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G = C23.4Q8order 64 = 26

4th non-split extension by C23 of Q8 acting via Q8/C2=C22

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

Aliases: C23.4Q8, C24.9C22, C23.82C23, (C2×C4).19D4, C2.4(C41D4), C22.75(C2×D4), C22.24(C2×Q8), C2.10(C22⋊Q8), C2.C4212C2, C22.42(C4○D4), (C22×C4).11C22, C2.9(C22.D4), (C2×C4⋊C4)⋊8C2, (C2×C22⋊C4).10C2, SmallGroup(64,80)

Series: Derived Chief Lower central Upper central Jennings

C1C23 — C23.4Q8
C1C2C22C23C24C2×C22⋊C4 — C23.4Q8
C1C23 — C23.4Q8
C1C23 — C23.4Q8
C1C23 — C23.4Q8

Generators and relations for C23.4Q8
 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=bd-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], C23.4Q8
Quotients: C1, C2 [×7], C22 [×7], D4 [×6], Q8 [×2], C23, C2×D4 [×3], C2×Q8, C4○D4 [×3], C22⋊Q8 [×3], C22.D4 [×3], C41D4, C23.4Q8

Character table of C23.4Q8

 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
ρ9222-2-2-22-2000-20000002000    orthogonal lifted from D4
ρ102-2-22-222-20000002000000-2    orthogonal lifted from D4
ρ112-2-22-222-2000000-20000002    orthogonal lifted from D4
ρ122-2-2-22-222002000000-20000    orthogonal lifted from D4
ρ13222-2-2-22-20002000000-2000    orthogonal lifted from D4
ρ142-2-2-22-22200-200000020000    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
ρ1722-2-222-2-200000002i-2i00000    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
ρ2022-2-222-2-20000000-2i2i00000    complex lifted from C4○D4
ρ2122-22-2-2-2200000-2i0000002i0    complex lifted from C4○D4
ρ222-22-2-22-2200002i000000-2i00    complex lifted from C4○D4

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

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

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

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

C23.4Q8 is a maximal subgroup of
C23.295C24  C42.162D4  C23.318C24  C23.323C24  C244Q8  C24.268C23  C24.269C23  C23.348C24  C23.349C24  C23.354C24  C24.276C23  C23.367C24  C24.290C23  C23.379C24  C23.382C24  C24.300C23  C23.398C24  C23.401C24  C23.416C24  C23.417C24  C23.419C24  C24.311C23  C23.422C24  C23.439C24  C24.326C23  C23.456C24  C24.583C23  C24.339C23  C42.178D4  C24.346C23  C23.496C24  C4222D4  C42.184D4  C24.355C23  C42.185D4  C24.589C23  C23.524C24  C245Q8  C42.187D4  C42.189D4  C42.190D4  C23.535C24  C4230D4  C24.375C23  C23.551C24  C23.554C24  C23.556C24  C24.378C23  C42.198D4  C23.567C24  C23.568C24  C23.571C24  C23.572C24  C23.574C24  C24.385C23  C23.585C24  C23.592C24  C24.401C23  C23.605C24  C23.606C24  C24.412C23  C23.611C24  C23.618C24  C23.620C24  C23.621C24  C24.418C23  C23.627C24  C23.630C24  C23.632C24  C23.635C24  C24.426C23  C23.640C24  C23.641C24  C24.428C23  C23.643C24  C24.432C23  C24.434C23  C23.652C24  C23.654C24  C23.668C24  C23.671C24  C23.673C24  C23.677C24  C24.448C23  C23.696C24  C23.698C24  C23.701C24  C23.707C24  C24.459C23  C23.716C24  C42.199D4  C42.200D4  C23.726C24  C23.727C24  C23.729C24  C23.734C24  C23.736C24  C23.737C24  C23.738C24  C23.741C24  C24.15Q8  C24.3A4
 C24.D2p: C24.4D4  C24.16D4  C24.17D4  C24.18D4  C24.18D6  C24.7D10  C24.7D14 ...
 C2p.(C41D4): C4216D4  C4219D4  C42.167D4  C42.196D4  (C2×C12).33D4  (C2×C12).290D4  (C2×C20).33D4  (C2×C20).289D4 ...
C23.4Q8 is a maximal quotient of
C24.5Q8  C24.634C23  C24.635C23
 (C2×C4p).D4: C24.11Q8  (C2×C8).168D4  (C2×C4).27D8  (C2×C8).169D4  (C2×C8).60D4  (C2×C8).170D4  (C2×C8).171D4  C42.10D4 ...

Matrix representation of C23.4Q8 in GL6(𝔽5)

100000
040000
001000
002400
000010
000004
,
400000
040000
004000
000400
000040
000004
,
400000
040000
001000
000100
000040
000004
,
010000
100000
002000
000200
000001
000040
,
040000
100000
003200
001200
000002
000030

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

C23.4Q8 in GAP, Magma, Sage, TeX

C_2^3._4Q_8
% in TeX

G:=Group("C2^3.4Q8");
// GroupNames label

G:=SmallGroup(64,80);
// by ID

G=gap.SmallGroup(64,80);
# by ID

G:=PCGroup([6,-2,2,2,-2,2,2,48,121,151,362,332]);
// 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=b*d^-1>;
// generators/relations

Export

Character table of C23.4Q8 in TeX

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