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G = Q88SD16order 128 = 27

2nd semidirect product of Q8 and SD16 acting through Inn(Q8)

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

Aliases: Q88SD16, C42.503C23, C4.242- 1+4, Q823C2, (C8×Q8)⋊22C2, C4⋊C4.276D4, Q88(C4○D4), Q82(Q8⋊C4), D42Q844C2, C4⋊SD1644C2, (C4×SD16)⋊47C2, C2.62(D4○D8), C4.4D832C2, (C2×Q8).268D4, Q86D4.8C2, C4.48(C2×SD16), C4⋊C4.430C23, C4⋊C8.350C22, (C4×C8).280C22, (C2×C8).369C23, (C2×C4).554C24, C4⋊Q8.183C22, C2.62(Q85D4), (C4×D4).194C22, (C2×D4).267C23, C41D4.96C22, (C4×Q8).308C22, (C2×Q8).402C23, 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

C1C2×C4 — Q88SD16
C1C2C4C2×C4C42C4×Q8Q82 — Q88SD16
C1C2C2×C4 — Q88SD16
C1C22C4×Q8 — Q88SD16
C1C2C2C2×C4 — Q88SD16

Generators and relations for Q88SD16
 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 >

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], C41D4 [×3], C4⋊Q8 [×3], C4⋊Q8 [×3], C2×SD16 [×3], C2×C4○D4, C4×SD16 [×3], C8×Q8, C4⋊SD16 [×3], D42Q8 [×3], C4.4D8 [×3], Q86D4, Q82, Q88SD16
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, D4○D8, Q88SD16

Smallest permutation representation of Q88SD16
On 64 points
Generators in S64
(1 51 59 25)(2 52 60 26)(3 53 61 27)(4 54 62 28)(5 55 63 29)(6 56 64 30)(7 49 57 31)(8 50 58 32)(9 22 35 46)(10 23 36 47)(11 24 37 48)(12 17 38 41)(13 18 39 42)(14 19 40 43)(15 20 33 44)(16 21 34 45)
(1 46 59 22)(2 23 60 47)(3 48 61 24)(4 17 62 41)(5 42 63 18)(6 19 64 43)(7 44 57 20)(8 21 58 45)(9 51 35 25)(10 26 36 52)(11 53 37 27)(12 28 38 54)(13 55 39 29)(14 30 40 56)(15 49 33 31)(16 32 34 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 35)(2 38)(3 33)(4 36)(5 39)(6 34)(7 37)(8 40)(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)| (1,51,59,25)(2,52,60,26)(3,53,61,27)(4,54,62,28)(5,55,63,29)(6,56,64,30)(7,49,57,31)(8,50,58,32)(9,22,35,46)(10,23,36,47)(11,24,37,48)(12,17,38,41)(13,18,39,42)(14,19,40,43)(15,20,33,44)(16,21,34,45), (1,46,59,22)(2,23,60,47)(3,48,61,24)(4,17,62,41)(5,42,63,18)(6,19,64,43)(7,44,57,20)(8,21,58,45)(9,51,35,25)(10,26,36,52)(11,53,37,27)(12,28,38,54)(13,55,39,29)(14,30,40,56)(15,49,33,31)(16,32,34,50), (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), (1,35)(2,38)(3,33)(4,36)(5,39)(6,34)(7,37)(8,40)(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( (1,51,59,25)(2,52,60,26)(3,53,61,27)(4,54,62,28)(5,55,63,29)(6,56,64,30)(7,49,57,31)(8,50,58,32)(9,22,35,46)(10,23,36,47)(11,24,37,48)(12,17,38,41)(13,18,39,42)(14,19,40,43)(15,20,33,44)(16,21,34,45), (1,46,59,22)(2,23,60,47)(3,48,61,24)(4,17,62,41)(5,42,63,18)(6,19,64,43)(7,44,57,20)(8,21,58,45)(9,51,35,25)(10,26,36,52)(11,53,37,27)(12,28,38,54)(13,55,39,29)(14,30,40,56)(15,49,33,31)(16,32,34,50), (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), (1,35)(2,38)(3,33)(4,36)(5,39)(6,34)(7,37)(8,40)(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([(1,51,59,25),(2,52,60,26),(3,53,61,27),(4,54,62,28),(5,55,63,29),(6,56,64,30),(7,49,57,31),(8,50,58,32),(9,22,35,46),(10,23,36,47),(11,24,37,48),(12,17,38,41),(13,18,39,42),(14,19,40,43),(15,20,33,44),(16,21,34,45)], [(1,46,59,22),(2,23,60,47),(3,48,61,24),(4,17,62,41),(5,42,63,18),(6,19,64,43),(7,44,57,20),(8,21,58,45),(9,51,35,25),(10,26,36,52),(11,53,37,27),(12,28,38,54),(13,55,39,29),(14,30,40,56),(15,49,33,31),(16,32,34,50)], [(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)], [(1,35),(2,38),(3,33),(4,36),(5,39),(6,34),(7,37),(8,40),(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)])

35 conjugacy classes

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

35 irreducible representations

dim11111111222244
type++++++++++-+
imageC1C2C2C2C2C2C2C2D4D4SD16C4○D42- 1+4D4○D8
kernelQ88SD16C4×SD16C8×Q8C4⋊SD16D42Q8C4.4D8Q86D4Q82C4⋊C4C2×Q8Q8Q8C4C2
# reps13133311318412

Matrix representation of Q88SD16 in GL4(𝔽17) generated by

16000
01600
0001
00160
,
1000
0100
0040
00013
,
12500
121200
0004
00130
,
01600
16000
00013
0040
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] >;

Q88SD16 in GAP, Magma, Sage, TeX

Q_8\rtimes_8{\rm SD}_{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

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