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G = C85D4order 64 = 26

2nd semidirect product of C8 and D4 acting via D4/C4=C2

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

Aliases: C85D4, C41SD16, C42.79C22, C4⋊Q87C2, (C4×C8)⋊12C2, C4.1(C2×D4), (C2×C4).76D4, C41D4.6C2, (C2×SD16)⋊14C2, C2.5(C41D4), (C2×C8).92C22, C2.16(C2×SD16), (C2×C4).117C23, (C2×D4).28C22, C22.113(C2×D4), (C2×Q8).24C22, SmallGroup(64,173)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C85D4
C1C2C22C2×C4C42C4×C8 — C85D4
C1C2C2×C4 — C85D4
C1C22C42 — C85D4
C1C2C2C2×C4 — C85D4

Generators and relations for C85D4
 G = < a,b,c | a8=b4=c2=1, ab=ba, cac=a3, cbc=b-1 >

Subgroups: 145 in 71 conjugacy classes, 33 normal (9 characteristic)
C1, C2, C2 [×2], C2 [×2], C4 [×6], C4 [×2], C22, C22 [×6], C8 [×4], C2×C4, C2×C4 [×2], C2×C4 [×2], D4 [×8], Q8 [×4], C23 [×2], C42, C4⋊C4 [×2], C2×C8 [×2], SD16 [×8], C2×D4 [×2], C2×D4 [×2], C2×Q8 [×2], C4×C8, C41D4, C4⋊Q8, C2×SD16 [×4], C85D4
Quotients: C1, C2 [×7], C22 [×7], D4 [×6], C23, SD16 [×4], C2×D4 [×3], C41D4, C2×SD16 [×2], C85D4

Character table of C85D4

 class 12A2B2C2D2E4A4B4C4D4E4F4G4H8A8B8C8D8E8F8G8H
 size 1111882222228822222222
ρ11111111111111111111111    trivial
ρ211111-1-1-11-11-1-11-11-111-1-11    linear of order 2
ρ3111111111111-1-1-1-1-1-1-1-1-1-1    linear of order 2
ρ411111-1-1-11-11-11-11-11-1-111-1    linear of order 2
ρ51111-1-1111111-1-111111111    linear of order 2
ρ61111-11-1-11-11-11-1-11-111-1-11    linear of order 2
ρ71111-1-111111111-1-1-1-1-1-1-1-1    linear of order 2
ρ81111-11-1-11-11-1-111-11-1-111-1    linear of order 2
ρ92-2-22000020-2000-202002-20    orthogonal lifted from D4
ρ102222002-2-2-2-220000000000    orthogonal lifted from D4
ρ112-2-22000020-200020-200-220    orthogonal lifted from D4
ρ122-2-220000-2020000202-200-2    orthogonal lifted from D4
ρ132-2-220000-2020000-20-22002    orthogonal lifted from D4
ρ14222200-22-22-2-20000000000    orthogonal lifted from D4
ρ1522-2-20020000-200--2--2--2-2-2-2-2--2    complex lifted from SD16
ρ1622-2-20020000-200-2-2-2--2--2--2--2-2    complex lifted from SD16
ρ1722-2-200-20000200--2-2--2--2--2-2-2-2    complex lifted from SD16
ρ182-22-2000-2020000--2-2-2--2-2--2-2--2    complex lifted from SD16
ρ192-22-200020-20000--2--2-2-2--2--2-2-2    complex lifted from SD16
ρ2022-2-200-20000200-2--2-2-2-2--2--2--2    complex lifted from SD16
ρ212-22-200020-20000-2-2--2--2-2-2--2--2    complex lifted from SD16
ρ222-22-2000-2020000-2--2--2-2--2-2--2-2    complex lifted from SD16

Smallest permutation representation of C85D4
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 12 29 23)(2 13 30 24)(3 14 31 17)(4 15 32 18)(5 16 25 19)(6 9 26 20)(7 10 27 21)(8 11 28 22)
(1 23)(2 18)(3 21)(4 24)(5 19)(6 22)(7 17)(8 20)(9 28)(10 31)(11 26)(12 29)(13 32)(14 27)(15 30)(16 25)

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

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

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

C85D4 is a maximal subgroup of
C8.29D8  (C4×C8)⋊6C4  (C4×C8).C4  D4⋊SD16  Q8⋊SD16  C42.185C23  D42SD16  C42.191C23  Q82SD16  C88D8  C8⋊D8  C8212C2  C42.664C23  C42.666C23  C83D8  C8.2D8  M4(2)⋊7D4  C42.365D4  C42.386C23  C42.259D4  C42.264D4  C42.265D4  C42.266D4  C42.268D4  C42.407C23  C42.408C23  D84D4  D4×SD16  Q89SD16  C42.528C23  C42.72C23  C42.75C23
 C4p⋊SD16: C814SD16  C8⋊SD16  C88SD16  C85D8  C85SD16  C86SD16  C85D12  C124SD16 ...
 C4p.(C2×D4): C42.360D4  M4(2)⋊8D4  M4(2)⋊10D4  Q165D4  SD1611D4  D86D4  C2415D4  C4015D4 ...
C85D4 is a maximal quotient of
C85Q16  C8212C2  C8.9SD16  C42.58Q8  C42.431D4  C42.432D4  (C2×C4)⋊9SD16  (C2×C8).169D4  (C2×C8).170D4
 C4p⋊SD16: C88SD16  C85D8  C85SD16  C86SD16  C85D12  C124SD16  C126SD16  C85D20 ...
 (C2×D4).D2p: (C2×C4)⋊3SD16  (C2×C8)⋊20D4  C2415D4  C4015D4  C5615D4 ...

Matrix representation of C85D4 in GL4(𝔽17) generated by

12500
121200
00160
00016
,
16000
01600
0001
00160
,
16000
0100
0001
0010
G:=sub<GL(4,GF(17))| [12,12,0,0,5,12,0,0,0,0,16,0,0,0,0,16],[16,0,0,0,0,16,0,0,0,0,0,16,0,0,1,0],[16,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0] >;

C85D4 in GAP, Magma, Sage, TeX

C_8\rtimes_5D_4
% in TeX

G:=Group("C8:5D4");
// GroupNames label

G:=SmallGroup(64,173);
// by ID

G=gap.SmallGroup(64,173);
# by ID

G:=PCGroup([6,-2,2,2,-2,2,-2,121,55,362,86,963,117]);
// Polycyclic

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

Export

Character table of C85D4 in TeX

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