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G = Q8⋊3SD16  order 128 = 27

2nd semidirect product of Q8 and SD16 acting via SD16/D4=C2

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

Aliases: Q8⋊3SD16, C42.203C23, D4⋊C8⋊21C2, Q8⋊C8⋊25C2, C4⋊C4.294D4, (C2×D4).26D4, C4.4D8⋊6C2, Q8⋊Q8⋊31C2, C4.82(C4○D8), C4.D8⋊13C2, (C4×C8).24C22, Q8⋊6D4.3C2, (C2×Q8).200D4, C4.32(C2×SD16), C4⋊Q8.23C22, D4.D4⋊31C2, C4⋊C8.166C22, C4.64(C8⋊C22), (C4×D4).33C22, (C4×Q8).33C22, C2.20(D4⋊D4), C4⋊1D4.20C22, C2.15(D4.9D4), C22.169C22≀C2, C2.15(C22⋊SD16), (C2×C4).960(C2×D4), SmallGroup(128,374)

Series: Derived ►Chief ►Lower central ►Upper central ►Jennings

Derived series
C1 — C42 — Q8⋊3SD16
Chief series
C1 — C2 — C22 — C2×C4 — C42 — C4×D4 — Q8⋊6D4 — Q8⋊3SD16
Lower central
C1 — C22 — C42 — Q8⋊3SD16
Upper central
C1 — C22 — C42 — Q8⋊3SD16
Jennings
C1 — C22 — C22 — C42 — Q8⋊3SD16

Generators and relations for Q8⋊3SD16
 G = < a,b,c,d | a4=c8=d2=1, b2=a2, bab-1=cac-1=a-1, ad=da, cbc-1=ab, dbd=a2b, dcd=c3 >

Subgroups: 328 in 125 conjugacy classes, 36 normal (32 characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C2×C4, C2×C4, D4, Q8, Q8, C23, C42, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, SD16, C22×C4, C2×D4, C2×D4, C2×D4, C2×Q8, C2×Q8, C4○D4, C4×C8, D4⋊C4, Q8⋊C4, C4⋊C8, C4.Q8, C4×D4, C4×D4, C4×Q8, C4⋊D4, C4⋊1D4, C4⋊1D4, C4⋊Q8, C2×SD16, C2×C4○D4, D4⋊C8, Q8⋊C8, C4.D8, D4.D4, Q8⋊Q8, C4.4D8, Q8⋊6D4, Q8⋊3SD16
Quotients: C1, C2, C22, D4, C23, SD16, C2×D4, C22≀C2, C2×SD16, C4○D8, C8⋊C22, D4⋊D4, C22⋊SD16, D4.9D4, Q8⋊3SD16

Character table of Q8⋊3SD16

 class 12A2B2C2D2E2F4A4B4C4D4E4F4G4H4I4J4K8A8B8C8D8E8F8G8H
 size 111188822224444481644448888
ρ111111111111111111111111111    trivial
ρ21111-11-11111-111-11-11-1-1-1-11-1-11    linear of order 2
ρ31111-11-11111-111-11-1-11111-111-1    linear of order 2
ρ411111111111111111-1-1-1-1-1-1-1-1-1    linear of order 2
ρ51111-1-1-111111-1-1111-111111-1-11    linear of order 2
ρ611111-111111-1-1-1-11-1-1-1-1-1-11111    linear of order 2
ρ711111-111111-1-1-1-11-111111-1-1-1-1    linear of order 2
ρ81111-1-1-111111-1-11111-1-1-1-1-111-1    linear of order 2
ρ92222-202-2-2-2-2000020000000000    orthogonal lifted from D4
ρ1022220002-2-22-200-2-22000000000    orthogonal lifted from D4
ρ1122220-20-222-20220-20000000000    orthogonal lifted from D4
ρ122222020-222-20-2-20-20000000000    orthogonal lifted from D4
ρ13222220-2-2-2-2-2000020000000000    orthogonal lifted from D4
ρ1422220002-2-222002-2-2000000000    orthogonal lifted from D4
ρ152-2-2200002-2002i-2i0000-√2√2-√2√20-√-2√-20    complex lifted from C4○D8
ρ162-2-2200002-2002i-2i0000√2-√2√2-√20√-2-√-20    complex lifted from C4○D8
ρ1722-2-2000200-2-2002000√-2√-2-√-2-√-2-√-200√-2    complex lifted from SD16
ρ182-2-2200002-200-2i2i0000√2-√2√2-√20-√-2√-20    complex lifted from C4○D8
ρ192-2-2200002-200-2i2i0000-√2√2-√2√20√-2-√-20    complex lifted from C4○D8
ρ2022-2-2000200-2200-2000-√-2-√-2√-2√-2-√-200√-2    complex lifted from SD16
ρ2122-2-2000200-2-2002000-√-2-√-2√-2√-2√-200-√-2    complex lifted from SD16
ρ2222-2-2000200-2200-2000√-2√-2-√-2-√-2√-200-√-2    complex lifted from SD16
ρ234-4-440000-440000000000000000    orthogonal lifted from C8⋊C22
ρ2444-4-4000-4004000000000000000    orthogonal lifted from C8⋊C22
ρ254-44-400000000000000-2i2i2i-2i0000    complex lifted from D4.9D4
ρ264-44-4000000000000002i-2i-2i2i0000    complex lifted from D4.9D4

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

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

Matrix representation of Q8⋊3SD16 ►in GL4(𝔽17) generated by

11500
11600
0010
0001
,
13000
13400
0010
0001
,
10700
5700
00010
001210
,
13800
13400
00115
00016
G:=sub<GL(4,GF(17))| [1,1,0,0,15,16,0,0,0,0,1,0,0,0,0,1],[13,13,0,0,0,4,0,0,0,0,1,0,0,0,0,1],[10,5,0,0,7,7,0,0,0,0,0,12,0,0,10,10],[13,13,0,0,8,4,0,0,0,0,1,0,0,0,15,16] >;
 

Q8⋊3SD16 in GAP, Magma, Sage, TeX 

Q_8\rtimes_3{\rm SD}_{16}
 
% in TeX
 
G:=Group("Q8:3SD16");
 
// GroupNames label
 
G:=SmallGroup(128,374);
 
// by ID
 
G=gap.SmallGroup(128,374);
 
# by ID
 
G:=PCGroup([7,-2,2,2,-2,2,-2,2,448,141,422,520,1123,570,521,136,2804,1411,718,172]);
 
// 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=a^-1,a*d=d*a,c*b*c^-1=a*b,d*b*d=a^2*b,d*c*d=c^3>;
 
// generators/relations
 

Export

Character table of Q8⋊3SD16 in TeX

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