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G = C8.2D4order 64 = 26

2nd non-split extension by C8 of D4 acting via D4/C2=C22

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

Aliases: C8.2D4, C42.32C22, C4⋊Q88C2, C8⋊C45C2, C4.6(C2×D4), (C2×Q16)⋊9C2, (C2×C4).45D4, (C2×C8).21C22, (C2×SD16).2C2, C4.4D4.7C2, C2.10(C41D4), (C2×C4).122C23, (C2×D4).32C22, C22.118(C2×D4), (C2×Q8).28C22, C2.22(C8.C22), SmallGroup(64,178)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C8.2D4
C1C2C22C2×C4C42C8⋊C4 — C8.2D4
C1C2C2×C4 — C8.2D4
C1C22C42 — C8.2D4
C1C2C2C2×C4 — C8.2D4

Generators and relations for C8.2D4
 G = < a,b,c | a8=c2=1, b4=a4, bab-1=a5, cac=a3, cbc=b3 >

Subgroups: 113 in 62 conjugacy classes, 29 normal (11 characteristic)
C1, C2, C2 [×2], C2, C4 [×2], C4 [×5], C22, C22 [×3], C8 [×4], C2×C4, C2×C4 [×2], C2×C4 [×3], D4 [×2], Q8 [×6], C23, C42, C22⋊C4 [×2], C4⋊C4 [×2], C2×C8 [×2], SD16 [×4], Q16 [×4], C2×D4, C2×Q8, C2×Q8 [×2], C8⋊C4, C4.4D4, C4⋊Q8, C2×SD16 [×2], C2×Q16 [×2], C8.2D4
Quotients: C1, C2 [×7], C22 [×7], D4 [×6], C23, C2×D4 [×3], C41D4, C8.C22 [×2], C8.2D4

Character table of C8.2D4

 class 12A2B2C2D4A4B4C4D4E4F4G8A8B8C8D
 size 1111822448884444
ρ11111111111111111    trivial
ρ21111-1111111-1-1-1-1-1    linear of order 2
ρ31111111-1-11-1-11-11-1    linear of order 2
ρ41111-111-1-11-11-11-11    linear of order 2
ρ51111-111-1-1-1111-11-1    linear of order 2
ρ61111111-1-1-11-1-11-11    linear of order 2
ρ71111-11111-1-1-11111    linear of order 2
ρ8111111111-1-11-1-1-1-1    linear of order 2
ρ92-22-202-200000-2020    orthogonal lifted from D4
ρ1022220-2-2-220000000    orthogonal lifted from D4
ρ112-22-20-2200000020-2    orthogonal lifted from D4
ρ1222220-2-22-20000000    orthogonal lifted from D4
ρ132-22-202-20000020-20    orthogonal lifted from D4
ρ142-22-20-22000000-202    orthogonal lifted from D4
ρ154-4-44000000000000    symplectic lifted from C8.C22, Schur index 2
ρ1644-4-4000000000000    symplectic lifted from C8.C22, Schur index 2

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

C8.2D4 is a maximal subgroup of
(C2×D4).D4  C42.5C23  C42.6C23  C42.7C23  M4(2)⋊10D4  C42.390C23  C42.409C23  C42.73C23  C42.75C23  C42.532C23
 C42.D2p: C42.247D4  M4(2)⋊8D4  M4(2)⋊9D4  C42.256D4  C42.258D4  C42.273D4  C42.274D4  C42.276D4 ...
 (C2×C8).D2p: M4(2).20D4  C24.31D4  C24.37D4  C40.31D4  C40.37D4  C56.31D4  C56.37D4 ...
 D4.pD4⋊C2: C42.408C23  C42.411C23  SD166D4  D810D4  SD168D4  Q169D4  D8.13D4  C42.531C23 ...
C8.2D4 is a maximal quotient of
 C42.D2p: C42.110D4  C42.111D4  C8.D12  C42.65D6  C42.80D6  C8.D20  C42.65D10  C42.80D10 ...
 (C2×C8).D2p: C85SD16  C8.9SD16  C42.665C23  C42.666C23  C42.667C23  C8.2D8  C83Q16  C42.26Q8 ...

Matrix representation of C8.2D4 in GL6(𝔽17)

010000
1600000
000040
000004
0001300
004000
,
1600000
0160000
000010
000001
000100
0016000
,
100000
0160000
0001600
0016000
000010
0000016

G:=sub<GL(6,GF(17))| [0,16,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,13,0,0,0,4,0,0,0,0,0,0,4,0,0],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,0,16,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,1,0,0],[1,0,0,0,0,0,0,16,0,0,0,0,0,0,0,16,0,0,0,0,16,0,0,0,0,0,0,0,1,0,0,0,0,0,0,16] >;

C8.2D4 in GAP, Magma, Sage, TeX

C_8._2D_4
% in TeX

G:=Group("C8.2D4");
// GroupNames label

G:=SmallGroup(64,178);
// by ID

G=gap.SmallGroup(64,178);
# by ID

G:=PCGroup([6,-2,2,2,-2,2,-2,192,121,55,362,332,86,963,117]);
// Polycyclic

G:=Group<a,b,c|a^8=c^2=1,b^4=a^4,b*a*b^-1=a^5,c*a*c=a^3,c*b*c=b^3>;
// generators/relations

Export

Character table of C8.2D4 in TeX

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