Copied to
clipboard

G = C8.12D4order 64 = 26

8th non-split extension by C8 of D4 acting via D4/C4=C2

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

Aliases: C8.12D4, C42.82C22, (C4×C8)⋊9C2, C4.4(C2×D4), (C2×Q16)⋊5C2, (C2×D8).3C2, (C2×C4).58D4, C4.4D44C2, (C2×SD16)⋊15C2, C2.18(C4○D8), C2.8(C41D4), (C2×C8).80C22, (C2×C4).120C23, (C2×D4).30C22, C22.116(C2×D4), (C2×Q8).26C22, SmallGroup(64,176)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C8.12D4
C1C2C22C2×C4C42C4×C8 — C8.12D4
C1C2C2×C4 — C8.12D4
C1C22C42 — C8.12D4
C1C2C2C2×C4 — C8.12D4

Generators and relations for C8.12D4
 G = < a,b,c | a8=b4=1, c2=a4, ab=ba, cac-1=a3, cbc-1=a4b-1 >

Subgroups: 129 in 65 conjugacy classes, 29 normal (13 characteristic)
C1, C2, C2 [×2], C2 [×2], C4 [×2], C4 [×4], C22, C22 [×6], C8 [×4], C2×C4, C2×C4 [×2], C2×C4 [×2], D4 [×4], Q8 [×4], C23 [×2], C42, C22⋊C4 [×4], C2×C8 [×2], D8 [×2], SD16 [×4], Q16 [×2], C2×D4 [×2], C2×Q8 [×2], C4×C8, C4.4D4 [×2], C2×D8, C2×SD16 [×2], C2×Q16, C8.12D4
Quotients: C1, C2 [×7], C22 [×7], D4 [×6], C23, C2×D4 [×3], C41D4, C4○D8 [×2], C8.12D4

Character table of C8.12D4

 class 12A2B2C2D2E4A4B4C4D4E4F4G4H8A8B8C8D8E8F8G8H
 size 1111882222228822222222
ρ11111111111111111111111    trivial
ρ21111-11111111-11-1-1-1-1-1-1-1-1    linear of order 2
ρ31111-1-1-1-11-11-111-11-11-11-11    linear of order 2
ρ411111-1-1-11-11-1-111-11-11-11-1    linear of order 2
ρ51111-1-1111111-1-111111111    linear of order 2
ρ611111-11111111-1-1-1-1-1-1-1-1-1    linear of order 2
ρ7111111-1-11-11-1-1-1-11-11-11-11    linear of order 2
ρ81111-11-1-11-11-11-11-11-11-11-1    linear of order 2
ρ92-2-22000020-2000-202020-20    orthogonal lifted from D4
ρ102222002-2-2-2-220000000000    orthogonal lifted from D4
ρ112-2-22000020-200020-20-2020    orthogonal lifted from D4
ρ122-2-220000-202000020-20-202    orthogonal lifted from D4
ρ132-2-220000-2020000-202020-2    orthogonal lifted from D4
ρ14222200-22-22-2-20000000000    orthogonal lifted from D4
ρ152-22-2000-2i02i0000--2-2-22--2-2-22    complex lifted from C4○D8
ρ1622-2-200-2i00002i00-2-2-2-2--22--22    complex lifted from C4○D8
ρ172-22-20002i0-2i0000-2-2--22-2-2--22    complex lifted from C4○D8
ρ1822-2-2002i0000-2i00-22-22--2-2--2-2    complex lifted from C4○D8
ρ1922-2-2002i0000-2i00--2-2--2-2-22-22    complex lifted from C4○D8
ρ2022-2-200-2i00002i00--22--22-2-2-2-2    complex lifted from C4○D8
ρ212-22-20002i0-2i0000--22-2-2--22-2-2    complex lifted from C4○D8
ρ222-22-2000-2i02i0000-22--2-2-22--2-2    complex lifted from C4○D8

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

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

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

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

C8.12D4 is a maximal subgroup of
C42.410C23  C42.411C23  C42.533C23
 C8.D4p: C8.30D8  C8.3D8  C8.21D8  C163D4  C8.7D8  C8.8D12  C8.8D20  C8.8D28 ...
 C42.D2p: D8.5D4  Q16.5D4  C8.22SD16  C8.12SD16  C42.360D4  M4(2)⋊9D4  C42.308D4  C42.260D4 ...
 (C2p×Q16)⋊C2: C42.387C23  Q164D4  Q1612D4  C42.527C23  C42.530C23  C42.532C23  C24.28D4  C40.28D4 ...
 (C2p×SD16)⋊C2: C42.385C23  SD162D4  SD1610D4  C42.528C23  C42.531C23  C24.43D4  C40.43D4  C56.43D4 ...
 C4p.(C2×D4): M4(2)⋊10D4  M4(2)⋊11D4  M4(2).20D4  D85D4  D812D4  D8○SD16  C24.22D4  C40.22D4 ...
C8.12D4 is a maximal quotient of
 C42.D2p: C42.433D4  C8.8D12  C42.214D6  C8.8D20  C42.214D10  C8.8D28  C42.214D14 ...
 (C2×D4).D2p: (C22×D8).C2  (C2×C8).41D4  C24.22D4  C24.43D4  C40.22D4  C40.43D4  C56.22D4  C56.43D4 ...
 (C2×C8).D2p: C85D8  C85Q16  C8212C2  C825C2  C8.7Q16  C823C2  C42.664C23  C42.665C23 ...

Matrix representation of C8.12D4 in GL4(𝔽17) generated by

12500
121200
00125
001212
,
01300
4000
0040
0004
,
01300
13000
0040
00013
G:=sub<GL(4,GF(17))| [12,12,0,0,5,12,0,0,0,0,12,12,0,0,5,12],[0,4,0,0,13,0,0,0,0,0,4,0,0,0,0,4],[0,13,0,0,13,0,0,0,0,0,4,0,0,0,0,13] >;

C8.12D4 in GAP, Magma, Sage, TeX

C_8._{12}D_4
% in TeX

G:=Group("C8.12D4");
// GroupNames label

G:=SmallGroup(64,176);
// by ID

G=gap.SmallGroup(64,176);
# by ID

G:=PCGroup([6,-2,2,2,-2,2,-2,121,55,362,230,963,117]);
// Polycyclic

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

Export

Character table of C8.12D4 in TeX

׿
×
𝔽