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G = C23⋊Q8order 64 = 26

1st semidirect product of C23 and Q8 acting via Q8/C2=C22

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

Aliases: C231Q8, C24.5C22, C23.76C23, (C2×C4).14D4, C2.5C22≀C2, (C22×Q8)⋊1C2, C22.69(C2×D4), C2.6(C22⋊Q8), C22.20(C2×Q8), C2.5(C4.4D4), C2.C4210C2, C22.36(C4○D4), (C22×C4).25C22, (C2×C22⋊C4).7C2, SmallGroup(64,74)

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=d2, eae-1=ab=ba, dad-1=ac=ca, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede-1=d-1 >

Subgroups: 181 in 101 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], Q8 [×8], C23, C23 [×2], C23 [×6], C22⋊C4 [×6], C22×C4 [×6], C2×Q8 [×6], C24, C2.C42 [×3], C2×C22⋊C4 [×3], C22×Q8, C23⋊Q8
Quotients: C1, C2 [×7], C22 [×7], D4 [×6], Q8 [×2], C23, C2×D4 [×3], C2×Q8, C4○D4 [×3], C22≀C2, C22⋊Q8 [×3], C4.4D4 [×3], 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
ρ1022-2-222-2-200000002-200000    orthogonal lifted from D4
ρ1122-22-2-2-22000002000000-20    orthogonal lifted from D4
ρ1222-22-2-2-2200000-200000020    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
ρ172-2-22-222-20000002i000000-2i    complex lifted from C4○D4
ρ182-2-2-22-222002i000000-2i0000    complex lifted from C4○D4
ρ192-22-2-22-220000-2i0000002i00    complex lifted from C4○D4
ρ202-2-22-222-2000000-2i0000002i    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 24)(4 22)(5 32)(6 16)(7 30)(8 14)(9 15)(10 29)(11 13)(12 31)(18 28)(20 26)
(1 25)(2 26)(3 27)(4 28)(5 15)(6 16)(7 13)(8 14)(9 32)(10 29)(11 30)(12 31)(17 21)(18 22)(19 23)(20 24)
(1 23)(2 24)(3 21)(4 22)(5 9)(6 10)(7 11)(8 12)(13 30)(14 31)(15 32)(16 29)(17 27)(18 28)(19 25)(20 26)
(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 10 3 12)(2 9 4 11)(5 22 7 24)(6 21 8 23)(13 20 15 18)(14 19 16 17)(25 29 27 31)(26 32 28 30)

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

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

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

C23⋊Q8 is a maximal subgroup of
C42.162D4  C42.163D4  C23.304C24  C24.262C23  C244Q8  C24.267C23  C23.350C24  C23.352C24  C24.279C23  C23.359C24  C24.283C23  C24.285C23  C23.369C24  C23.372C24  C23.377C24  C23.388C24  C23.391C24  C23.392C24  C24.313C23  C24.315C23  C23.432C24  C23.457C24  C24.332C23  C23.461C24  C42.173D4  C24.583C23  C23.472C24  C24.338C23  C42.178D4  C24.346C23  C42.182D4  C42.183D4  C42.184D4  C24.355C23  C4226D4  C23.514C24  C245Q8  C42.187D4  C4229D4  C42.189D4  C42.192D4  C24.374C23  C24.592C23  C42.193D4  C23.550C24  C23.553C24  C4232D4  C24.378C23  C24.379C23  C23.570C24  C23.574C24  C23.576C24  C24.385C23  C23.581C24  C23.584C24  C24.393C23  C24.394C23  C23.592C24  C24.403C23  C23.600C24  C23.602C24  C24.412C23  C23.612C24  C23.616C24  C24.418C23  C24.421C23  C23.631C24  C23.636C24  C23.637C24  C24.428C23  C23.645C24  C23.651C24  C23.654C24  C23.659C24  C23.660C24  C23.663C24  C23.664C24  C23.675C24  C24.450C23  C23.685C24  C23.688C24  C23.698C24  C24.456C23  C23.708C24  C23.714C24  C23.716C24  C4234D4  C42.200D4  C23.724C24  C23.730C24  C23.731C24  C23.732C24  C23.735C24  C23.738C24  C246Q8  C23.741C24  C24.A4
 C24.D2p: C23.Q16  C232SD16  C23⋊Q16  C24.12D4  C232Dic6  C23⋊Dic10  C23⋊Dic14 ...
 C2p.C22≀C2: C23.288C24  C23.309C24  C24.565C23  C24.360C23  (C22×S3)⋊Q8  (C22×Q8)⋊9S3  (C2×C4).20D20  (C22×D5)⋊Q8 ...
C23⋊Q8 is a maximal quotient of
C42.8D4  C24⋊Q8
 (C2×C4).D4p: (C2×D4)⋊Q8  (C22×S3)⋊Q8  (C2×C4).20D20  (C2×C4).20D28 ...
 C24.D2p: C24.5Q8  C232Dic6  C23⋊Dic10  C23⋊Dic14 ...
 (C22×C4).D2p: C24.632C23  C24.636C23  (C2×Q8)⋊Q8  C4⋊C4.84D4  C4⋊C4.85D4  (C2×D4)⋊2Q8  (C2×Q8)⋊2Q8  C242Q8 ...

Matrix representation of C23⋊Q8 in GL6(𝔽5)

100000
040000
001000
000400
000010
000024
,
400000
040000
004000
000400
000010
000001
,
100000
010000
001000
000100
000040
000004
,
100000
010000
002000
000300
000023
000003
,
010000
100000
000100
004000
000020
000043

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

C23⋊Q8 in GAP, Magma, Sage, TeX

C_2^3\rtimes Q_8
% in TeX

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

G:=SmallGroup(64,74);
// by ID

G=gap.SmallGroup(64,74);
# by ID

G:=PCGroup([6,-2,2,2,-2,2,2,48,121,55,362,332]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^2=c^2=d^4=1,e^2=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

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Character table of C23⋊Q8 in TeX

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