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G = C23.8Q8order 64 = 26

3rd non-split extension by C23 of Q8 acting via Q8/C4=C2

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

Aliases: C23.8Q8, C23.35D4, C24.26C22, C23.61C23, C2.4(C4×D4), C22⋊C44C4, (C2×C4).98D4, C222(C4⋊C4), C2.2C22≀C2, (C23×C4).4C2, C23.14(C2×C4), C22.34(C2×D4), C2.2(C22⋊Q8), C22.12(C2×Q8), C2.C427C2, (C22×C4).4C22, C22.19(C4○D4), C22.34(C22×C4), C2.2(C22.D4), (C2×C4⋊C4)⋊2C2, (C2×C4)⋊2(C2×C4), C2.7(C2×C4⋊C4), (C2×C22⋊C4).5C2, SmallGroup(64,66)

Series: Derived Chief Lower central Upper central Jennings

C1C22 — C23.8Q8
C1C2C22C23C24C23×C4 — C23.8Q8
C1C22 — C23.8Q8
C1C23 — C23.8Q8
C1C23 — C23.8Q8

Generators and relations for C23.8Q8
 G = < a,b,c,d,e | a2=b2=c2=d4=1, e2=d2, ab=ba, eae-1=ac=ca, ad=da, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede-1=bd-1 >

Subgroups: 185 in 117 conjugacy classes, 53 normal (15 characteristic)
C1, C2 [×3], C2 [×4], C2 [×4], C4 [×10], C22 [×3], C22 [×8], C22 [×12], C2×C4 [×8], C2×C4 [×22], C23, C23 [×6], C23 [×4], C22⋊C4 [×4], C22⋊C4 [×2], C4⋊C4 [×4], C22×C4 [×2], C22×C4 [×4], C22×C4 [×6], C24, C2.C42 [×2], C2×C22⋊C4 [×2], C2×C4⋊C4 [×2], C23×C4, C23.8Q8
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×6], Q8 [×2], C23, C4⋊C4 [×4], C22×C4, C2×D4 [×3], C2×Q8, C4○D4 [×2], C2×C4⋊C4, C4×D4 [×2], C22≀C2, C22⋊Q8 [×2], C22.D4, C23.8Q8

Character table of C23.8Q8

 class 12A2B2C2D2E2F2G2H2I2J2K4A4B4C4D4E4F4G4H4I4J4K4L4M4N4O4P
 size 1111111122222222222244444444
ρ11111111111111111111111111111    trivial
ρ211111111-1-1-1-111-111-1-1-1-11-111-11-1    linear of order 2
ρ311111111-1-1-1-111-111-1-1-11-11-1-11-11    linear of order 2
ρ411111111111111111111-1-1-1-1-1-1-1-1    linear of order 2
ρ511111111-1-1-1-1-1-11-1-1111-1-1111-1-11    linear of order 2
ρ6111111111111-1-1-1-1-1-1-1-11-1-1111-1-1    linear of order 2
ρ7111111111111-1-1-1-1-1-1-1-1-111-1-1-111    linear of order 2
ρ811111111-1-1-1-1-1-11-1-111111-1-1-111-1    linear of order 2
ρ91-11-11-11-1-11-11i-ii-ii-ii-i-i1-1i-ii-11    linear of order 4
ρ101-11-11-11-1-11-11-ii-ii-ii-ii-i-11i-ii1-1    linear of order 4
ρ111-11-11-11-1-11-11i-ii-ii-ii-ii-11-ii-i1-1    linear of order 4
ρ121-11-11-11-1-11-11-ii-ii-ii-iii1-1-ii-i-11    linear of order 4
ρ131-11-11-11-11-11-1i-i-i-iii-iii11i-i-i-1-1    linear of order 4
ρ141-11-11-11-11-11-1-iiii-i-ii-ii-1-1i-i-i11    linear of order 4
ρ151-11-11-11-11-11-1i-i-i-iii-ii-i-1-1-iii11    linear of order 4
ρ161-11-11-11-11-11-1-iiii-i-ii-i-i11-iii-1-1    linear of order 4
ρ172-222-2-2-2222-2-20000000000000000    orthogonal lifted from D4
ρ182-2-2-222-220000-2-202200000000000    orthogonal lifted from D4
ρ1922-2-2-2-222000000-200-22200000000    orthogonal lifted from D4
ρ202-222-2-2-22-2-2220000000000000000    orthogonal lifted from D4
ρ212-2-2-222-220000220-2-200000000000    orthogonal lifted from D4
ρ2222-2-2-2-2220000002002-2-200000000    orthogonal lifted from D4
ρ23222-2-22-2-22-2-220000000000000000    symplectic lifted from Q8, Schur index 2
ρ24222-2-22-2-2-222-20000000000000000    symplectic lifted from Q8, Schur index 2
ρ2522-222-2-2-20000-2i2i0-2i2i00000000000    complex lifted from C4○D4
ρ262-2-22-222-20000002i00-2i-2i2i00000000    complex lifted from C4○D4
ρ2722-222-2-2-200002i-2i02i-2i00000000000    complex lifted from C4○D4
ρ282-2-22-222-2000000-2i002i2i-2i00000000    complex lifted from C4○D4

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

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

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

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

C23.8Q8 is a maximal subgroup of
C23.2M4(2)  C24.182C23  C4242D4  C23.178C24  C4×C22≀C2  C4×C22.D4  C4×C22⋊Q8  C23.199C24  C24.547C23  C23.203C24  C24.195C23  C4213D4  C24.198C23  C42.160D4  C23.211C24  C24.203C23  C24.204C23  C24.205C23  C23.224C24  C23.225C24  D4×C4⋊C4  C23.240C24  C23.241C24  C24.558C23  C24.215C23  C24.220C23  C23.250C24  C23.255C24  C24.225C23  C23.259C24  C23.261C24  C24.230C23  C24.243C23  C24.244C23  C23.309C24  C248D4  C23.311C24  C23.313C24  C23.318C24  C24.563C23  C23.322C24  C23.324C24  C24.258C23  C24.262C23  C24.264C23  C23.334C24  C244Q8  C24.567C23  C24.568C23  C24.569C23  C24.269C23  C23.344C24  C24.271C23  C23.349C24  C23.350C24  C23.352C24  C23.354C24  C24.276C23  C23.356C24  C24.279C23  C23.360C24  C24.282C23  C24.283C23  C23.364C24  C24.285C23  C24.286C23  C23.368C24  C24.289C23  C24.572C23  C23.374C24  C23.375C24  C24.293C23  C23.377C24  C24.295C23  C23.379C24  C23.380C24  C24.573C23  C23.382C24  C24.576C23  C24.300C23  C24.577C23  C23.402C24  C23.405C24  C23.410C24  C24.309C23  C23.417C24  C23.422C24  C24.313C23  C24.315C23  C23.430C24  C23.449C24  C24.326C23  C23.456C24  C23.458C24  C24.331C23  C24.583C23  C42.175D4  C24.584C23  C23.473C24  C24.338C23  C24.339C23  C24.340C23  C24.341C23  C23.478C24  C23.479C24  C42.178D4  C23.483C24  C24.345C23  C24.346C23  C24.347C23  C24.348C23  C4222D4  C42.183D4  C23.500C24  C4223D4  C23.502C24  C4224D4  C4225D4  C42.185D4  C24.589C23  C245Q8  C23.527C24  C23.530C24  C42.190D4  C4230D4  C24.374C23  C23.543C24  C23.546C24  C24.375C23  C24.376C23  C23.553C24  C23.571C24  C23.572C24  C23.574C24  C24.384C23  C23.576C24  C24.385C23  C23.580C24  C23.581C24  C24.389C23  C24.393C23  C24.394C23  C24.395C23  C23.589C24  C23.590C24  C23.591C24  C23.592C24  C23.593C24  C24.401C23  C23.595C24  C24.405C23  C24.407C23  C24.408C23  C23.606C24  C23.607C24  C23.608C24  C24.412C23  C23.611C24  C23.612C24  C24.413C23  C23.615C24  C23.617C24  C23.618C24  C23.620C24  C23.622C24  C24.418C23  C23.624C24  C24.420C23  C24.421C23  C23.632C24  C24.426C23  C24.427C23  C24.428C23  C23.643C24  C24.430C23  C23.645C24  C24.432C23  C23.647C24  C24.434C23  C23.649C24  C24.435C23  C23.651C24  C24.437C23  C24.438C23  C24.440C23  C23.663C24  C23.664C24  C24.443C23  C23.668C24  C24.445C23  C23.671C24  C23.678C24  C23.679C24  C24.448C23  C23.681C24  C23.682C24  C24.450C23  C23.686C24  C23.687C24  C23.688C24  C24.454C23  C23.696C24  C23.697C24  C24.456C23  C23.707C24  C23.724C24  C23.726C24  C23.727C24  C23.734C24  C23.735C24  C248Q8  C42.439D4  C4243D4  C23.753C24  C24.599C23
 C24.D2p: C24.22D4  C24.33D4  C24.91D4  C24.94D4  C24.95D4  C24.96D4  C24.97D4  C24.166D4 ...
 D2p⋊(C4⋊C4): C23.231C24  D6⋊C4⋊C4  D6⋊C46C4  D103(C4⋊C4)  D105(C4⋊C4)  D10⋊(C4⋊C4)  D14⋊C4⋊C4  D14⋊C46C4 ...
C23.8Q8 is a maximal quotient of
C24.626C23  C24.632C23  C24.634C23  C23.21M4(2)  (C2×C8).195D4  C24.10Q8  C4.10D42C4  M4(2).40D4  C429(C2×C4)  M4(2).41D4  Q8⋊(C4⋊C4)  Q8⋊C4⋊C4  M4(2).42D4  (C2×D4).Q8
 C24.D2p: C24.17Q8  C24.5Q8  C23.37D8  C24.159D4  C24.71D4  C24.21D4  C24.22D4  C24.57D6 ...
 D2p⋊(C4⋊C4): C4≀C2⋊C4  C2.(C4×D8)  D4⋊(C4⋊C4)  D6⋊C4⋊C4  D6⋊C46C4  D103(C4⋊C4)  D105(C4⋊C4)  D10⋊(C4⋊C4) ...

Matrix representation of C23.8Q8 in GL5(𝔽5)

10000
01000
00400
00040
00004
,
40000
04000
00400
00010
00001
,
10000
04000
00400
00010
00001
,
20000
04000
00100
00023
00003
,
30000
00100
01000
00041
00031

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

C23.8Q8 in GAP, Magma, Sage, TeX

C_2^3._8Q_8
% in TeX

G:=Group("C2^3.8Q8");
// GroupNames label

G:=SmallGroup(64,66);
// by ID

G=gap.SmallGroup(64,66);
# by ID

G:=PCGroup([6,-2,2,2,-2,2,2,192,121,199,362]);
// 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,e*a*e^-1=a*c=c*a,a*d=d*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 C23.8Q8 in TeX

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