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G = D8.2Q8order 128 = 27

2nd non-split extension by D8 of Q8 acting via Q8/C4=C2

p-group, metabelian, nilpotent (class 4), monomial

Aliases: D8.2Q8, Q16.2Q8, C8.11SD16, C42.152D4, M5(2).7C22, C8.Q84C2, C8.2(C2×Q8), C8○D8.6C2, C8.C85C2, (C2×C8).134D4, D82C4.2C2, C8.81(C4○D4), C8.5Q812C2, C4.68(C2×SD16), (C4×C8).166C22, C4.Q8.4C22, (C2×C8).242C23, C4○D8.22C22, C4.52(C22⋊Q8), C2.13(D42Q8), C22.28(C8⋊C22), C8.C4.22C22, (C2×C4).284(C2×D4), SmallGroup(128,963)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C8 — D8.2Q8
C1C2C4C8C2×C8C4○D8C8○D8 — D8.2Q8
C1C2C4C2×C8 — D8.2Q8
C1C2C2×C4C4×C8 — D8.2Q8
C1C2C2C2C2C4C4C2×C8 — D8.2Q8

Generators and relations for D8.2Q8
 G = < a,b,c,d | a8=b2=1, c4=a4, d2=a6c2, bab=a-1, ac=ca, dad-1=a3, bc=cb, dbd-1=a5b, dcd-1=a4c3 >

Subgroups: 132 in 60 conjugacy classes, 30 normal (22 characteristic)
C1, C2, C2 [×2], C4 [×2], C4 [×4], C22, C22, C8 [×4], C8, C2×C4, C2×C4 [×4], D4 [×2], Q8, C16 [×2], C42, C4⋊C4 [×4], C2×C8 [×2], C2×C8, M4(2) [×2], D8, SD16, Q16, C4○D4, C4×C8, C4≀C2, C4.Q8 [×2], C2.D8, C8.C4, M5(2) [×2], C42.C2, C8○D4, C4○D8, D82C4 [×2], C8.C8, C8.Q8 [×2], C8○D8, C8.5Q8, D8.2Q8
Quotients: C1, C2 [×7], C22 [×7], D4 [×2], Q8 [×2], C23, SD16 [×2], C2×D4, C2×Q8, C4○D4, C22⋊Q8, C2×SD16, C8⋊C22, D42Q8, D8.2Q8

Character table of D8.2Q8

 class 12A2B2C4A4B4C4D4E4F4G8A8B8C8D8E8F8G8H16A16B16C16D
 size 1128224481616222244888888
ρ111111111111111111111111    trivial
ρ2111111-1-111-111-1-11-1-1-11-1-11    linear of order 2
ρ3111111-1-11-1111-1-11-1-1-1-111-1    linear of order 2
ρ4111111111-1-111111111-1-1-1-1    linear of order 2
ρ5111-111-1-1-1-1111-1-11-1111-1-11    linear of order 2
ρ6111-11111-1-1-1111111-1-11111    linear of order 2
ρ7111-11111-111111111-1-1-1-1-1-1    linear of order 2
ρ8111-111-1-1-11-111-1-11-111-111-1    linear of order 2
ρ9222022-2-2000-2-222-22000000    orthogonal lifted from D4
ρ1022202222000-2-2-2-2-2-2000000    orthogonal lifted from D4
ρ1122-22-2200-200-2-20020000000    symplectic lifted from Q8, Schur index 2
ρ1222-2-2-2200200-2-20020000000    symplectic lifted from Q8, Schur index 2
ρ1322-20-22000002200-202i-2i0000    complex lifted from C4○D4
ρ1422-20-22000002200-20-2i2i0000    complex lifted from C4○D4
ρ1522-202-20000000-2-20200--2--2-2-2    complex lifted from SD16
ρ1622-202-20000000220-200-2--2-2--2    complex lifted from SD16
ρ1722-202-20000000220-200--2-2--2-2    complex lifted from SD16
ρ1822-202-20000000-2-20200-2-2--2--2    complex lifted from SD16
ρ194440-4-400000000000000000    orthogonal lifted from C8⋊C22
ρ204-400002i-2i000-2-22-222-2200000000    complex faithful
ρ214-40000-2i2i000-2-22-2-222200000000    complex faithful
ρ224-400002i-2i0002-2-2-2-222200000000    complex faithful
ρ234-40000-2i2i0002-2-2-222-2200000000    complex faithful

Smallest permutation representation of D8.2Q8
On 32 points
Generators in S32
(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)
(1 16)(2 15)(3 14)(4 13)(5 12)(6 11)(7 10)(8 9)(17 30)(18 29)(19 28)(20 27)(21 26)(22 25)(23 32)(24 31)
(1 17 3 19 5 21 7 23)(2 18 4 20 6 22 8 24)(9 31 15 29 13 27 11 25)(10 32 16 30 14 28 12 26)
(1 23)(2 18)(3 21)(4 24)(5 19)(6 22)(7 17)(8 20)(9 26 13 30)(10 29 14 25)(11 32 15 28)(12 27 16 31)

G:=sub<Sym(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), (1,16)(2,15)(3,14)(4,13)(5,12)(6,11)(7,10)(8,9)(17,30)(18,29)(19,28)(20,27)(21,26)(22,25)(23,32)(24,31), (1,17,3,19,5,21,7,23)(2,18,4,20,6,22,8,24)(9,31,15,29,13,27,11,25)(10,32,16,30,14,28,12,26), (1,23)(2,18)(3,21)(4,24)(5,19)(6,22)(7,17)(8,20)(9,26,13,30)(10,29,14,25)(11,32,15,28)(12,27,16,31)>;

G:=Group( (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), (1,16)(2,15)(3,14)(4,13)(5,12)(6,11)(7,10)(8,9)(17,30)(18,29)(19,28)(20,27)(21,26)(22,25)(23,32)(24,31), (1,17,3,19,5,21,7,23)(2,18,4,20,6,22,8,24)(9,31,15,29,13,27,11,25)(10,32,16,30,14,28,12,26), (1,23)(2,18)(3,21)(4,24)(5,19)(6,22)(7,17)(8,20)(9,26,13,30)(10,29,14,25)(11,32,15,28)(12,27,16,31) );

G=PermutationGroup([(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)], [(1,16),(2,15),(3,14),(4,13),(5,12),(6,11),(7,10),(8,9),(17,30),(18,29),(19,28),(20,27),(21,26),(22,25),(23,32),(24,31)], [(1,17,3,19,5,21,7,23),(2,18,4,20,6,22,8,24),(9,31,15,29,13,27,11,25),(10,32,16,30,14,28,12,26)], [(1,23),(2,18),(3,21),(4,24),(5,19),(6,22),(7,17),(8,20),(9,26,13,30),(10,29,14,25),(11,32,15,28),(12,27,16,31)])

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

51200
5500
001212
00512
,
001212
00512
51200
5500
,
141400
31400
001414
00314
,
141400
14300
00130
0004
G:=sub<GL(4,GF(17))| [5,5,0,0,12,5,0,0,0,0,12,5,0,0,12,12],[0,0,5,5,0,0,12,5,12,5,0,0,12,12,0,0],[14,3,0,0,14,14,0,0,0,0,14,3,0,0,14,14],[14,14,0,0,14,3,0,0,0,0,13,0,0,0,0,4] >;

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

D_8._2Q_8
% in TeX

G:=Group("D8.2Q8");
// GroupNames label

G:=SmallGroup(128,963);
// by ID

G=gap.SmallGroup(128,963);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,-2,280,141,512,422,2019,248,1684,998,102,4037,1027,124]);
// Polycyclic

G:=Group<a,b,c,d|a^8=b^2=1,c^4=a^4,d^2=a^6*c^2,b*a*b=a^-1,a*c=c*a,d*a*d^-1=a^3,b*c=c*b,d*b*d^-1=a^5*b,d*c*d^-1=a^4*c^3>;
// generators/relations

Export

Character table of D8.2Q8 in TeX

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