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G = Q87SD16order 128 = 27

1st semidirect product of Q8 and SD16 acting through Inn(Q8)

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

Aliases: Q87SD16, C42.500C23, C4.212- 1+4, (C8×Q8)⋊21C2, (D4×Q8).8C2, C4⋊C4.273D4, Q83Q82C2, Q83(D4⋊C4), Q8⋊Q845C2, (C2×Q8).267D4, C2.61(Q8○D8), C4.47(C2×SD16), D4.33(C4○D4), C4⋊C8.349C22, C4⋊C4.427C23, (C2×C4).551C24, (C4×C8).278C22, (C2×C8).367C23, C4.SD1631C2, (C4×SD16).12C2, C4⋊Q8.180C22, C2.59(Q85D4), D4.D4.12C2, (C4×D4).191C22, (C2×D4).428C23, (C4×Q8).305C22, (C2×Q8).250C23, C2.31(C22×SD16), C4.Q8.172C22, C22.811(C22×D4), D4⋊C4.217C22, Q8⋊C4.120C22, (C2×SD16).167C22, C4.252(C2×C4○D4), (C2×C4).1099(C2×D4), SmallGroup(128,2091)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — Q87SD16
C1C2C4C2×C4C42C4×D4D4×Q8 — Q87SD16
C1C2C2×C4 — Q87SD16
C1C22C4×Q8 — Q87SD16
C1C2C2C2×C4 — Q87SD16

Generators and relations for Q87SD16
 G = < a,b,c,d | a4=c8=d2=1, b2=a2, bab-1=a-1, ac=ca, ad=da, cbc-1=a2b, bd=db, dcd=c3 >

Subgroups: 328 in 185 conjugacy classes, 96 normal (18 characteristic)
C1, C2 [×3], C2 [×2], C4 [×2], C4 [×6], C4 [×10], C22, C22 [×4], C8 [×4], C2×C4, C2×C4 [×6], C2×C4 [×13], D4 [×2], D4, Q8 [×4], Q8 [×12], C23, C42 [×3], C42 [×3], C22⋊C4 [×3], C4⋊C4, C4⋊C4 [×6], C4⋊C4 [×12], C2×C8, C2×C8 [×3], SD16 [×6], C22×C4 [×3], C2×D4, C2×Q8, C2×Q8 [×3], C2×Q8 [×7], C4×C8 [×3], D4⋊C4, Q8⋊C4 [×9], C4⋊C8 [×3], C4.Q8 [×3], C4×D4 [×3], C4×Q8, C4×Q8 [×3], C4×Q8, C22⋊Q8 [×3], C42.C2 [×3], C4⋊Q8 [×6], C2×SD16 [×3], C22×Q8, C4×SD16 [×3], C8×Q8, D4.D4 [×3], Q8⋊Q8 [×3], C4.SD16 [×3], D4×Q8, Q83Q8, Q87SD16
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, Q85D4, C22×SD16, Q8○D8, Q87SD16

Smallest permutation representation of Q87SD16
On 64 points
Generators in S64
(1 30 51 45)(2 31 52 46)(3 32 53 47)(4 25 54 48)(5 26 55 41)(6 27 56 42)(7 28 49 43)(8 29 50 44)(9 62 37 19)(10 63 38 20)(11 64 39 21)(12 57 40 22)(13 58 33 23)(14 59 34 24)(15 60 35 17)(16 61 36 18)
(1 63 51 20)(2 21 52 64)(3 57 53 22)(4 23 54 58)(5 59 55 24)(6 17 56 60)(7 61 49 18)(8 19 50 62)(9 29 37 44)(10 45 38 30)(11 31 39 46)(12 47 40 32)(13 25 33 48)(14 41 34 26)(15 27 35 42)(16 43 36 28)
(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)
(2 4)(3 7)(6 8)(9 15)(11 13)(12 16)(17 19)(18 22)(21 23)(25 31)(27 29)(28 32)(33 39)(35 37)(36 40)(42 44)(43 47)(46 48)(49 53)(50 56)(52 54)(57 61)(58 64)(60 62)

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

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

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

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++++++++++--
imageC1C2C2C2C2C2C2C2D4D4C4○D4SD162- 1+4Q8○D8
kernelQ87SD16C4×SD16C8×Q8D4.D4Q8⋊Q8C4.SD16D4×Q8Q83Q8C4⋊C4C2×Q8D4Q8C4C2
# reps13133311314812

Matrix representation of Q87SD16 in GL4(𝔽17) generated by

0100
16000
00160
00016
,
101600
16700
00160
00016
,
0400
13000
00010
001210
,
1000
0100
0010
00116
G:=sub<GL(4,GF(17))| [0,16,0,0,1,0,0,0,0,0,16,0,0,0,0,16],[10,16,0,0,16,7,0,0,0,0,16,0,0,0,0,16],[0,13,0,0,4,0,0,0,0,0,0,12,0,0,10,10],[1,0,0,0,0,1,0,0,0,0,1,1,0,0,0,16] >;

Q87SD16 in GAP, Magma, Sage, TeX

Q_8\rtimes_7{\rm SD}_{16}
% in TeX

G:=Group("Q8:7SD16");
// GroupNames label

G:=SmallGroup(128,2091);
// by ID

G=gap.SmallGroup(128,2091);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,448,253,120,758,346,80,4037,1027,124]);
// 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,c*b*c^-1=a^2*b,b*d=d*b,d*c*d=c^3>;
// generators/relations

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