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G = C232D4order 64 = 26

1st semidirect product of C23 and D4 acting via D4/C2=C22

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

Aliases: C232D4, C24.4C22, C23.75C23, (C2×C4)⋊2D4, C2.4C22≀C2, (C22×D4)⋊1C2, C2.3(C41D4), C2.6(C4⋊D4), C22.68(C2×D4), C2.C429C2, (C22×C4).7C22, C22.35(C4○D4), (C2×C22⋊C4)⋊6C2, SmallGroup(64,73)

Series: Derived Chief Lower central Upper central Jennings

C1C23 — C232D4
C1C2C22C23C24C22×D4 — C232D4
C1C23 — C232D4
C1C23 — C232D4
C1C23 — C232D4

Generators and relations for C232D4
 G = < a,b,c,d,e | a2=b2=c2=d4=e2=1, eae=ab=ba, ac=ca, dad-1=abc, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede=d-1 >

Subgroups: 317 in 161 conjugacy classes, 43 normal (7 characteristic)
C1, C2, C2 [×6], C2 [×6], C4 [×7], C22, C22 [×6], C22 [×30], C2×C4 [×6], C2×C4 [×9], D4 [×24], C23, C23 [×6], C23 [×18], C22⋊C4 [×6], C22×C4, C22×C4 [×3], C2×D4 [×18], C24 [×3], C2.C42, C2×C22⋊C4 [×3], C22×D4 [×3], C232D4
Quotients: C1, C2 [×7], C22 [×7], D4 [×12], C23, C2×D4 [×6], C4○D4, C22≀C2 [×3], C4⋊D4 [×3], C41D4, C232D4

Character table of C232D4

 class 12A2B2C2D2E2F2G2H2I2J2K2L2M4A4B4C4D4E4F4G4H
 size 1111111144444444444444
ρ11111111111111111111111    trivial
ρ21111111111-111-1-11-11-1-1-1-1    linear of order 2
ρ3111111111-1-11-1-11-1-1-11-111    linear of order 2
ρ4111111111-111-11-1-11-1-11-1-1    linear of order 2
ρ511111111-111-111-1-1-1-11-11-1    linear of order 2
ρ611111111-11-1-11-11-11-1-11-11    linear of order 2
ρ711111111-1-1-1-1-1-1-1111111-1    linear of order 2
ρ811111111-1-11-1-1111-11-1-1-11    linear of order 2
ρ92-2222-2-2-2-20020000000000    orthogonal lifted from D4
ρ102-2222-2-2-2200-20000000000    orthogonal lifted from D4
ρ1122-22-2-2-22000000000020-20    orthogonal lifted from D4
ρ122-2-2-22-222000000020-20000    orthogonal lifted from D4
ρ13222-2-2-22-200-200200000000    orthogonal lifted from D4
ρ14222-2-2-22-200200-200000000    orthogonal lifted from D4
ρ152-2-2-22-2220000000-2020000    orthogonal lifted from D4
ρ1622-2-222-2-20-2002000000000    orthogonal lifted from D4
ρ1722-2-222-2-20200-2000000000    orthogonal lifted from D4
ρ182-22-2-22-2200000000200-200    orthogonal lifted from D4
ρ192-22-2-22-2200000000-200200    orthogonal lifted from D4
ρ2022-22-2-2-220000000000-2020    orthogonal lifted from D4
ρ212-2-22-222-20000002i000000-2i    complex lifted from C4○D4
ρ222-2-22-222-2000000-2i0000002i    complex lifted from C4○D4

Smallest permutation representation of C232D4
On 32 points
Generators in S32
(1 29)(2 12)(3 31)(4 10)(5 20)(6 21)(7 18)(8 23)(9 16)(11 14)(13 32)(15 30)(17 28)(19 26)(22 25)(24 27)
(1 5)(2 6)(3 7)(4 8)(9 22)(10 23)(11 24)(12 21)(13 26)(14 27)(15 28)(16 25)(17 30)(18 31)(19 32)(20 29)
(1 27)(2 28)(3 25)(4 26)(5 14)(6 15)(7 16)(8 13)(9 18)(10 19)(11 20)(12 17)(21 30)(22 31)(23 32)(24 29)
(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 4)(2 3)(5 8)(6 7)(9 17)(10 20)(11 19)(12 18)(13 14)(15 16)(21 31)(22 30)(23 29)(24 32)(25 28)(26 27)

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

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

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

C232D4 is a maximal subgroup of
C4215D4  C4216D4  C247D4  C23.304C24  C23.308C24  C248D4  C24.249C23  C23.324C24  C23.328C24  C24.262C23  C23.333C24  C24.276C23  C23.356C24  C24.278C23  C23.359C24  C24.282C23  C24.283C23  C23.364C24  C23.367C24  C24.290C23  C4217D4  C23.439C24  C4219D4  C4220D4  C23.443C24  C24.327C23  C23.455C24  C23.457C24  C4222D4  C23.500C24  C23.502C24  C4224D4  C4226D4  C249D4  C2410D4  C4227D4  C23.530C24  C4229D4  C23.535C24  C4230D4  C23.548C24  C23.556C24  C4231D4  C24.377C23  C4232D4  C23.568C24  C23.569C24  C23.570C24  C23.571C24  C23.573C24  C24.384C23  C23.576C24  C23.578C24  C24.389C23  C24.395C23  C23.593C24  C24.403C23  C24.406C23  C23.603C24  C23.605C24  C23.606C24  C24.412C23  C23.612C24  C24.413C23  C23.630C24  C23.633C24  C23.649C24  C23.652C24  C24.437C23  C23.656C24  C23.660C24  C23.678C24  C23.697C24  C23.703C24  C2411D4  C24.459C23  C23.715C24  C4233D4  C4234D4  C23.724C24  C23.725C24  C23.726C24  C23.728C24  C23.729C24  C232D4⋊C3
 C24.D2p: C24.D4  C23⋊D8  C23⋊SD16  C24.9D4  C233D12  C24.32D6  C232D20  C24.21D10 ...
 C2p.C22≀C2: C23.288C24  C24.244C23  C24.263C23  C24.360C23  (C2×C12)⋊5D4  (C2×C20)⋊5D4  (C2×C28)⋊5D4 ...
C232D4 is a maximal quotient of
C24.50D4  C24.5Q8  C24.634C23  C429D4  C42.129D4  C4210D4  C42.130D4  M4(2)⋊D4  M4(2)⋊4D4  M4(2).D4  (C2×C4)⋊3SD16  (C2×C4)⋊2Q16  (C2×C8).2D4  M4(2).4D4  C24⋊D4  C24.31D4
 (C2×C4p)⋊D4: C232D8  C233SD16  M4(2)⋊5D4  C422D4  (C2×C4)⋊2D8  (C2×C8)⋊20D4  (C2×C12)⋊5D4  C233D12 ...
 C2.(D4.pD4): C232Q16  (C22×D8).C2  (C2×C8).41D4  M4(2).5D4  M4(2).6D4 ...

Matrix representation of C232D4 in GL6(𝔽5)

400000
010000
001000
000100
000042
000001
,
400000
040000
001000
000100
000040
000004
,
100000
010000
001000
000100
000040
000004
,
010000
400000
000100
004000
000034
000002
,
010000
100000
000100
001000
000034
000032

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

C232D4 in GAP, Magma, Sage, TeX

C_2^3\rtimes_2D_4
% in TeX

G:=Group("C2^3:2D4");
// GroupNames label

G:=SmallGroup(64,73);
// by ID

G=gap.SmallGroup(64,73);
# by ID

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

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

Export

Character table of C232D4 in TeX

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