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G = C23.23D4order 64 = 26

2nd non-split extension by C23 of D4 acting via D4/C4=C2

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

Aliases: C23.23D4, C23.62C23, C24.27C22, (C2×C4)⋊9D4, (C2×D4)⋊3C4, C2.5(C4×D4), C232(C2×C4), (C23×C4)⋊1C2, C2.3C22≀C2, C2.2(C4⋊D4), (C22×D4).1C2, C22.35(C2×D4), C221(C22⋊C4), C2.C428C2, C22.20(C4○D4), C22.35(C22×C4), (C22×C4).90C22, C2.3(C22.D4), (C2×C4)⋊3(C2×C4), (C2×C22⋊C4)⋊2C2, C2.7(C2×C22⋊C4), SmallGroup(64,67)

Series: Derived Chief Lower central Upper central Jennings

C1C22 — C23.23D4
C1C2C22C23C24C23×C4 — C23.23D4
C1C22 — C23.23D4
C1C23 — C23.23D4
C1C23 — C23.23D4

Generators and relations for C23.23D4
 G = < a,b,c,d,e | a2=b2=c2=d4=e2=1, eae=ab=ba, ac=ca, ad=da, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede=cd-1 >

Subgroups: 249 in 143 conjugacy classes, 53 normal (15 characteristic)
C1, C2 [×3], C2 [×4], C2 [×6], C4 [×8], C22 [×3], C22 [×8], C22 [×22], C2×C4 [×6], C2×C4 [×20], D4 [×8], C23, C23 [×8], C23 [×10], C22⋊C4 [×6], C22×C4, C22×C4 [×4], C22×C4 [×6], C2×D4 [×4], C2×D4 [×4], C24 [×2], C2.C42 [×2], C2×C22⋊C4, C2×C22⋊C4 [×2], C23×C4, C22×D4, C23.23D4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×8], C23, C22⋊C4 [×4], C22×C4, C2×D4 [×4], C4○D4 [×2], C2×C22⋊C4, C4×D4 [×2], C22≀C2, C4⋊D4 [×2], C22.D4, C23.23D4

Character table of C23.23D4

 class 12A2B2C2D2E2F2G2H2I2J2K2L2M4A4B4C4D4E4F4G4H4I4J4K4L4M4N
 size 1111111122224422222222444444
ρ11111111111111111111111111111    trivial
ρ211111111111111-1-1-1-1-1-1-1-1-11-1-1-11    linear of order 2
ρ311111111-1-1-1-1-1-1-1-1-1-11111111-1-11    linear of order 2
ρ411111111-1-1-1-1-1-11111-1-1-1-1-11-1111    linear of order 2
ρ511111111-1-1-1-1111111-1-1-1-11-11-1-1-1    linear of order 2
ρ611111111-1-1-1-111-1-1-1-11111-1-1-111-1    linear of order 2
ρ7111111111111-1-111111111-1-1-1-1-1-1    linear of order 2
ρ8111111111111-1-1-1-1-1-1-1-1-1-11-1111-1    linear of order 2
ρ91-11-11-11-1-11-11-11-iii-ii-ii-ii1-ii-i-1    linear of order 4
ρ101-11-11-11-11-11-1-11-iii-i-ii-iii-1-i-ii1    linear of order 4
ρ111-11-11-11-1-11-111-1i-i-ii-ii-iii-1-ii-i1    linear of order 4
ρ121-11-11-11-11-11-11-1i-i-iii-ii-ii1-i-ii-1    linear of order 4
ρ131-11-11-11-11-11-1-11i-i-iii-ii-i-i-1ii-i1    linear of order 4
ρ141-11-11-11-1-11-11-11i-i-ii-ii-ii-i1i-ii-1    linear of order 4
ρ151-11-11-11-11-11-11-1-iii-i-ii-ii-i1ii-i-1    linear of order 4
ρ161-11-11-11-1-11-111-1-iii-ii-ii-i-i-1i-ii1    linear of order 4
ρ172-2-22-222-2000000000022-2-2000000    orthogonal lifted from D4
ρ182-2-22-222-20000000000-2-222000000    orthogonal lifted from D4
ρ192-222-2-2-2222-2-20000000000000000    orthogonal lifted from D4
ρ202-2-2-222-22000000-2-2220000000000    orthogonal lifted from D4
ρ21222-2-22-2-22-2-220000000000000000    orthogonal lifted from D4
ρ222-2-2-222-2200000022-2-20000000000    orthogonal lifted from D4
ρ23222-2-22-2-2-222-20000000000000000    orthogonal lifted from D4
ρ242-222-2-2-22-2-2220000000000000000    orthogonal lifted from D4
ρ2522-222-2-2-2000000-2i2i-2i2i0000000000    complex lifted from C4○D4
ρ2622-222-2-2-20000002i-2i2i-2i0000000000    complex lifted from C4○D4
ρ2722-2-2-2-22200000000002i-2i-2i2i000000    complex lifted from C4○D4
ρ2822-2-2-2-2220000000000-2i2i2i-2i000000    complex lifted from C4○D4

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

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

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

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

C23.23D4 is a maximal subgroup of
C23.179C24  C4×C22≀C2  C4×C4⋊D4  C4×C22.D4  C24.542C23  C24.547C23  C23.203C24  C24.195C23  C24.198C23  C23.215C24  C24.204C23  C24.205C23  D4×C22⋊C4  C24.549C23  C23.235C24  C23.240C24  C23.241C24  C24.215C23  C24.218C23  C24.219C23  C24.221C23  C24.223C23  C23.257C24  C24.225C23  C23.259C24  C23.261C24  C23.262C24  C247D4  C23.304C24  C24.243C23  C24.244C23  C23.309C24  C248D4  C23.311C24  C23.313C24  C23.316C24  C24.252C23  C23.318C24  C24.563C23  C24.254C23  C23.324C24  C24.258C23  C23.327C24  C23.328C24  C24.262C23  C24.263C23  C24.565C23  C24.269C23  C23.344C24  C23.345C24  C23.349C24  C23.350C24  C23.356C24  C24.278C23  C23.359C24  C24.282C23  C24.283C23  C23.364C24  C23.367C24  C24.289C23  C24.290C23  C23.372C24  C23.374C24  C24.293C23  C23.377C24  C23.379C24  C24.573C23  C24.300C23  C23.388C24  C23.398C24  C23.404C24  C23.410C24  C23.416C24  C23.418C24  C24.311C23  C23.426C24  C23.431C24  C23.434C24  C4217D4  C23.443C24  C4221D4  C24.326C23  C24.327C23  C23.455C24  C23.457C24  C24.331C23  C24.332C23  C24.583C23  C23.472C24  C24.340C23  C23.478C24  C23.479C24  C23.491C24  C24.347C23  C24.348C23  C4222D4  C23.500C24  C4223D4  C23.502C24  C4224D4  C4225D4  C4226D4  C2410D4  C24.587C23  C24.589C23  C23.530C24  C4229D4  C23.535C24  C4230D4  C24.592C23  C23.543C24  C23.548C24  C24.375C23  C23.553C24  C23.568C24  C23.569C24  C23.571C24  C23.572C24  C23.573C24  C23.576C24  C23.578C24  C23.581C24  C23.583C24  C23.584C24  C24.393C23  C24.394C23  C24.395C23  C23.591C24  C23.592C24  C23.593C24  C23.595C24  C24.403C23  C23.597C24  C24.406C23  C23.600C24  C24.407C23  C23.602C24  C23.603C24  C23.605C24  C23.606C24  C23.608C24  C24.411C23  C24.412C23  C23.612C24  C24.413C23  C23.615C24  C23.617C24  C24.418C23  C24.420C23  C23.630C24  C23.632C24  C23.637C24  C23.640C24  C23.641C24  C23.643C24  C24.432C23  C24.434C23  C23.649C24  C24.435C23  C23.651C24  C23.652C24  C24.437C23  C23.656C24  C24.438C23  C23.660C24  C24.440C23  C23.678C24  C23.679C24  C24.448C23  C23.681C24  C23.682C24  C24.450C23  C23.686C24  C23.696C24  C23.697C24  C24.456C23  C23.724C24  C23.725C24  C23.726C24  C23.727C24  C2413D4  C4246D4  C4243D4  C23.753C24  C24.598C23
 C24.D2p: C24.5D4  2+ 1+42C4  C24.22D4  C24.33D4  C24.90D4  C24.94D4  C24.95D4  C24.97D4 ...
 C2p.(C4×D4): C4242D4  C4213D4  C4214D4  C23.234C24  C24.217C23  (C2×C4)⋊9D12  (C2×C4)⋊9D20  (C2×F5)⋊D4 ...
C23.23D4 is a maximal quotient of
C24.626C23  C232C42  C24.631C23  C23.22M4(2)  C232M4(2)  M4(2).43D4  (C2×SD16)⋊15C4  M4(2).44D4  C8.C22⋊C4  C8⋊C22⋊C4  M4(2)⋊19D4  C4⋊Q815C4  C4.4D413C4  (C2×C8)⋊D4  C24.174C23  M4(2)⋊20D4  M4(2).45D4  M4(2).46D4  M4(2).47D4  C42.5D4  C42.6D4  C42.426D4  C4.(C4×D4)  (C2×C8)⋊4D4  C42.7D4  M4(2)⋊21D4  M4(2).50D4  (C2×F5)⋊D4
 C24.D2p: C24.17Q8  C24.50D4  C24.5Q8  C24.52D4  C24.72D4  C24.160D4  C24.73D4  C23.38D8 ...
 (C2×C4)⋊D4p: (C2×C4)⋊9D8  (C2×C4)⋊9D12  (C2×C4)⋊9D20  (C2×C4)⋊9D28 ...
 C2.(D4.pD4): (C2×SD16)⋊14C4  (C2×C4)⋊9Q16  M4(2).48D4  M4(2).49D4 ...

Matrix representation of C23.23D4 in GL5(𝔽5)

10000
00100
01000
00001
00010
,
10000
04000
00400
00040
00004
,
40000
04000
00400
00010
00001
,
30000
03000
00300
00003
00030
,
40000
01000
00400
00040
00001

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

C23.23D4 in GAP, Magma, Sage, TeX

C_2^3._{23}D_4
% in TeX

G:=Group("C2^3.23D4");
// GroupNames label

G:=SmallGroup(64,67);
// by ID

G=gap.SmallGroup(64,67);
# by ID

G:=PCGroup([6,-2,2,2,-2,2,2,192,121,247,362]);
// 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,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=c*d^-1>;
// generators/relations

Export

Character table of C23.23D4 in TeX

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