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G = Q8.2D8  order 128 = 27

2nd non-split extension by Q8 of D8 acting via D8/C8=C2

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

Aliases: Q8.2D8, C42.233C23, (C8×Q8)⋊6C2, C8⋊1C8⋊9C2, C4.36(C2×D8), C4⋊C4.208D4, C4⋊2Q16⋊6C2, (C2×C8).316D4, C4.89(C4○D8), C4⋊C8.26C22, (C4×C8).58C22, C4⋊SD16.8C2, (C2×Q8).154D4, C4.D8.3C2, C4.4D8.5C2, C4⋊Q8.56C22, C4.10D8⋊25C2, C2.11(C8⋊7D4), C4⋊1D4.31C22, C4.93(C8.C22), (C4×Q8).269C22, C2.13(D4.3D4), C2.10(Q8.D4), C22.194(C4⋊D4), (C2×C4).18(C4○D4), (C2×C4).1268(C2×D4), SmallGroup(128,414)

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

Derived series
C1 — C42 — Q8.2D8
Chief series
C1 — C2 — C22 — C2×C4 — C42 — C4×Q8 — C8×Q8 — Q8.2D8
Lower central
C1 — C22 — C42 — Q8.2D8
Upper central
C1 — C22 — C42 — Q8.2D8
Jennings
C1 — C22 — C22 — C42 — Q8.2D8

Generators and relations for Q8.2D8
 G = < a,b,c,d | a4=c8=1, b2=d2=a2, bab-1=dad-1=a-1, ac=ca, bc=cb, dbd-1=a-1b, dcd-1=a2c-1 >

Subgroups: 208 in 83 conjugacy classes, 34 normal (32 characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C2×C4, C2×C4, D4, Q8, Q8, C23, C42, C42, C4⋊C4, C4⋊C4, C2×C8, C2×C8, SD16, Q16, C2×D4, C2×Q8, C2×Q8, C4×C8, C4×C8, D4⋊C4, Q8⋊C4, C4⋊C8, C4⋊C8, C4×Q8, C4⋊1D4, C4⋊Q8, C2×SD16, C2×Q16, C4.D8, C4.10D8, C8⋊1C8, C8×Q8, C4⋊SD16, C4⋊2Q16, C4.4D8, Q8.2D8
Quotients: C1, C2, C22, D4, C23, D8, C2×D4, C4○D4, C4⋊D4, C2×D8, C4○D8, C8.C22, Q8.D4, C8⋊7D4, D4.3D4, Q8.2D8

Character table of Q8.2D8

 class 12A2B2C2D4A4B4C4D4E4F4G4H4I4J8A8B8C8D8E8F8G8H8I8J8K8L8M8N
 size 1111162222444441622224444448888
ρ111111111111111111111111111111    trivial
ρ21111-1111111111-11111111111-1-1-1-1    linear of order 2
ρ31111-11111-1-11-1-1-11111-1-1-111-11111    linear of order 2
ρ4111111111-1-11-1-111111-1-1-111-1-1-1-1-1    linear of order 2
ρ51111-11111-1-11-1-11-1-1-1-1111-1-11-11-11    linear of order 2
ρ6111111111-1-11-1-1-1-1-1-1-1111-1-111-11-1    linear of order 2
ρ711111111111111-1-1-1-1-1-1-1-1-1-1-1-11-11    linear of order 2
ρ81111-11111111111-1-1-1-1-1-1-1-1-1-11-11-1    linear of order 2
ρ922220-22-2200-2000-2-2-2-20002200000    orthogonal lifted from D4
ρ1022220-22-2200-20002222000-2-200000    orthogonal lifted from D4
ρ11222202-22-2-22-2-22000000000000000    orthogonal lifted from D4
ρ12222202-22-22-2-22-2000000000000000    orthogonal lifted from D4
ρ1322-2-2020-200-20020√2-√2√2-√2√2-√2-√2-√2√2√20000    orthogonal lifted from D8
ρ1422-2-2020-200200-20-√2√2-√2√2√2-√2-√2√2-√2√20000    orthogonal lifted from D8
ρ1522-2-2020-200-20020-√2√2-√2√2-√2√2√2√2-√2-√20000    orthogonal lifted from D8
ρ1622-2-2020-200200-20√2-√2√2-√2-√2√2√2-√2√2-√20000    orthogonal lifted from D8
ρ1722220-2-2-2-20020000000-2i-2i2i002i0000    complex lifted from C4○D4
ρ182-22-20020-2000000-2i2i2i-2i000000-√2-√-2√2√-2    complex lifted from C4○D8
ρ1922-2-20-2020-2i002i00√2-√2√2-√2√-2-√-2√-2√2-√2-√-20000    complex lifted from C4○D8
ρ2022-2-20-20202i00-2i00-√2√2-√2√2√-2-√-2√-2-√2√2-√-20000    complex lifted from C4○D8
ρ212-22-20020-2000000-2i2i2i-2i000000√2√-2-√2-√-2    complex lifted from C4○D8
ρ2222220-2-2-2-200200000002i2i-2i00-2i0000    complex lifted from C4○D4
ρ232-22-20020-20000002i-2i-2i2i000000√2-√-2-√2√-2    complex lifted from C4○D8
ρ2422-2-20-20202i00-2i00√2-√2√2-√2-√-2√-2-√-2√2-√2√-20000    complex lifted from C4○D8
ρ252-22-20020-20000002i-2i-2i2i000000-√2√-2√2-√-2    complex lifted from C4○D8
ρ2622-2-20-2020-2i002i00-√2√2-√2√2-√-2√-2-√-2-√2√2√-20000    complex lifted from C4○D8
ρ274-44-400-40400000000000000000000    symplectic lifted from C8.C22, Schur index 2
ρ284-4-44000000000002√-22√-2-2√-2-2√-20000000000    complex lifted from D4.3D4
ρ294-4-4400000000000-2√-2-2√-22√-22√-20000000000    complex lifted from D4.3D4

Smallest permutation representation of Q8.2D8
►On 64 points
Generators in S64
(1 61 20 13)(2 62 21 14)(3 63 22 15)(4 64 23 16)(5 57 24 9)(6 58 17 10)(7 59 18 11)(8 60 19 12)(25 41 54 36)(26 42 55 37)(27 43 56 38)(28 44 49 39)(29 45 50 40)(30 46 51 33)(31 47 52 34)(32 48 53 35)

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

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

Matrix representation of Q8.2D8 ►in GL4(𝔽17) generated by

16000
01600
00115
00116
,
01300
4000
00710
001210
,
14300
141400
00130
00013
,
14300
3300
00130
00134
G:=sub<GL(4,GF(17))| [16,0,0,0,0,16,0,0,0,0,1,1,0,0,15,16],[0,4,0,0,13,0,0,0,0,0,7,12,0,0,10,10],[14,14,0,0,3,14,0,0,0,0,13,0,0,0,0,13],[14,3,0,0,3,3,0,0,0,0,13,13,0,0,0,4] >;
 

Q8.2D8 in GAP, Magma, Sage, TeX 

Q_8._2D_8
 
% in TeX
 
G:=Group("Q8.2D8");
 
// GroupNames label
 
G:=SmallGroup(128,414);
 
// by ID
 
G=gap.SmallGroup(128,414);
 
# by ID
 
G:=PCGroup([7,-2,2,2,-2,2,-2,2,141,736,422,352,1123,136,2804,718,172]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^4=c^8=1,b^2=d^2=a^2,b*a*b^-1=d*a*d^-1=a^-1,a*c=c*a,b*c=c*b,d*b*d^-1=a^-1*b,d*c*d^-1=a^2*c^-1>;
 
// generators/relations
 

Export

Character table of Q8.2D8 in TeX

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