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G = Q8.D8order 128 = 27

1st non-split extension by Q8 of D8 acting via D8/D4=C2

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

Aliases: Q8.6D8, C42.202C23, D4⋊C89C2, Q8⋊C811C2, C4⋊C4.55D4, C4.29(C2×D8), D4⋊Q82C2, C42Q162C2, (C2×D4).51D4, C4.D83C2, C4.4D85C2, C4.59(C4○D8), C4⋊C8.11C22, (C4×C8).23C22, Q86D4.2C2, (C2×Q8).199D4, C4⋊Q8.22C22, C4.63(C8⋊C22), (C4×D4).32C22, (C4×Q8).32C22, C2.15(C22⋊D8), C41D4.19C22, C4.64(C8.C22), C22.168C22≀C2, C2.20(D4.7D4), C2.14(D4.9D4), (C2×C4).959(C2×D4), SmallGroup(128,373)

Series: Derived Chief Lower central Upper central Jennings

C1C42 — Q8.D8
C1C2C22C2×C4C42C4×Q8Q86D4 — Q8.D8
C1C22C42 — Q8.D8
C1C22C42 — Q8.D8
C1C22C22C42 — Q8.D8

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

Subgroups: 328 in 125 conjugacy classes, 36 normal (32 characteristic)
C1, C2 [×3], C2 [×3], C4 [×4], C4 [×6], C22, C22 [×9], C8 [×3], C2×C4 [×3], C2×C4 [×10], D4 [×14], Q8 [×2], Q8 [×3], C23 [×3], C42, C42, C22⋊C4 [×3], C4⋊C4 [×2], C4⋊C4 [×3], C2×C8 [×3], Q16 [×2], C22×C4 [×3], C2×D4, C2×D4 [×2], C2×D4 [×6], C2×Q8, C2×Q8, C4○D4 [×4], C4×C8, D4⋊C4 [×3], Q8⋊C4, C4⋊C8 [×2], C2.D8, C4×D4, C4×D4, C4×Q8, C4⋊D4 [×3], C41D4, C41D4, C4⋊Q8, C2×Q16, C2×C4○D4, D4⋊C8, Q8⋊C8, C4.D8, C42Q16, D4⋊Q8, C4.4D8, Q86D4, Q8.D8
Quotients: C1, C2 [×7], C22 [×7], D4 [×6], C23, D8 [×2], C2×D4 [×3], C22≀C2, C2×D8, C4○D8, C8⋊C22, C8.C22, C22⋊D8, D4.7D4, D4.9D4, Q8.D8

Character table of Q8.D8

 class 12A2B2C2D2E2F4A4B4C4D4E4F4G4H4I4J4K8A8B8C8D8E8F8G8H
 size 111188822224444481644448888
ρ111111111111111111111111111    trivial
ρ211111111111111111-1-1-1-1-1-1-1-1-1    linear of order 2
ρ31111-1-1-1111111-11-111-1-1-1-11-11-1    linear of order 2
ρ41111-1-1-1111111-11-11-11111-11-11    linear of order 2
ρ51111-11111111-1-1-1-1-1-1-1-1-1-11111    linear of order 2
ρ61111-11111111-1-1-1-1-111111-1-1-1-1    linear of order 2
ρ711111-1-111111-11-11-1-111111-11-1    linear of order 2
ρ811111-1-111111-11-11-11-1-1-1-1-11-11    linear of order 2
ρ922222002-22-2-20-20-20000000000    orthogonal lifted from D4
ρ102222000-22-22-22020-2000000000    orthogonal lifted from D4
ρ112222-2002-22-2-202020000000000    orthogonal lifted from D4
ρ1222220-22-2-2-2-2200000000000000    orthogonal lifted from D4
ρ132222000-22-22-2-20-202000000000    orthogonal lifted from D4
ρ14222202-2-2-2-2-2200000000000000    orthogonal lifted from D4
ρ152-22-2000020-20-2020002-22-2020-2    orthogonal lifted from D8
ρ162-22-2000020-20-202000-22-220-202    orthogonal lifted from D8
ρ172-22-2000020-2020-2000-22-22020-2    orthogonal lifted from D8
ρ182-22-2000020-2020-20002-22-20-202    orthogonal lifted from D8
ρ1922-2-200020-20002i0-2i00-2-2--2--220-20    complex lifted from C4○D8
ρ2022-2-200020-2000-2i02i00--2--2-2-220-20    complex lifted from C4○D8
ρ2122-2-200020-2000-2i02i00-2-2--2--2-2020    complex lifted from C4○D8
ρ2222-2-200020-20002i0-2i00--2--2-2-2-2020    complex lifted from C4○D8
ρ234-44-40000-404000000000000000    orthogonal lifted from C8⋊C22
ρ2444-4-4000-4040000000000000000    symplectic lifted from C8.C22, Schur index 2
ρ254-4-44000000000000002i-2i-2i2i0000    complex lifted from D4.9D4
ρ264-4-4400000000000000-2i2i2i-2i0000    complex lifted from D4.9D4

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

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

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

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

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

0100
16000
0010
0001
,
13000
0400
0010
0001
,
12500
121200
0066
00140
,
12500
5500
0066
001411
G:=sub<GL(4,GF(17))| [0,16,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[13,0,0,0,0,4,0,0,0,0,1,0,0,0,0,1],[12,12,0,0,5,12,0,0,0,0,6,14,0,0,6,0],[12,5,0,0,5,5,0,0,0,0,6,14,0,0,6,11] >;

Q8.D8 in GAP, Magma, Sage, TeX

Q_8.D_8
% in TeX

G:=Group("Q8.D8");
// GroupNames label

G:=SmallGroup(128,373);
// by ID

G=gap.SmallGroup(128,373);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,2,448,141,422,352,1123,570,521,136,2804,1411,718,172]);
// 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,c*b*c^-1=d*b*d^-1=a^-1*b,d*c*d^-1=a^2*c^-1>;
// generators/relations

Export

Character table of Q8.D8 in TeX

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