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G = C23.Q8order 64 = 26

3rd non-split extension by C23 of Q8 acting via Q8/C2=C22

p-group, metabelian, nilpotent (class 2), monomial

Aliases: C23.3Q8, C24.7C22, C23.79C23, (C2×C4).16D4, C2.8(C4⋊D4), C22.72(C2×D4), C2.8(C22⋊Q8), C22.22(C2×Q8), C2.C424C2, (C22×C4).8C22, C2.4(C422C2), C22.39(C4○D4), (C2×C4⋊C4)⋊6C2, (C2×C22⋊C4).8C2, SmallGroup(64,77)

Series: Derived Chief Lower central Upper central Jennings

C1C23 — C23.Q8
C1C2C22C23C24C2×C22⋊C4 — C23.Q8
C1C23 — C23.Q8
C1C23 — C23.Q8
C1C23 — C23.Q8

Generators and relations for C23.Q8
 G = < a,b,c,d,e | a2=b2=c2=d4=1, e2=bd2, eae-1=ab=ba, dad-1=ac=ca, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede-1=d-1 >

Subgroups: 165 in 93 conjugacy classes, 39 normal (7 characteristic)
C1, C2, C2 [×6], C2 [×2], C4 [×9], C22, C22 [×6], C22 [×10], C2×C4 [×6], C2×C4 [×15], C23, C23 [×2], C23 [×6], C22⋊C4 [×6], C4⋊C4 [×6], C22×C4 [×6], C24, C2.C42, C2×C22⋊C4 [×3], C2×C4⋊C4 [×3], C23.Q8
Quotients: C1, C2 [×7], C22 [×7], D4 [×6], Q8 [×2], C23, C2×D4 [×3], C2×Q8, C4○D4 [×3], C4⋊D4 [×3], C22⋊Q8 [×3], C422C2, C23.Q8

Character table of C23.Q8

 class 12A2B2C2D2E2F2G2H2I4A4B4C4D4E4F4G4H4I4J4K4L
 size 1111111144444444444444
ρ11111111111111111111111    trivial
ρ211111111-1-11-111-1-1-11-111-1    linear of order 2
ρ31111111111-1-1-111-1-1-1-1-111    linear of order 2
ρ411111111-1-1-11-11-111-11-11-1    linear of order 2
ρ511111111111-1-1-1-1111-1-1-1-1    linear of order 2
ρ611111111-1-111-1-11-1-111-1-11    linear of order 2
ρ71111111111-111-1-1-1-1-111-1-1    linear of order 2
ρ811111111-1-1-1-11-1111-1-11-11    linear of order 2
ρ922-2-222-2-20000000-2200000    orthogonal lifted from D4
ρ102-2-22-222-20000002000000-2    orthogonal lifted from D4
ρ1122-2-222-2-200000002-200000    orthogonal lifted from D4
ρ122-2-22-222-2000000-20000002    orthogonal lifted from D4
ρ13222-2-2-22-2000-20000002000    orthogonal lifted from D4
ρ14222-2-2-22-20002000000-2000    orthogonal lifted from D4
ρ152-2222-2-2-2-22000000000000    symplectic lifted from Q8, Schur index 2
ρ162-2222-2-2-22-2000000000000    symplectic lifted from Q8, Schur index 2
ρ172-2-2-22-222002i000000-2i0000    complex lifted from C4○D4
ρ182-22-2-22-220000-2i0000002i00    complex lifted from C4○D4
ρ1922-22-2-2-22000002i000000-2i0    complex lifted from C4○D4
ρ2022-22-2-2-2200000-2i0000002i0    complex lifted from C4○D4
ρ212-22-2-22-2200002i000000-2i00    complex lifted from C4○D4
ρ222-2-2-22-22200-2i0000002i0000    complex lifted from C4○D4

Smallest permutation representation of C23.Q8
On 32 points
Generators in S32
(2 26)(4 28)(5 32)(6 16)(7 30)(8 14)(9 15)(10 29)(11 13)(12 31)(18 24)(20 22)
(1 21)(2 22)(3 23)(4 24)(5 15)(6 16)(7 13)(8 14)(9 32)(10 29)(11 30)(12 31)(17 27)(18 28)(19 25)(20 26)
(1 25)(2 26)(3 27)(4 28)(5 9)(6 10)(7 11)(8 12)(13 30)(14 31)(15 32)(16 29)(17 23)(18 24)(19 21)(20 22)
(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 29 23 12)(2 32 24 11)(3 31 21 10)(4 30 22 9)(5 28 13 20)(6 27 14 19)(7 26 15 18)(8 25 16 17)

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

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

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

C23.Q8 is a maximal subgroup of
C42.162D4  C23.301C24  C23.311C24  C23.313C24  C23.316C24  C23.324C24  C24.258C23  C244Q8  C24.267C23  C24.268C23  C23.349C24  C23.350C24  C23.352C24  C23.354C24  C23.356C24  C24.278C23  C23.360C24  C24.282C23  C24.285C23  C24.286C23  C23.369C24  C24.289C23  C24.290C23  C23.374C24  C23.375C24  C23.380C24  C24.573C23  C23.397C24  C23.412C24  C24.309C23  C23.418C24  C23.419C24  C23.422C24  C23.425C24  C23.426C24  C23.429C24  C23.434C24  C24.327C23  C23.456C24  C23.458C24  C42.175D4  C23.473C24  C24.338C23  C24.340C23  C23.479C24  C24.345C23  C23.494C24  C4223D4  C23.508C24  C4225D4  C24.587C23  C245Q8  C42.188D4  C23.530C24  C4230D4  C42.192D4  C23.543C24  C23.544C24  C23.548C24  C24.375C23  C23.550C24  C23.551C24  C24.376C23  C23.554C24  C4232D4  C42.198D4  C24.379C23  C23.567C24  C23.572C24  C23.574C24  C24.384C23  C23.576C24  C23.578C24  C23.581C24  C24.389C23  C23.583C24  C23.585C24  C23.591C24  C23.592C24  C23.593C24  C24.401C23  C23.595C24  C23.597C24  C23.600C24  C24.407C23  C23.602C24  C24.408C23  C23.605C24  C23.606C24  C23.607C24  C23.611C24  C23.615C24  C23.616C24  C23.618C24  C23.620C24  C23.621C24  C23.625C24  C23.627C24  C24.421C23  C23.630C24  C23.631C24  C23.632C24  C23.635C24  C24.427C23  C23.640C24  C23.643C24  C24.430C23  C24.434C23  C23.649C24  C24.435C23  C24.438C23  C24.440C23  C24.443C23  C23.668C24  C23.672C24  C23.675C24  C23.677C24  C23.679C24  C24.448C23  C23.681C24  C23.683C24  C24.450C23  C23.686C24  C23.688C24  C24.454C23  C23.693C24  C23.695C24  C23.700C24  C23.701C24  C24.459C23  C23.714C24  C42.199D4  C4235D4  C23.725C24  C23.726C24  C23.727C24  C23.729C24  C23.730C24  C23.731C24  C23.736C24  C23.737C24  C23.738C24  C23.741C24  C24.15Q8  C24.2A4
 C2p.(C4⋊D4): C4215D4  C24.252C23  C24.259C23  C23.329C24  C42.166D4  C42.168D4  C4228D4  (C22×C4).37D6 ...
C23.Q8 is a maximal quotient of
C24.631C23  C24.633C23
 C24.D2p: C24.5Q8  C24.17D6  C24.6D10  C24.6D14 ...
 (C2×C4p).D4: M4(2).12D4  M4(2).13D4  C4⋊C4.106D4  (C2×Q8).8Q8  (C2×C4).23D8  (C2×C8).52D4  (C2×C12).56D4  (C2×C20).56D4 ...
 C2p.(C4⋊D4): C24.Q8  M4(2).15D4  (C22×C4).37D6  (C22×D5).Q8  (C22×D7).Q8 ...

Matrix representation of C23.Q8 in GL6(𝔽5)

100000
040000
001000
000400
000010
000004
,
100000
010000
004000
000400
000040
000004
,
400000
040000
004000
000400
000040
000004
,
010000
400000
000100
004000
000001
000010
,
300000
020000
000400
004000
000002
000020

G:=sub<GL(6,GF(5))| [1,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,4],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[0,4,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[3,0,0,0,0,0,0,2,0,0,0,0,0,0,0,4,0,0,0,0,4,0,0,0,0,0,0,0,0,2,0,0,0,0,2,0] >;

C23.Q8 in GAP, Magma, Sage, TeX

C_2^3.Q_8
% in TeX

G:=Group("C2^3.Q8");
// GroupNames label

G:=SmallGroup(64,77);
// by ID

G=gap.SmallGroup(64,77);
# by ID

G:=PCGroup([6,-2,2,2,-2,2,2,144,121,55,362,332]);
// Polycyclic

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

Export

Character table of C23.Q8 in TeX

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