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G = C86D4order 64 = 26

3rd semidirect product of C8 and D4 acting via D4/C4=C2

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

Aliases: C86D4, C41M4(2), C42.70C22, C4⋊C816C2, (C4×C8)⋊15C2, C4⋊C4.8C4, (C2×D4).9C4, (C4×D4).3C2, C2.11(C4×D4), C4.78(C2×D4), C22⋊C814C2, C2.8(C8○D4), C22⋊C4.5C4, C4.53(C4○D4), C23.12(C2×C4), (C2×M4(2))⋊15C2, (C2×C4).155C23, (C2×C8).101C22, C2.10(C2×M4(2)), C22.48(C22×C4), (C22×C4).41C22, (C2×C4).29(C2×C4), SmallGroup(64,117)

Series: Derived Chief Lower central Upper central Jennings

C1C22 — C86D4
C1C2C4C2×C4C2×C8C2×M4(2) — C86D4
C1C22 — C86D4
C1C2×C4 — C86D4
C1C2C2C2×C4 — C86D4

Generators and relations for C86D4
 G = < a,b,c | a8=b4=c2=1, ab=ba, cac=a5, cbc=b-1 >

Subgroups: 89 in 61 conjugacy classes, 37 normal (19 characteristic)
C1, C2 [×3], C2 [×2], C4 [×2], C4 [×2], C4 [×3], C22, C22 [×6], C8 [×2], C8 [×3], C2×C4 [×3], C2×C4 [×2], C2×C4 [×4], D4 [×2], C23 [×2], C42, C22⋊C4 [×2], C4⋊C4, C2×C8 [×2], C2×C8 [×2], M4(2) [×4], C22×C4 [×2], C2×D4, C4×C8, C22⋊C8 [×2], C4⋊C8, C4×D4, C2×M4(2) [×2], C86D4
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, C86D4

Character table of C86D4

 class 12A2B2C2D2E4A4B4C4D4E4F4G4H4I4J8A8B8C8D8E8F8G8H8I8J8K8L
 size 1111441111222244222222224444
ρ11111111111111111111111111111    trivial
ρ21111-1-111111111-1-111111111-1-1-1-1    linear of order 2
ρ311111-11111-1-1-1-1-111-11-111-1-11-11-1    linear of order 2
ρ41111-111111-1-1-1-11-11-11-111-1-1-11-11    linear of order 2
ρ51111-1-111111111-1-1-1-1-1-1-1-1-1-11111    linear of order 2
ρ61111111111111111-1-1-1-1-1-1-1-1-1-1-1-1    linear of order 2
ρ71111-111111-1-1-1-11-1-11-11-1-1111-11-1    linear of order 2
ρ811111-11111-1-1-1-1-11-11-11-1-111-11-11    linear of order 2
ρ91111-11-1-1-1-11-1-11-11-iiii-ii-i-ii-i-ii    linear of order 4
ρ1011111-1-1-1-1-11-1-111-1-iiii-ii-i-i-iii-i    linear of order 4
ρ111111-11-1-1-1-11-1-11-11i-i-i-ii-iii-iii-i    linear of order 4
ρ1211111-1-1-1-1-11-1-111-1i-i-i-ii-iiii-i-ii    linear of order 4
ρ131111-1-1-1-1-1-1-111-111-i-ii-i-iiiiii-i-i    linear of order 4
ρ14111111-1-1-1-1-111-1-1-1-i-ii-i-iiii-i-iii    linear of order 4
ρ151111-1-1-1-1-1-1-111-111ii-iii-i-i-i-i-iii    linear of order 4
ρ16111111-1-1-1-1-111-1-1-1ii-iii-i-i-iii-i-i    linear of order 4
ρ172-22-200-22-2200000020-20-22000000    orthogonal lifted from D4
ρ182-22-200-22-22000000-20202-2000000    orthogonal lifted from D4
ρ1922-2-2002i-2i-2i2i-2i2-22i00000000000000    complex lifted from M4(2)
ρ2022-2-200-2i2i2i-2i2i2-2-2i00000000000000    complex lifted from M4(2)
ρ2122-2-200-2i2i2i-2i-2i-222i00000000000000    complex lifted from M4(2)
ρ2222-2-2002i-2i-2i2i2i-22-2i00000000000000    complex lifted from M4(2)
ρ232-22-2002-22-20000002i02i0-2i-2i000000    complex lifted from C4○D4
ρ242-22-2002-22-2000000-2i0-2i02i2i000000    complex lifted from C4○D4
ρ252-2-2200-2i-2i2i2i000000085080087830000    complex lifted from C8○D4
ρ262-2-22002i2i-2i-2i000000083087008850000    complex lifted from C8○D4
ρ272-2-2200-2i-2i2i2i000000080850083870000    complex lifted from C8○D4
ρ282-2-22002i2i-2i-2i000000087083008580000    complex lifted from C8○D4

Smallest permutation representation of C86D4
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 14 18 25)(2 15 19 26)(3 16 20 27)(4 9 21 28)(5 10 22 29)(6 11 23 30)(7 12 24 31)(8 13 17 32)
(1 25)(2 30)(3 27)(4 32)(5 29)(6 26)(7 31)(8 28)(9 17)(10 22)(11 19)(12 24)(13 21)(14 18)(15 23)(16 20)

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

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

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

C86D4 is a maximal subgroup of
C8.31D8  C815SD16  Q82M4(2)  C8.D8  C83SD16  C84SD16  C8.8SD16  C42.248C23  C42.250C23  C42.252C23  C42.254C23  C42.265C23  C42.681C23  M4(2)⋊22D4  D4×M4(2)  M4(2)⋊23D4  C42.290C23  C42.291C23  C42.292C23  C42.294C23  Q86M4(2)  D47M4(2)  C42.693C23  C42.297C23  C42.299C23  C42.694C23  C42.301C23  C42.698C23  Q87M4(2)  C42.308C23  C42.309C23  C42.310C23  D84D4  D85D4  SD161D4  SD162D4  SD163D4  Q164D4  Q165D4  C42.471C23  C42.472C23  C42.473C23  C42.474C23  C42.475C23  C42.476C23  C42.477C23  C42.478C23  C42.479C23  C42.480C23  C42.481C23  C42.482C23  C42.492C23  C42.493C23  C42.494C23  C42.495C23  C42.496C23  C42.497C23  C42.498C23  Dic52M4(2)  C4018D4
 C8⋊D4p: C89D8  C86D8  C8⋊D8  C82D8  C86D12  C86D20  C86D28 ...
 C4p⋊M4(2): C89SD16  C8⋊M4(2)  C83M4(2)  C122M4(2)  C123M4(2)  C206M4(2)  C207M4(2)  C20⋊M4(2) ...
 D2p⋊C8⋊C2: D42M4(2)  Dic3⋊M4(2)  C2421D4  Dic5⋊M4(2)  Dic7⋊M4(2)  C5618D4 ...
C86D4 is a maximal quotient of
C86Q16  C8.M4(2)  (C2×C8).195D4  C232M4(2)  (C2×C8).Q8  C22⋊C44C8  C23.9M4(2)  C42.61Q8  C42.325D4
 C8⋊D4p: C86D8  C86D12  C86D20  C86D28 ...
 C4p⋊M4(2): C89SD16  C8⋊M4(2)  C83M4(2)  C122M4(2)  C123M4(2)  C206M4(2)  C207M4(2)  C20⋊M4(2) ...
 C2p.(C4×D4): C23.17C42  C4⋊C813C4  Dic3⋊M4(2)  C2421D4  Dic52M4(2)  C4018D4  Dic5⋊M4(2)  Dic7⋊M4(2) ...

Matrix representation of C86D4 in GL4(𝔽17) generated by

51500
21200
00130
00013
,
101300
4700
0018
00416
,
101300
12700
0018
00016
G:=sub<GL(4,GF(17))| [5,2,0,0,15,12,0,0,0,0,13,0,0,0,0,13],[10,4,0,0,13,7,0,0,0,0,1,4,0,0,8,16],[10,12,0,0,13,7,0,0,0,0,1,0,0,0,8,16] >;

C86D4 in GAP, Magma, Sage, TeX

C_8\rtimes_6D_4
% in TeX

G:=Group("C8:6D4");
// GroupNames label

G:=SmallGroup(64,117);
// by ID

G=gap.SmallGroup(64,117);
# by ID

G:=PCGroup([6,-2,2,2,-2,2,-2,96,121,650,332,86,88]);
// Polycyclic

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

Export

Character table of C86D4 in TeX

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