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G = D83Q8order 128 = 27

3rd semidirect product of D8 and Q8 acting via Q8/C4=C2

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

Aliases: D83Q8, Q163Q8, C8.10SD16, C42.151D4, M5(2).6C22, C8.Q83C2, C8.1(C2×Q8), C83Q83C2, C8○D8.5C2, C8.C84C2, (C2×C8).133D4, D82C4.1C2, C8.80(C4○D4), C4.67(C2×SD16), (C2×C8).241C23, C4.Q8.3C22, (C4×C8).165C22, C4○D8.21C22, C4.51(C22⋊Q8), C2.12(D42Q8), C22.27(C8⋊C22), C8.C4.21C22, (C2×C4).283(C2×D4), SmallGroup(128,962)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C8 — D83Q8
C1C2C4C8C2×C8C4○D8C8○D8 — D83Q8
C1C2C4C2×C8 — D83Q8
C1C2C2×C4C4×C8 — D83Q8
C1C2C2C2C2C4C4C2×C8 — D83Q8

Generators and relations for D83Q8
 G = < a,b,c,d | a8=b2=c4=1, d2=c2, bab=a-1, ac=ca, dad-1=a3, cbc-1=a2b, dbd-1=a5b, dcd-1=c-1 >

Subgroups: 148 in 63 conjugacy classes, 30 normal (22 characteristic)
C1, C2, C2 [×2], C4 [×2], C4 [×5], C22, C22, C8 [×4], C8, C2×C4, C2×C4 [×4], D4 [×2], Q8 [×3], C16 [×2], C42, C4⋊C4 [×3], C2×C8 [×2], C2×C8, M4(2) [×2], D8, SD16, Q16, C2×Q8, C4○D4, C4×C8, C4≀C2, C4.Q8 [×2], C4.Q8, C8.C4, M5(2) [×2], C4⋊Q8, C8○D4, C4○D8, D82C4 [×2], C8.C8, C8.Q8 [×2], C8○D8, C83Q8, D83Q8
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, D83Q8

Character table of D83Q8

 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
ρ204-40000-22000-2-22-22-2-2-200000000    complex faithful
ρ214-40000-220002-2-2-2-2-22-200000000    complex faithful
ρ224-400002-20002-2-2-22-2-2-200000000    complex faithful
ρ234-400002-2000-2-22-2-2-22-200000000    complex faithful

Permutation representations of D83Q8
On 16 points - transitive group 16T351
Generators in S16
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)
(1 16)(2 15)(3 14)(4 13)(5 12)(6 11)(7 10)(8 9)
(1 5)(2 6)(3 7)(4 8)(9 15 13 11)(10 16 14 12)
(2 4)(3 7)(6 8)(9 10 13 14)(11 16 15 12)

G:=sub<Sym(16)| (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16), (1,16)(2,15)(3,14)(4,13)(5,12)(6,11)(7,10)(8,9), (1,5)(2,6)(3,7)(4,8)(9,15,13,11)(10,16,14,12), (2,4)(3,7)(6,8)(9,10,13,14)(11,16,15,12)>;

G:=Group( (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16), (1,16)(2,15)(3,14)(4,13)(5,12)(6,11)(7,10)(8,9), (1,5)(2,6)(3,7)(4,8)(9,15,13,11)(10,16,14,12), (2,4)(3,7)(6,8)(9,10,13,14)(11,16,15,12) );

G=PermutationGroup([(1,2,3,4,5,6,7,8),(9,10,11,12,13,14,15,16)], [(1,16),(2,15),(3,14),(4,13),(5,12),(6,11),(7,10),(8,9)], [(1,5),(2,6),(3,7),(4,8),(9,15,13,11),(10,16,14,12)], [(2,4),(3,7),(6,8),(9,10,13,14),(11,16,15,12)])

G:=TransitiveGroup(16,351);

Matrix representation of D83Q8 in GL4(𝔽3) generated by

1002
0210
0220
1001
,
0120
2001
1001
0220
,
0001
0100
0010
2000
,
1001
0100
0020
1002
G:=sub<GL(4,GF(3))| [1,0,0,1,0,2,2,0,0,1,2,0,2,0,0,1],[0,2,1,0,1,0,0,2,2,0,0,2,0,1,1,0],[0,0,0,2,0,1,0,0,0,0,1,0,1,0,0,0],[1,0,0,1,0,1,0,0,0,0,2,0,1,0,0,2] >;

D83Q8 in GAP, Magma, Sage, TeX

D_8\rtimes_3Q_8
% in TeX

G:=Group("D8:3Q8");
// GroupNames label

G:=SmallGroup(128,962);
// by ID

G=gap.SmallGroup(128,962);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,-2,280,141,64,422,2019,248,1684,998,102,4037,1027,124]);
// Polycyclic

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

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Character table of D83Q8 in TeX

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