Copied to
clipboard

G = C4.4D8order 64 = 26

4th non-split extension by C4 of D8 acting via D8/C8=C2

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

Aliases: C4.4D8, C4.3SD16, C42.76C22, (C4×C8)⋊5C2, C4⋊Q85C2, C2.9(C2×D8), (C2×C4).74D4, D4⋊C43C2, C41D4.4C2, C4.13(C4○D4), C4⋊C4.16C22, (C2×C8).66C22, C2.14(C2×SD16), (C2×C4).111C23, C2.9(C4.4D4), (C2×D4).24C22, C22.107(C2×D4), SmallGroup(64,167)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C4.4D8
C1C2C4C2×C4C2×C8C4×C8 — C4.4D8
C1C2C2×C4 — C4.4D8
C1C22C42 — C4.4D8
C1C2C2C2×C4 — C4.4D8

Generators and relations for C4.4D8
 G = < a,b,c | a4=b8=1, c2=a2, ab=ba, cac-1=a-1, cbc-1=a2b-1 >

Subgroups: 129 in 59 conjugacy classes, 29 normal (13 characteristic)
C1, C2 [×3], C2 [×2], C4 [×6], C4 [×2], C22, C22 [×6], C8 [×2], C2×C4 [×3], C2×C4 [×2], D4 [×8], Q8 [×2], C23 [×2], C42, C4⋊C4 [×2], C4⋊C4, C2×C8 [×2], C2×D4 [×2], C2×D4 [×2], C2×Q8, C4×C8, D4⋊C4 [×4], C41D4, C4⋊Q8, C4.4D8
Quotients: C1, C2 [×7], C22 [×7], D4 [×2], C23, D8 [×2], SD16 [×2], C2×D4, C4○D4 [×2], C4.4D4, C2×D8, C2×SD16, C4.4D8

Character table of C4.4D8

 class 12A2B2C2D2E4A4B4C4D4E4F4G4H8A8B8C8D8E8F8G8H
 size 1111882222228822222222
ρ11111111111111111111111    trivial
ρ211111-1-1-11-11-11-1-11-11-11-11    linear of order 2
ρ311111-1-1-11-11-1-111-11-11-11-1    linear of order 2
ρ4111111111111-1-1-1-1-1-1-1-1-1-1    linear of order 2
ρ51111-1-1111111-1-111111111    linear of order 2
ρ61111-11-1-11-11-1-11-11-11-11-11    linear of order 2
ρ71111-11-1-11-11-11-11-11-11-11-1    linear of order 2
ρ81111-1-111111111-1-1-1-1-1-1-1-1    linear of order 2
ρ922220022-2-2-2-20000000000    orthogonal lifted from D4
ρ10222200-2-2-22-220000000000    orthogonal lifted from D4
ρ112-2-220000020-200-22-2-22-222    orthogonal lifted from D8
ρ122-2-2200000-20200-2-2-22222-2    orthogonal lifted from D8
ρ132-2-220000020-2002-222-22-2-2    orthogonal lifted from D8
ρ142-2-2200000-20200222-2-2-2-22    orthogonal lifted from D8
ρ152-22-20000-2020002i0-2i0-2i02i0    complex lifted from C4○D4
ρ1622-2-200-22000000--2--2-2-2--2--2-2-2    complex lifted from SD16
ρ172-22-2000020-20000-2i0-2i02i02i    complex lifted from C4○D4
ρ182-22-20000-202000-2i02i02i0-2i0    complex lifted from C4○D4
ρ1922-2-2002-2000000-2--2--2-2-2--2--2-2    complex lifted from SD16
ρ202-22-2000020-200002i02i0-2i0-2i    complex lifted from C4○D4
ρ2122-2-200-22000000-2-2--2--2-2-2--2--2    complex lifted from SD16
ρ2222-2-2002-2000000--2-2-2--2--2-2-2--2    complex lifted from SD16

Smallest permutation representation of C4.4D8
On 32 points
Generators in S32
(1 21 27 13)(2 22 28 14)(3 23 29 15)(4 24 30 16)(5 17 31 9)(6 18 32 10)(7 19 25 11)(8 20 26 12)
(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 27 8)(2 7 28 25)(3 32 29 6)(4 5 30 31)(9 16 17 24)(10 23 18 15)(11 14 19 22)(12 21 20 13)

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

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

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

C4.4D8 is a maximal subgroup of
D43D8  Q86SD16  Q83D8  C42.189C23  D4.D8  C42.201C23  Q8.D8  Q83SD16  D4.2SD16  Q8.2SD16  C42.248C23  C42.249C23  C823C2  C42.355D4  C42.366C23  C42.240D4  C42.243D4  C42.263D4  C42.278D4  C42.279D4  C42.280D4  D47SD16  C42.469C23  C42.473C23  C42.482C23  Q84D8  C42.502C23  Q88SD16  C42.506C23  C42.507C23  C42.509C23  C42.514C23  C42.516C23
 C4p.D8: C825C2  C8.2D8  C4.5D24  C12.16D8  C12.D8  C4.5D40  C20.16D8  C20.D8 ...
 C4p⋊Q8⋊C2: C42.391C23  C42.423C23  Dic3.SD16  Dic5.5D8  Dic7.SD16 ...
 C8pD4⋊C2: C85D8  C86SD16  C42.664C23  C83D8  C42.365D4  C42.366D4  C42.259D4  C42.261D4 ...
 C2.(C8pD4): D4.2D8  Q8.2D8  C42.252C23  C42.253C23  C8212C2  C85SD16  C42.666C23  C42.667C23 ...
C4.4D8 is a maximal quotient of
C42.55Q8  C2.(C87D4)  C42.432D4  C42.436D4  C4⋊C47D4  (C2×C4).24D8  (C2×C4).28D8
 C4.D8p: C4.4D16  C4.5D24  C4.5D40  C4.5D56 ...
 C4p.SD16: C4.SD32  C8.22SD16  C8.12SD16  C8.13SD16  C8.14SD16  C12.16D8  C12.D8  C20.16D8 ...
 C4⋊C4.D2p: (C2×D4)⋊Q8  Dic3.SD16  Dic5.5D8  Dic7.SD16 ...

Matrix representation of C4.4D8 in GL4(𝔽17) generated by

16200
16100
00013
00130
,
71000
12000
0001
0010
,
71000
121000
0001
00160
G:=sub<GL(4,GF(17))| [16,16,0,0,2,1,0,0,0,0,0,13,0,0,13,0],[7,12,0,0,10,0,0,0,0,0,0,1,0,0,1,0],[7,12,0,0,10,10,0,0,0,0,0,16,0,0,1,0] >;

C4.4D8 in GAP, Magma, Sage, TeX

C_4._4D_8
% in TeX

G:=Group("C4.4D8");
// GroupNames label

G:=SmallGroup(64,167);
// by ID

G=gap.SmallGroup(64,167);
# by ID

G:=PCGroup([6,-2,2,2,-2,2,-2,121,103,362,50,963,117,1444,88]);
// Polycyclic

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

Export

Character table of C4.4D8 in TeX

׿
×
𝔽