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G = C23.7Q8order 64 = 26

2nd non-split extension by C23 of Q8 acting via Q8/C4=C2

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

Aliases: C23.7Q8, C23.22D4, C23.56C23, C24.24C22, (C22×C4)⋊8C4, C221(C4⋊C4), C42(C22⋊C4), (C2×C4).115D4, (C23×C4).6C2, C22.9(C2×Q8), C2.1(C4⋊D4), C23.25(C2×C4), C22.30(C2×D4), C2.1(C22⋊Q8), C2.C421C2, C2.5(C42⋊C2), C22.15(C4○D4), C22.29(C22×C4), (C22×C4).87C22, (C2×C4⋊C4)⋊1C2, C2.4(C2×C4⋊C4), (C2×C4).69(C2×C4), C2.5(C2×C22⋊C4), (C2×C22⋊C4).3C2, SmallGroup(64,61)

Series: Derived Chief Lower central Upper central Jennings

C1C22 — C23.7Q8
C1C2C22C23C24C23×C4 — C23.7Q8
C1C22 — C23.7Q8
C1C23 — C23.7Q8
C1C23 — C23.7Q8

Generators and relations for C23.7Q8
 G = < a,b,c,d,e | a2=b2=c2=d4=1, e2=bd2, ab=ba, eae-1=ac=ca, ad=da, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede-1=d-1 >

Subgroups: 185 in 117 conjugacy classes, 57 normal (13 characteristic)
C1, C2 [×3], C2 [×4], C2 [×4], C4 [×4], C4 [×6], C22 [×3], C22 [×8], C22 [×12], C2×C4 [×8], C2×C4 [×22], C23, C23 [×6], C23 [×4], C22⋊C4 [×4], C4⋊C4 [×4], C22×C4 [×2], C22×C4 [×8], C22×C4 [×4], C24, C2.C42 [×2], C2×C22⋊C4 [×2], C2×C4⋊C4 [×2], C23×C4, C23.7Q8
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×6], Q8 [×2], C23, C22⋊C4 [×4], C4⋊C4 [×4], C22×C4, C2×D4 [×3], C2×Q8, C4○D4 [×2], C2×C22⋊C4, C2×C4⋊C4, C42⋊C2, C4⋊D4 [×2], C22⋊Q8 [×2], C23.7Q8

Character table of C23.7Q8

 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
ρ91-11-11-11-1-11-111-11-11-11-1-i-iii-iii-i    linear of order 4
ρ101-11-11-11-11-11-11-1-1-111-11i-i-ii-i-iii    linear of order 4
ρ111-11-11-11-1-11-111-11-11-11-1ii-i-ii-i-ii    linear of order 4
ρ121-11-11-11-11-11-11-1-1-111-11-iii-iii-i-i    linear of order 4
ρ131-11-11-11-1-11-11-11-11-11-11-ii-ii-ii-ii    linear of order 4
ρ141-11-11-11-11-11-1-1111-1-11-1iiii-i-i-i-i    linear of order 4
ρ151-11-11-11-1-11-11-11-11-11-11i-ii-ii-ii-i    linear of order 4
ρ161-11-11-11-11-11-1-1111-1-11-1-i-i-i-iiiii    linear of order 4
ρ1722-222-2-2-20000-220-2200000000000    orthogonal lifted from D4
ρ182-222-2-2-2222-2-20000000000000000    orthogonal lifted from D4
ρ192-2-2-222-220000220-2-200000000000    orthogonal lifted from D4
ρ2022-222-2-2-200002-202-200000000000    orthogonal lifted from D4
ρ212-222-2-2-22-2-2220000000000000000    orthogonal lifted from D4
ρ222-2-2-222-220000-2-202200000000000    orthogonal lifted from D4
ρ23222-2-22-2-2-222-20000000000000000    symplectic lifted from Q8, Schur index 2
ρ24222-2-22-2-22-2-220000000000000000    symplectic lifted from Q8, Schur index 2
ρ252-2-22-222-20000002i00-2i-2i2i00000000    complex lifted from C4○D4
ρ2622-2-2-2-2220000002i002i-2i-2i00000000    complex lifted from C4○D4
ρ2722-2-2-2-222000000-2i00-2i2i2i00000000    complex lifted from C4○D4
ρ282-2-22-222-2000000-2i002i2i-2i00000000    complex lifted from C4○D4

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

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

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

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

C23.7Q8 is a maximal subgroup of
C23.C42  C24.167C23  C24.175C23  C24.176C23  C25.85C22  C23.165C24  C23.167C24  C4×C4⋊D4  C4×C22⋊Q8  C24.542C23  C23.194C24  C23.195C24  C24.192C23  C24.545C23  C24.547C23  C23.201C24  C42.159D4  C4213D4  C24.198C23  C23.211C24  C24.204C23  D4×C22⋊C4  C24.549C23  Q8×C22⋊C4  C23.223C24  C23.224C24  C23.226C24  C23.227C24  C24.208C23  C23.229C24  D4×C4⋊C4  C23.234C24  C23.235C24  C23.236C24  C24.225C23  C23.259C24  C24.227C23  C24.244C23  C23.308C24  C23.309C24  C24.249C23  C23.315C24  C23.316C24  C24.252C23  C23.318C24  C24.563C23  C24.254C23  C23.321C24  C23.322C24  C23.323C24  C24.258C23  C23.327C24  C244Q8  C24.567C23  C24.267C23  C24.568C23  C24.268C23  C24.569C23  C23.350C24  C23.354C24  C24.278C23  C23.360C24  C23.367C24  C23.368C24  C24.572C23  C23.380C24  C24.573C23  C23.385C24  C24.299C23  C24.300C23  C24.301C23  C23.390C24  C23.391C24  C23.392C24  C24.577C23  C24.304C23  C23.395C24  C23.396C24  C23.397C24  C23.398C24  C23.405C24  C23.410C24  C24.311C23  C23.422C24  C23.430C24  C23.431C24  C23.434C24  C4217D4  C42.165D4  C4218D4  C42.166D4  C42.170D4  C23.449C24  C24.326C23  C42.172D4  C42.173D4  C24.583C23  C24.584C23  C24.338C23  C24.341C23  C23.478C24  C23.479C24  C24.360C23  C24.361C23  C2410D4  C24.587C23  C23.524C24  C23.525C24  C245Q8  C23.527C24  C42.187D4  C42.188D4  C23.530C24  C23.546C24  C23.559C24  C24.379C23  C23.567C24  C23.571C24  C23.572C24  C24.393C23  C24.394C23  C23.590C24  C23.591C24  C23.592C24  C23.593C24  C24.401C23  C23.595C24  C24.403C23  C23.602C24  C23.603C24  C24.408C23  C23.605C24  C23.606C24  C23.607C24  C23.608C24  C23.615C24  C23.617C24  C23.622C24  C23.637C24  C24.426C23  C24.427C23  C23.640C24  C23.641C24  C24.428C23  C23.643C24  C24.430C23  C23.645C24  C24.432C23  C23.647C24  C24.434C23  C24.435C23  C23.656C24  C23.668C24  C24.445C23  C23.678C24  C23.679C24  C24.448C23  C23.681C24  C24.450C23  C23.686C24  C23.687C24  C23.688C24  C24.459C23  C23.714C24  C23.741C24  C2413D4  C248Q8  C42.439D4  C24.598C23  C24.599C23  C42.440D4
 C24.D2p: C23.8D8  C23.30D8  C24.58D4  C24.60D4  C24.61D4  C23.35D8  C24.155D4  C24.65D4 ...
 D2p⋊(C4⋊C4): C23.231C24  D6⋊(C4⋊C4)  C4⋊(D6⋊C4)  D102(C4⋊C4)  D104(C4⋊C4)  D106(C4⋊C4)  D14⋊(C4⋊C4)  C4⋊(D14⋊C4) ...
C23.7Q8 is a maximal quotient of
C24.624C23  C24.625C23  C24.631C23  C42.425D4  C42.95D4  C24.167C23  C42.96D4  C42.97D4  C42.99D4  C42.100D4  C42.101D4  C24.19Q8  C24.9Q8  (C2×D4).24Q8  (C2×C8).103D4  C8○D4⋊C4  C4○D4.4Q8  C4○D4.5Q8
 C24.D2p: C24.17Q8  C24.5Q8  C24.133D4  C23.22D8  C24.67D4  C24.55D6  C24.75D6  C24.44D10 ...
 D2p⋊(C4⋊C4): C42.98D4  C42.102D4  D6⋊(C4⋊C4)  C4⋊(D6⋊C4)  D102(C4⋊C4)  D104(C4⋊C4)  D106(C4⋊C4)  D14⋊(C4⋊C4) ...

Matrix representation of C23.7Q8 in GL5(𝔽5)

10000
01000
00100
00010
00014
,
40000
04000
00400
00010
00001
,
10000
01000
00100
00040
00004
,
10000
01300
01400
00010
00001
,
20000
04000
04100
00013
00004

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

C23.7Q8 in GAP, Magma, Sage, TeX

C_2^3._7Q_8
% in TeX

G:=Group("C2^3.7Q8");
// GroupNames label

G:=SmallGroup(64,61);
// by ID

G=gap.SmallGroup(64,61);
# by ID

G:=PCGroup([6,-2,2,2,-2,2,2,192,121,55,362]);
// Polycyclic

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

Export

Character table of C23.7Q8 in TeX

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