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G = C89D4order 64 = 26

3rd semidirect product of C8 and D4 acting via D4/C22=C2

p-group, metabelian, nilpotent (class 2), monomial

Aliases: C89D4, C221M4(2), C42.8C22, C4⋊C815C2, C4⋊C4.7C4, C8⋊C49C2, (C4×D4).2C2, (C2×D4).8C4, C2.10(C4×D4), C4.77(C2×D4), C22⋊C813C2, (C22×C8)⋊11C2, C2.7(C8○D4), C22⋊C4.4C4, C4.52(C4○D4), C23.11(C2×C4), C2.9(C2×M4(2)), (C2×M4(2))⋊14C2, (C2×C8).100C22, (C2×C4).154C23, C22.47(C22×C4), (C22×C4).96C22, (C2×C4).36(C2×C4), SmallGroup(64,116)

Series: Derived Chief Lower central Upper central Jennings

C1C22 — C89D4
C1C2C4C2×C4C2×C8C22×C8 — C89D4
C1C22 — C89D4
C1C2×C4 — C89D4
C1C2C2C2×C4 — C89D4

Generators and relations for C89D4
 G = < a,b,c | a8=b4=c2=1, bab-1=cac=a5, cbc=b-1 >

Subgroups: 89 in 62 conjugacy classes, 37 normal (33 characteristic)
C1, C2 [×3], C2 [×3], C4 [×2], C4 [×4], C22, C22 [×2], C22 [×5], C8 [×2], C8 [×3], C2×C4 [×5], C2×C4 [×4], D4 [×2], C23 [×2], C42, C22⋊C4 [×2], C4⋊C4, C2×C8 [×4], C2×C8 [×2], M4(2) [×2], C22×C4 [×2], C2×D4, C8⋊C4, C22⋊C8 [×2], C4⋊C8, C4×D4, C22×C8, C2×M4(2), C89D4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×2], C23, M4(2) [×2], C22×C4, C2×D4, C4○D4, C4×D4, C2×M4(2), C8○D4, C89D4

Character table of C89D4

 class 12A2B2C2D2E2F4A4B4C4D4E4F4G4H4I8A8B8C8D8E8F8G8H8I8J8K8L
 size 1111224111122444222222224444
ρ11111111111111111111111111111    trivial
ρ21111-1-111111-1-1-11-1-1111-1-1-1111-1-1    linear of order 2
ρ31111-1-1-11111-1-11-111-1-1-1111-111-1-1    linear of order 2
ρ4111111-1111111-1-1-1-1-1-1-1-1-1-1-11111    linear of order 2
ρ51111-1-111111-1-1-11-11-1-1-1111-1-1-111    linear of order 2
ρ61111111111111111-1-1-1-1-1-1-1-1-1-1-1-1    linear of order 2
ρ7111111-1111111-1-1-111111111-1-1-1-1    linear of order 2
ρ81111-1-1-11111-1-11-11-1111-1-1-11-1-111    linear of order 2
ρ91111-1-1-1-1-1-1-11111-1i-iii-ii-i-ii-ii-i    linear of order 4
ρ10111111-1-1-1-1-1-1-1-111ii-i-i-ii-ii-iii-i    linear of order 4
ρ11111111-1-1-1-1-1-1-1-111-i-iiii-ii-ii-i-ii    linear of order 4
ρ121111-1-1-1-1-1-1-11111-1-ii-i-ii-iii-ii-ii    linear of order 4
ρ131111111-1-1-1-1-1-11-1-1-i-iiii-ii-i-iii-i    linear of order 4
ρ141111-1-11-1-1-1-111-1-11-ii-i-ii-iiii-ii-i    linear of order 4
ρ151111-1-11-1-1-1-111-1-11i-iii-ii-i-i-ii-ii    linear of order 4
ρ161111111-1-1-1-1-1-11-1-1ii-i-i-ii-iii-i-ii    linear of order 4
ρ172-2-22000-222-2000002000-2-2200000    orthogonal lifted from D4
ρ182-2-22000-222-200000-200022-200000    orthogonal lifted from D4
ρ192-2-220002-2-2200000-2i000-2i2i2i00000    complex lifted from C4○D4
ρ202-22-2-220-2i-2i2i2i2i-2i000000000000000    complex lifted from M4(2)
ρ212-22-22-20-2i-2i2i2i-2i2i000000000000000    complex lifted from M4(2)
ρ222-22-2-2202i2i-2i-2i-2i2i000000000000000    complex lifted from M4(2)
ρ232-2-220002-2-22000002i0002i-2i-2i00000    complex lifted from C4○D4
ρ242-22-22-202i2i-2i-2i2i-2i000000000000000    complex lifted from M4(2)
ρ2522-2-20002i-2i2i-2i00000088387000850000    complex lifted from C8○D4
ρ2622-2-2000-2i2i-2i2i00000083885000870000    complex lifted from C8○D4
ρ2722-2-20002i-2i2i-2i00000085878300080000    complex lifted from C8○D4
ρ2822-2-2000-2i2i-2i2i00000087858000830000    complex lifted from C8○D4

Smallest permutation representation of C89D4
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 11)(2 31 24 16)(3 28 17 13)(4 25 18 10)(5 30 19 15)(6 27 20 12)(7 32 21 9)(8 29 22 14)
(2 6)(4 8)(9 32)(10 29)(11 26)(12 31)(13 28)(14 25)(15 30)(16 27)(18 22)(20 24)

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

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

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

C89D4 is a maximal subgroup of
C42.290C23  C42.291C23  C42.292C23  C42.293C23  C42.294C23  C233M4(2)  C42.297C23  C42.298C23  C42.299C23  C42.300C23  C42.301C23  C42.698C23  C42.307C23  C42.308C23  C42.309C23  C42.310C23  D89D4  SD16⋊D4  SD166D4  D810D4  SD167D4  SD168D4  Q169D4  Q1610D4  C42.41C23  C42.42C23  C42.43C23  C42.44C23  C42.45C23  C42.46C23  C42.47C23  C42.48C23  C42.49C23  C42.50C23  C42.51C23  C42.52C23  C42.53C23  C42.54C23  C42.55C23  C42.56C23  C42.57C23  C42.58C23  C42.59C23  C42.60C23  C42.61C23  C42.62C23  C42.63C23  C42.64C23  C8⋊S4
 D2p⋊M4(2): D46M4(2)  D47M4(2)  D48M4(2)  C89D12  D62M4(2)  D63M4(2)  C24⋊D4  C89D20 ...
 C2p.(C4×D4): C42.264C23  C42.265C23  C42.266C23  M4(2)⋊22D4  D4×M4(2)  M4(2)⋊23D4  C3⋊C826D4  C42.47D6 ...
C89D4 is a maximal quotient of
C812SD16  C815SD16  C89Q16  D4.M4(2)  Q8.M4(2)  Q82M4(2)  C23.21M4(2)  (C2×C8).195D4  C23.22M4(2)  C232M4(2)  C4⋊C43C8  (C2×C8).Q8  C22⋊C44C8  C23.9M4(2)
 D2p⋊M4(2): C89D8  D42M4(2)  C89D12  D62M4(2)  D63M4(2)  C24⋊D4  C89D20  D104M4(2) ...
 C42.D2p: C42.27Q8  C42.109D4  C42.47D6  C42.47D10  C42.47D14 ...
 C2p.(C4×D4): C23.36C42  C23.17C42  C4⋊C814C4  C3⋊C826D4  C2433D4  C52C826D4  C4032D4  C5⋊C8⋊D4 ...

Matrix representation of C89D4 in GL4(𝔽17) generated by

131500
6400
0001
0040
,
1000
131600
0009
00150
,
1000
131600
0010
00016
G:=sub<GL(4,GF(17))| [13,6,0,0,15,4,0,0,0,0,0,4,0,0,1,0],[1,13,0,0,0,16,0,0,0,0,0,15,0,0,9,0],[1,13,0,0,0,16,0,0,0,0,1,0,0,0,0,16] >;

C89D4 in GAP, Magma, Sage, TeX

C_8\rtimes_9D_4
% in TeX

G:=Group("C8:9D4");
// GroupNames label

G:=SmallGroup(64,116);
// by ID

G=gap.SmallGroup(64,116);
# by ID

G:=PCGroup([6,-2,2,2,-2,2,-2,96,121,650,86,88]);
// Polycyclic

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

Export

Character table of C89D4 in TeX

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