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G = C232Q8order 64 = 26

2nd semidirect product of C23 and Q8 acting via Q8/C2=C22

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

Aliases: C232Q8, C24.18C22, C23.40C23, C22.37C24, C2.102+ 1+4, C4⋊C44C22, C22⋊Q810C2, (C2×Q8)⋊4C22, C22.4(C2×Q8), C2.7(C22×Q8), (C2×C4).24C23, C22⋊C4.18C22, (C22×C4).66C22, (C2×C22⋊C4).13C2, SmallGroup(64,224)

Series: Derived Chief Lower central Upper central Jennings

C1C22 — C232Q8
C1C2C22C23C22×C4C2×C22⋊C4 — C232Q8
C1C22 — C232Q8
C1C22 — C232Q8
C1C22 — C232Q8

Generators and relations for C232Q8
 G = < a,b,c,d,e | a2=b2=c2=d4=1, e2=d2, ab=ba, dad-1=eae-1=ac=ca, ebe-1=bc=cb, bd=db, cd=dc, ce=ec, ede-1=d-1 >

Subgroups: 193 in 121 conjugacy classes, 81 normal (5 characteristic)
C1, C2, C2 [×2], C2 [×6], C4 [×12], C22, C22 [×6], C22 [×10], C2×C4 [×12], C2×C4 [×6], Q8 [×4], C23 [×7], C23 [×2], C22⋊C4 [×12], C4⋊C4 [×12], C22×C4 [×6], C2×Q8 [×4], C24, C2×C22⋊C4 [×3], C22⋊Q8 [×12], C232Q8
Quotients: C1, C2 [×15], C22 [×35], Q8 [×4], C23 [×15], C2×Q8 [×6], C24, C22×Q8, 2+ 1+4 [×2], C232Q8

Character table of C232Q8

 class 12A2B2C2D2E2F2G2H2I4A4B4C4D4E4F4G4H4I4J4K4L
 size 1111222222444444444444
ρ11111111111111111111111    trivial
ρ211111-11-1-1-1-111-11-11-1-11-11    linear of order 2
ρ311111-11-1-1-11-11-1-11-111-1-11    linear of order 2
ρ41111111111-1-111-1-1-1-1-1-111    linear of order 2
ρ51111-11-1-11-1-11-11-111-11-1-11    linear of order 2
ρ61111-1-1-11-1111-1-1-1-111-1-111    linear of order 2
ρ71111-1-1-11-11-1-1-1-111-1-11111    linear of order 2
ρ81111-11-1-11-11-1-111-1-11-11-11    linear of order 2
ρ911111-11-1-1-1-11-111-1-111-11-1    linear of order 2
ρ10111111111111-1-111-1-1-1-1-1-1    linear of order 2
ρ111111111111-1-1-1-1-1-11111-1-1    linear of order 2
ρ1211111-11-1-1-11-1-11-111-1-111-1    linear of order 2
ρ131111-1-1-11-111111-1-1-1-111-1-1    linear of order 2
ρ141111-11-1-11-1-111-1-11-11-111-1    linear of order 2
ρ151111-11-1-11-11-11-11-11-11-11-1    linear of order 2
ρ161111-1-1-11-11-1-1111111-1-1-1-1    linear of order 2
ρ172-22-2-2-2222-2000000000000    symplectic lifted from Q8, Schur index 2
ρ182-22-22-2-2-222000000000000    symplectic lifted from Q8, Schur index 2
ρ192-22-222-22-2-2000000000000    symplectic lifted from Q8, Schur index 2
ρ202-22-2-222-2-22000000000000    symplectic lifted from Q8, Schur index 2
ρ2144-4-4000000000000000000    orthogonal lifted from 2+ 1+4
ρ224-4-44000000000000000000    orthogonal lifted from 2+ 1+4

Permutation representations of C232Q8
On 16 points - transitive group 16T97
Generators in S16
(2 11)(4 9)(5 14)(7 16)
(5 14)(6 15)(7 16)(8 13)
(1 10)(2 11)(3 12)(4 9)(5 14)(6 15)(7 16)(8 13)
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)
(1 14 3 16)(2 13 4 15)(5 12 7 10)(6 11 8 9)

G:=sub<Sym(16)| (2,11)(4,9)(5,14)(7,16), (5,14)(6,15)(7,16)(8,13), (1,10)(2,11)(3,12)(4,9)(5,14)(6,15)(7,16)(8,13), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16), (1,14,3,16)(2,13,4,15)(5,12,7,10)(6,11,8,9)>;

G:=Group( (2,11)(4,9)(5,14)(7,16), (5,14)(6,15)(7,16)(8,13), (1,10)(2,11)(3,12)(4,9)(5,14)(6,15)(7,16)(8,13), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16), (1,14,3,16)(2,13,4,15)(5,12,7,10)(6,11,8,9) );

G=PermutationGroup([(2,11),(4,9),(5,14),(7,16)], [(5,14),(6,15),(7,16),(8,13)], [(1,10),(2,11),(3,12),(4,9),(5,14),(6,15),(7,16),(8,13)], [(1,2,3,4),(5,6,7,8),(9,10,11,12),(13,14,15,16)], [(1,14,3,16),(2,13,4,15),(5,12,7,10),(6,11,8,9)])

G:=TransitiveGroup(16,97);

C232Q8 is a maximal subgroup of
C23⋊SD16  C23⋊Q16  C24.14D4  C24.17D4  C242Q8  C24.180C23  C24⋊Q8  C24.182C23  C22.47C25  C22.79C25  C22.80C25  C22.124C25  C22.153C25  C24.7A4
 C2p.2+ 1+4: C22.48C25  C22.90C25  C22.125C25  C22.150C25  C233Dic6  C6.512+ 1+4  C232Dic10  C10.512+ 1+4 ...
C232Q8 is a maximal quotient of
C23.211C24  C244Q8  C23.449C24  C24.584C23  C24.355C23  C23.508C24  C245Q8  C23.546C24  C23.559C24  C24.379C23  C23.567C24  C23.632C24  C24.434C23  C24.448C23  C24.450C23  C23.699C24  C24.456C23  C23.706C24  C23.709C24  C23.711C24  C246Q8  C23.741C24  C24.15Q8
 C24.D2p: C24.91D4  C233Dic6  C232Dic10  C232Dic14 ...
 C4⋊C4⋊D2p: C23.583C24  C23.592C24  C6.512+ 1+4  C10.512+ 1+4  C14.512+ 1+4 ...

Matrix representation of C232Q8 in GL6(𝔽5)

100000
010000
001000
000400
000040
000001
,
100000
010000
001000
000100
000040
000004
,
100000
010000
004000
000400
000040
000004
,
040000
100000
000100
001000
000001
000010
,
030000
300000
000010
000001
001000
000100

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

C232Q8 in GAP, Magma, Sage, TeX

C_2^3\rtimes_2Q_8
% in TeX

G:=Group("C2^3:2Q8");
// GroupNames label

G:=SmallGroup(64,224);
// by ID

G=gap.SmallGroup(64,224);
# by ID

G:=PCGroup([6,-2,2,2,2,-2,2,192,217,650,188,158,579]);
// Polycyclic

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

Export

Character table of C232Q8 in TeX

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