C4×Q8, C4×Q8 [×3], C4×Q8, C22⋊Q8 [×3], C4⋊Q8 [×6], C4⋊Q8 [×3], C2×SD16, C22×Q8, C4×SD16 [×3], C8×Q8, Q8⋊Q8 [×3], D42Q8 [×3], C83Q8 [×3], D4×Q8, Q82, Q8×SD16
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], Q8 [×4], C23 [×15], SD16 [×4], C2×D4 [×6], C2×Q8 [×6], C24, C2×SD16 [×6], C22×D4, C22×Q8, 2- 1+4, D4×Q8, C22×SD16, D4○SD16, Q8×SD16

Smallest permutation representation of Q8×SD16
On 64 points
Generators in S64
(1 30 22 55)(2 31 23 56)(3 32 24 49)(4 25 17 50)(5 26 18 51)(6 27 19 52)(7 28 20 53)(8 29 21 54)(9 38 63 46)(10 39 64 47)(11 40 57 48)(12 33 58 41)(13 34 59 42)(14 35 60 43)(15 36 61 44)(16 37 62 45)
(1 34 22 42)(2 35 23 43)(3 36 24 44)(4 37 17 45)(5 38 18 46)(6 39 19 47)(7 40 20 48)(8 33 21 41)(9 51 63 26)(10 52 64 27)(11 53 57 28)(12 54 58 29)(13 55 59 30)(14 56 60 31)(15 49 61 32)(16 50 62 25)
(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 18)(2 21)(3 24)(4 19)(5 22)(6 17)(7 20)(8 23)(9 59)(10 62)(11 57)(12 60)(13 63)(14 58)(15 61)(16 64)(25 52)(26 55)(27 50)(28 53)(29 56)(30 51)(31 54)(32 49)(33 43)(34 46)(35 41)(36 44)(37 47)(38 42)(39 45)(40 48)

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

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

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

35 conjugacy classes

class 1 2A2B2C2D2E4A···4H4I···4M4N···4S8A8B8C8D8E···8J
order1222224···44···44···488888···8
size1111442···24···48···822224···4

35 irreducible representations

dim11111111222244
type+++++++++-+-
imageC1C2C2C2C2C2C2C2D4Q8D4SD162- 1+4D4○SD16
kernelQ8×SD16C4×SD16C8×Q8Q8⋊Q8D42Q8C83Q8D4×Q8Q82C4⋊C4SD16C2×Q8Q8C4C2
# reps13133311341812

Matrix representation of Q8×SD16 in GL4(𝔽17) generated by

1000
0100
0042
00013
,
1000
0100
00315
00514
,
51200
5500
0010
0001
,
16000
0100
00160
00016
G:=sub<GL(4,GF(17))| [1,0,0,0,0,1,0,0,0,0,4,0,0,0,2,13],[1,0,0,0,0,1,0,0,0,0,3,5,0,0,15,14],[5,5,0,0,12,5,0,0,0,0,1,0,0,0,0,1],[16,0,0,0,0,1,0,0,0,0,16,0,0,0,0,16] >;

Q8×SD16 in GAP, Magma, Sage, TeX

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

G:=Group("Q8xSD16");
// GroupNames label

G:=SmallGroup(128,2111);
// by ID

G=gap.SmallGroup(128,2111);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,560,253,120,758,346,80,4037,1027,124]);
// 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,b*c=c*b,b*d=d*b,d*c*d=c^3>;
// generators/relations

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