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G = C83D4order 64 = 26

3rd semidirect product of C8 and D4 acting via D4/C2=C22

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

Aliases: C83D4, C42.31C22, (C2×D8)⋊9C2, C8⋊C44C2, C4.5(C2×D4), C41D44C2, (C2×C4).44D4, (C2×SD16)⋊3C2, C4.4D45C2, C2.9(C41D4), (C2×C8).20C22, C2.22(C8⋊C22), (C2×C4).121C23, (C2×D4).31C22, C22.117(C2×D4), (C2×Q8).27C22, SmallGroup(64,177)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C83D4
C1C2C22C2×C4C42C8⋊C4 — C83D4
C1C2C2×C4 — C83D4
C1C22C42 — C83D4
C1C2C2C2×C4 — C83D4

Generators and relations for C83D4
 G = < a,b,c | a8=b4=c2=1, bab-1=a5, cac=a-1, cbc=b-1 >

Subgroups: 161 in 72 conjugacy classes, 29 normal (11 characteristic)
C1, C2, C2 [×2], C2 [×3], C4 [×2], C4 [×3], C22, C22 [×9], C8 [×4], C2×C4, C2×C4 [×2], C2×C4, D4 [×10], Q8 [×2], C23 [×3], C42, C22⋊C4 [×2], C2×C8 [×2], D8 [×4], SD16 [×4], C2×D4, C2×D4 [×2], C2×D4 [×2], C2×Q8, C8⋊C4, C4.4D4, C41D4, C2×D8 [×2], C2×SD16 [×2], C83D4
Quotients: C1, C2 [×7], C22 [×7], D4 [×6], C23, C2×D4 [×3], C41D4, C8⋊C22 [×2], C83D4

Character table of C83D4

 class 12A2B2C2D2E2F4A4B4C4D4E8A8B8C8D
 size 1111888224484444
ρ11111111111111111    trivial
ρ2111111-11111-1-1-1-1-1    linear of order 2
ρ31111-1-1-11111-11111    linear of order 2
ρ41111-1-1111111-1-1-1-1    linear of order 2
ρ511111-1-111-1-11-1-111    linear of order 2
ρ611111-1111-1-1-111-1-1    linear of order 2
ρ71111-11111-1-1-1-1-111    linear of order 2
ρ81111-11-111-1-1111-1-1    linear of order 2
ρ92-22-20002-2000-2200    orthogonal lifted from D4
ρ102-22-20002-20002-200    orthogonal lifted from D4
ρ112-22-2000-2200000-22    orthogonal lifted from D4
ρ122222000-2-22-200000    orthogonal lifted from D4
ρ132222000-2-2-2200000    orthogonal lifted from D4
ρ142-22-2000-22000002-2    orthogonal lifted from D4
ρ154-4-44000000000000    orthogonal lifted from C8⋊C22
ρ1644-4-4000000000000    orthogonal lifted from C8⋊C22

Smallest permutation representation of C83D4
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 21 31 10)(2 18 32 15)(3 23 25 12)(4 20 26 9)(5 17 27 14)(6 22 28 11)(7 19 29 16)(8 24 30 13)
(1 31)(2 30)(3 29)(4 28)(5 27)(6 26)(7 25)(8 32)(9 11)(12 16)(13 15)(18 24)(19 23)(20 22)

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

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

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

C83D4 is a maximal subgroup of
C42.531C23
 C42.D2p: C42.247D4  M4(2)⋊7D4  M4(2)⋊9D4  C42.255D4  C42.257D4  C42.271D4  C42.272D4  C42.275D4 ...
 (C2p×D8)⋊C2: C42.391C23  C42.72C23  C42.533C23  C2411D4  C4011D4  C5611D4 ...
 C8pD4⋊C2: C42.C23  C42.2C23  M4(2)⋊10D4  M4(2)⋊11D4  C42.406C23  C42.407C23  C42.410C23  D89D4 ...
 (C2×D4).D2p: C8⋊C4⋊C4  C42.3C23  C249D4  C409D4  C569D4 ...
C83D4 is a maximal quotient of
(C2×C4)⋊2D8  (C2×C8)⋊20D4
 C8⋊D4p: C83D8  C8⋊D12  C8⋊D20  C8⋊D28 ...
 C42.D2p: C42.110D4  C42.112D4  C42.64D6  C42.74D6  C42.64D10  C42.74D10  C42.64D14  C42.74D14 ...
 (C2×C8).D2p: C85SD16  C86SD16  C42.664C23  C42.666C23  C42.667C23  C83Q16  C42.26Q8  (C2×D8)⋊10C4 ...

Matrix representation of C83D4 in GL6(𝔽17)

0160000
100000
005544
00125134
0013131212
00413512
,
010000
1600000
000010
000001
001000
000100
,
100000
0160000
001000
0001600
000010
0000016

G:=sub<GL(6,GF(17))| [0,1,0,0,0,0,16,0,0,0,0,0,0,0,5,12,13,4,0,0,5,5,13,13,0,0,4,13,12,5,0,0,4,4,12,12],[0,16,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,1,0,0],[1,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,16] >;

C83D4 in GAP, Magma, Sage, TeX

C_8\rtimes_3D_4
% in TeX

G:=Group("C8:3D4");
// GroupNames label

G:=SmallGroup(64,177);
// by ID

G=gap.SmallGroup(64,177);
# by ID

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

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

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Character table of C83D4 in TeX

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