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G = C23.11D4order 64 = 26

4th non-split extension by C23 of D4 acting via D4/C2=C22

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

Aliases: C23.11D4, C24.8C22, C23.80C23, (C2×C4).17D4, C2.9(C4⋊D4), C22.73(C2×D4), C2.C425C2, C2.7(C4.4D4), (C22×C4).9C22, C2.5(C422C2), C22.40(C4○D4), C2.7(C22.D4), (C2×C4⋊C4)⋊7C2, (C2×C22⋊C4).9C2, SmallGroup(64,78)

Series: Derived Chief Lower central Upper central Jennings

C1C23 — C23.11D4
C1C2C22C23C22×C4C2×C22⋊C4 — C23.11D4
C1C23 — C23.11D4
C1C23 — C23.11D4
C1C23 — C23.11D4

Generators and relations for C23.11D4
 G = < a,b,c,d,e | a2=b2=c2=d4=1, e2=b, dad-1=ab=ba, ac=ca, eae-1=abc, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede-1=d-1 >

Subgroups: 157 in 85 conjugacy classes, 35 normal (15 characteristic)
C1, C2 [×3], C2 [×4], C2 [×2], C4 [×7], C22 [×3], C22 [×4], C22 [×10], C2×C4 [×2], C2×C4 [×17], C23, C23 [×2], C23 [×6], C22⋊C4 [×6], C4⋊C4 [×2], C22×C4 [×2], C22×C4 [×4], C24, C2.C42, C2.C42 [×2], C2×C22⋊C4, C2×C22⋊C4 [×2], C2×C4⋊C4, C23.11D4
Quotients: C1, C2 [×7], C22 [×7], D4 [×4], C23, C2×D4 [×2], C4○D4 [×5], C4⋊D4, C22.D4 [×3], C4.4D4, C422C2 [×2], C23.11D4

Character table of C23.11D4

 class 12A2B2C2D2E2F2G2H2I4A4B4C4D4E4F4G4H4I4J4K4L
 size 1111111144444444444444
ρ11111111111111111111111    trivial
ρ211111111-1-11-1-1111-1-1-1-111    linear of order 2
ρ311111111111-1-1-1-1111-1-1-1-1    linear of order 2
ρ411111111-1-1111-1-11-1-111-1-1    linear of order 2
ρ511111111-1-1-1-111-1-111-111-1    linear of order 2
ρ61111111111-11-11-1-1-1-11-11-1    linear of order 2
ρ711111111-1-1-11-1-11-1111-1-11    linear of order 2
ρ81111111111-1-11-11-1-1-1-11-11    linear of order 2
ρ922-2-222-2-2-22000000000000    orthogonal lifted from D4
ρ10222-2-2-22-2000-20000002000    orthogonal lifted from D4
ρ1122-2-222-2-22-2000000000000    orthogonal lifted from D4
ρ12222-2-2-22-20002000000-2000    orthogonal lifted from D4
ρ132-2-22-222-20000002i000000-2i    complex lifted from C4○D4
ρ142-2-2-22-222000000002i-2i0000    complex lifted from C4○D4
ρ152-2-2-22-22200000000-2i2i0000    complex lifted from C4○D4
ρ162-22-2-22-220000-2i0000002i00    complex lifted from C4○D4
ρ172-2222-2-2-2002i0000-2i000000    complex lifted from C4○D4
ρ1822-22-2-2-22000002i000000-2i0    complex lifted from C4○D4
ρ192-2-22-222-2000000-2i0000002i    complex lifted from C4○D4
ρ2022-22-2-2-2200000-2i0000002i0    complex lifted from C4○D4
ρ212-2222-2-2-200-2i00002i000000    complex lifted from C4○D4
ρ222-22-2-22-2200002i000000-2i00    complex lifted from C4○D4

Smallest permutation representation of C23.11D4
On 32 points
Generators in S32
(1 3)(2 22)(4 24)(5 18)(6 32)(7 20)(8 30)(9 31)(10 19)(11 29)(12 17)(13 15)(14 28)(16 26)(21 23)(25 27)
(1 23)(2 24)(3 21)(4 22)(5 9)(6 10)(7 11)(8 12)(13 25)(14 26)(15 27)(16 28)(17 30)(18 31)(19 32)(20 29)
(1 27)(2 28)(3 25)(4 26)(5 20)(6 17)(7 18)(8 19)(9 29)(10 30)(11 31)(12 32)(13 21)(14 22)(15 23)(16 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 6 23 10)(2 5 24 9)(3 8 21 12)(4 7 22 11)(13 32 25 19)(14 31 26 18)(15 30 27 17)(16 29 28 20)

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

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

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

C23.11D4 is a maximal subgroup of
C23.295C24  C42.163D4  C23.301C24  C23.318C24  C24.563C23  C24.254C23  C23.321C24  C23.322C24  C24.258C23  C24.269C23  C23.344C24  C24.271C23  C23.348C24  C23.360C24  C24.286C23  C24.289C23  C24.290C23  C23.372C24  C23.374C24  C23.375C24  C23.377C24  C24.295C23  C23.380C24  C24.573C23  C23.382C24  C23.385C24  C24.299C23  C23.388C24  C24.301C23  C23.390C24  C24.304C23  C23.395C24  C23.396C24  C23.398C24  C24.308C23  C23.404C24  C23.410C24  C23.412C24  C23.413C24  C24.309C23  C23.416C24  C23.417C24  C23.418C24  C24.311C23  C24.313C23  C23.425C24  C23.426C24  C24.315C23  C23.429C24  C23.430C24  C23.431C24  C23.432C24  C23.443C24  C24.326C23  C23.457C24  C24.332C23  C23.461C24  C23.473C24  C24.339C23  C24.340C23  C24.341C23  C23.478C24  C42.182D4  C23.494C24  C24.347C23  C23.496C24  C24.348C23  C42.183D4  C23.500C24  C23.502C24  C4224D4  C42.185D4  C2410D4  C24.589C23  C23.524C24  C23.525C24  C23.530C24  C4229D4  C42.190D4  C42.192D4  C24.374C23  C24.592C23  C42.193D4  C23.543C24  C23.544C24  C23.548C24  C24.375C23  C23.550C24  C23.551C24  C24.376C23  C23.553C24  C23.554C24  C24.377C23  C24.378C23  C42.198D4  C23.568C24  C23.574C24  C23.578C24  C23.580C24  C23.584C24  C24.393C23  C24.394C23  C24.395C23  C23.589C24  C23.590C24  C23.591C24  C23.593C24  C24.401C23  C23.595C24  C24.403C23  C23.597C24  C24.405C23  C23.600C24  C24.407C23  C23.603C24  C23.606C24  C23.607C24  C23.608C24  C24.412C23  C23.612C24  C24.413C23  C23.615C24  C23.616C24  C23.617C24  C23.621C24  C23.622C24  C24.418C23  C23.624C24  C23.625C24  C24.420C23  C23.630C24  C23.635C24  C23.636C24  C23.637C24  C24.426C23  C24.427C23  C23.640C24  C23.641C24  C24.430C23  C23.645C24  C24.432C23  C23.647C24  C23.649C24  C24.435C23  C23.651C24  C23.652C24  C24.437C23  C23.654C24  C23.656C24  C24.438C23  C23.659C24  C24.440C23  C23.664C24  C24.443C23  C24.445C23  C23.671C24  C23.672C24  C23.673C24  C23.675C24  C23.677C24  C23.678C24  C23.679C24  C23.681C24  C23.682C24  C23.683C24  C23.686C24  C23.687C24  C23.693C24  C23.696C24  C23.697C24  C23.698C24  C23.700C24  C23.703C24  C23.708C24  C2411D4  C23.714C24  C23.715C24  C24.462C23  C42.199D4  C42.200D4  C23.724C24  C23.725C24  C23.726C24  C23.727C24  C23.728C24  C23.729C24  C23.730C24  C23.731C24  C23.732C24  C23.734C24  C23.735C24  C23.736C24  C23.737C24  C23.738C24
 C24.D2p: C23.4D8  C23.5D8  C24.14D4  C24.15D4  C24.95D4  C24.96D4  C24.97D4  C24.20D6 ...
 C2p.(C4⋊D4): C4215D4  C23.315C24  C23.327C24  C4217D4  C42.170D4  C42.186D4  (C2×C4).21D12  C6.(C4⋊D4) ...
C23.11D4 is a maximal quotient of
C24.632C23  C24.633C23  C24.635C23  (C2×C4).19Q16  C428C4⋊C2  (C2×Q8).109D4
 C24.D2p: C24.5Q8  C24.52D4  C23.12D8  C24.88D4  C24.89D4  C24.20D6  C24.21D6  C24.9D10 ...
 (C2×C4).D4p: (C2×C4).24D8  (C2×C4).21D12  (C2×C4).21D20  (C2×C4).21D28 ...
 (C2×C4p).D4: (C2×C8).55D4  (C2×C8).165D4  (C2×C8).D4  (C2×C8).6D4  (C2×C12).289D4  (C2×C20).290D4  (C2×C28).290D4 ...
 C2p.(C4⋊D4): C42.9D4  C6.(C4⋊D4)  C10.(C4⋊D4)  (C22×D7).9D4 ...

Matrix representation of C23.11D4 in GL6(𝔽5)

100000
040000
001000
001400
000040
000004
,
400000
040000
004000
000400
000010
000001
,
400000
040000
001000
000100
000010
000001
,
030000
200000
004200
000100
000020
000043
,
200000
020000
002100
000300
000032
000012

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

C23.11D4 in GAP, Magma, Sage, TeX

C_2^3._{11}D_4
% in TeX

G:=Group("C2^3.11D4");
// GroupNames label

G:=SmallGroup(64,78);
// by ID

G=gap.SmallGroup(64,78);
# by ID

G:=PCGroup([6,-2,2,2,-2,2,2,96,121,362,332,50]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^2=c^2=d^4=1,e^2=b,d*a*d^-1=a*b=b*a,a*c=c*a,e*a*e^-1=a*b*c,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 C23.11D4 in TeX

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