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G = C8.18D4order 64 = 26

5th non-split extension by C8 of D4 acting via D4/C22=C2

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

Aliases: C8.18D4, C221Q16, C23.28D4, C2.D83C2, (C2×Q16)⋊4C2, (C2×C4).56D4, C4.55(C2×D4), C2.6(C2×Q16), Q8⋊C42C2, C4.9(C4○D4), C4⋊C4.7C22, (C22×C8).9C2, C22⋊Q8.3C2, C2.12(C4○D8), (C2×C4).95C23, (C2×C8).77C22, C22.91(C2×D4), C2.19(C4⋊D4), (C2×Q8).12C22, (C22×C4).119C22, SmallGroup(64,148)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C8.18D4
C1C2C22C2×C4C22×C4C22×C8 — C8.18D4
C1C2C2×C4 — C8.18D4
C1C22C22×C4 — C8.18D4
C1C2C2C2×C4 — C8.18D4

Generators and relations for C8.18D4
 G = < a,b,c | a8=b4=1, c2=a4, bab-1=cac-1=a-1, cbc-1=a4b-1 >

Subgroups: 93 in 57 conjugacy classes, 29 normal (17 characteristic)
C1, C2 [×3], C2 [×2], C4 [×2], C4 [×5], C22, C22 [×2], C22 [×2], C8 [×2], C8, C2×C4 [×2], C2×C4 [×6], Q8 [×4], C23, C22⋊C4 [×2], C4⋊C4 [×2], C4⋊C4 [×2], C2×C8 [×2], C2×C8 [×2], Q16 [×2], C22×C4, C2×Q8 [×2], Q8⋊C4 [×2], C2.D8, C22⋊Q8 [×2], C22×C8, C2×Q16, C8.18D4
Quotients: C1, C2 [×7], C22 [×7], D4 [×4], C23, Q16 [×2], C2×D4 [×2], C4○D4, C4⋊D4, C2×Q16, C4○D8, C8.18D4

Character table of C8.18D4

 class 12A2B2C2D2E4A4B4C4D4E4F4G4H8A8B8C8D8E8F8G8H
 size 1111222222888822222222
ρ11111111111111111111111    trivial
ρ21111-1-1-111-11-1-1111-11-11-1-1    linear of order 2
ρ3111111111111-1-1-1-1-1-1-1-1-1-1    linear of order 2
ρ41111-1-1-111-11-11-1-1-11-11-111    linear of order 2
ρ51111-1-1-111-1-11-11-1-11-11-111    linear of order 2
ρ61111111111-1-111-1-1-1-1-1-1-1-1    linear of order 2
ρ71111-1-1-111-1-111-111-11-11-1-1    linear of order 2
ρ81111111111-1-1-1-111111111    linear of order 2
ρ9222222-2-2-2-2000000000000    orthogonal lifted from D4
ρ102-2-220002-200000002020-2-2    orthogonal lifted from D4
ρ112222-2-22-2-22000000000000    orthogonal lifted from D4
ρ122-2-220002-20000000-20-2022    orthogonal lifted from D4
ρ132-22-22-200000000-22-2-222-22    symplectic lifted from Q16, Schur index 2
ρ142-22-2-22000000002-2-222-2-22    symplectic lifted from Q16, Schur index 2
ρ152-22-22-2000000002-222-2-22-2    symplectic lifted from Q16, Schur index 2
ρ162-22-2-2200000000-222-2-222-2    symplectic lifted from Q16, Schur index 2
ρ1722-2-200-2i002i0000--2-22-2-2--2-22    complex lifted from C4○D8
ρ1822-2-2002i00-2i0000--2-2-2-22--22-2    complex lifted from C4○D8
ρ1922-2-2002i00-2i0000-2--22--2-2-2-22    complex lifted from C4○D8
ρ2022-2-200-2i002i0000-2--2-2--22-22-2    complex lifted from C4○D8
ρ212-2-22000-2200000-2i-2i02i02i00    complex lifted from C4○D4
ρ222-2-22000-22000002i2i0-2i0-2i00    complex lifted from C4○D4

Smallest permutation representation of C8.18D4
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 26 23 12)(2 25 24 11)(3 32 17 10)(4 31 18 9)(5 30 19 16)(6 29 20 15)(7 28 21 14)(8 27 22 13)
(1 16 5 12)(2 15 6 11)(3 14 7 10)(4 13 8 9)(17 28 21 32)(18 27 22 31)(19 26 23 30)(20 25 24 29)

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

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

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

C8.18D4 is a maximal subgroup of
C23.32D8  C23.13SD16  D8.9D4  Q16.8D4  D8.10D4  C168D4  C162D4  C23.50D8  C23.51D8  C23.20D8  C24.144D4  M4(2)⋊15D4  C8.D4⋊C2  (C2×C8)⋊14D4  M4(2)⋊17D4  C42.308D4  C42.387C23  C42.389C23  C42.390C23  C42.258D4  C233Q16  C24.123D4  C24.124D4  C4.162+ 1+4  C4.172+ 1+4  C4.182+ 1+4  C42.296D4  C42.298D4  C42.25C23  C42.28C23  C42.29C23  SD166D4  SD168D4  D812D4  D813D4  Q1612D4  D4×Q16  C42.461C23  C42.465C23  C42.467C23  C42.47C23  C42.48C23  C42.49C23  C42.50C23  C42.485C23  C42.491C23  C42.58C23  C42.63C23  A4⋊Q16
 D2p⋊Q16: D45Q16  D46Q16  D61Q16  D62Q16  D63Q16  D10⋊Q16  D102Q16  D103Q16 ...
 C8p.D4: C16.19D4  C16.D4  C24.82D4  C40.82D4  C56.82D4 ...
 (C2×C2p)⋊Q16: C42.367D4  C42.297D4  C3⋊C8.29D4  (C2×C10)⋊Q16  C7⋊C8.29D4 ...
C8.18D4 is a maximal quotient of
D4.1Q16  Q8.1Q16  C23.22D8  C24.135D4  C2.(C88D4)  C85(C4⋊C4)  C24.86D4  C24.88D4  (C2×C4).21Q16
 D2p⋊Q16: C8.28D8  D61Q16  D62Q16  D63Q16  D10⋊Q16  D102Q16  D103Q16  D14⋊Q16 ...
 (C2×C2p)⋊Q16: (C2×C4)⋊6Q16  (C2×C4)⋊3Q16  C3⋊C8.29D4  C24.82D4  (C2×C10)⋊Q16  C40.82D4  C7⋊C8.29D4  C56.82D4 ...
 C2.(D4.pD4): Q81Q16  D4.Q16  Q8.2Q16  C232Q16  (C2×C8).52D4 ...

Matrix representation of C8.18D4 in GL4(𝔽17) generated by

15000
0800
0080
00015
,
0100
1000
0001
00160
,
0100
16000
0001
00160
G:=sub<GL(4,GF(17))| [15,0,0,0,0,8,0,0,0,0,8,0,0,0,0,15],[0,1,0,0,1,0,0,0,0,0,0,16,0,0,1,0],[0,16,0,0,1,0,0,0,0,0,0,16,0,0,1,0] >;

C8.18D4 in GAP, Magma, Sage, TeX

C_8._{18}D_4
% in TeX

G:=Group("C8.18D4");
// GroupNames label

G:=SmallGroup(64,148);
// by ID

G=gap.SmallGroup(64,148);
# by ID

G:=PCGroup([6,-2,2,2,-2,2,-2,192,121,247,362,963,117]);
// Polycyclic

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

Export

Character table of C8.18D4 in TeX

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