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G = C23.37D4  order 64 = 26

8th non-split extension by C23 of D4 acting via D4/C22=C2

p-group, metabelian, nilpotent (class 3), monomial

Aliases: C23.37D4, (C2×D4)⋊8C4, C4⋊C4⋊8C22, (C2×C8)⋊8C22, D4.6(C2×C4), C4.51(C2×D4), (C2×C4).124D4, C4.5(C22×C4), D4⋊C4⋊15C2, C42⋊C2⋊3C2, C2.2(C8⋊C22), (C2×C4).63C23, (C22×D4).7C2, C22.45(C2×D4), C4.14(C22⋊C4), (C2×M4(2))⋊11C2, (C2×D4).49C22, (C22×C4).35C22, C22.17(C22⋊C4), (C2×C4).22(C2×C4), C2.21(C2×C22⋊C4), SmallGroup(64,99)

Series: Derived ►Chief ►Lower central ►Upper central ►Jennings

Derived series
C1 — C4 — C23.37D4
Chief series
C1 — C2 — C22 — C2×C4 — C22×C4 — C22×D4 — C23.37D4
Lower central
C1 — C2 — C4 — C23.37D4
Upper central
C1 — C22 — C22×C4 — C23.37D4
Jennings
C1 — C2 — C2 — C2×C4 — C23.37D4

Generators and relations for C23.37D4
 G = < a,b,c,d,e | a2=b2=c2=1, d4=c, e2=b, ab=ba, dad-1=eae-1=ac=ca, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede-1=bcd3 >

Subgroups: 193 in 95 conjugacy classes, 41 normal (13 characteristic)
C1, C2, C2, C2, C4, C4, C4, C22, C22, C22, C8, C2×C4, C2×C4, C2×C4, D4, D4, C23, C23, C42, C22⋊C4, C4⋊C4, C2×C8, M4(2), C22×C4, C2×D4, C2×D4, C24, D4⋊C4, C42⋊C2, C2×M4(2), C22×D4, C23.37D4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22⋊C4, C22×C4, C2×D4, C2×C22⋊C4, C8⋊C22, C23.37D4

Character table of C23.37D4

 class 12A2B2C2D2E2F2G2H2I4A4B4C4D4E4F4G4H8A8B8C8D
 size 1111224444222244444444
ρ11111111111111111111111    trivial
ρ2111111-1-1-1-111111111-1-1-1-1    linear of order 2
ρ311111111111111-1-1-1-1-1-1-1-1    linear of order 2
ρ4111111-1-1-1-11111-1-1-1-11111    linear of order 2
ρ51111-1-1-11-111-1-11-111-1-111-1    linear of order 2
ρ61111-1-11-11-11-1-11-111-11-1-11    linear of order 2
ρ71111-1-1-11-111-1-111-1-111-1-11    linear of order 2
ρ81111-1-11-11-11-1-111-1-11-111-1    linear of order 2
ρ91-11-11-111-1-1-11-11ii-i-ii-ii-i    linear of order 4
ρ101-11-11-1-1-111-11-11ii-i-i-ii-ii    linear of order 4
ρ111-11-11-111-1-1-11-11-i-iii-ii-ii    linear of order 4
ρ121-11-11-1-1-111-11-11-i-iiii-ii-i    linear of order 4
ρ131-11-1-11-111-1-1-111-ii-ii-i-iii    linear of order 4
ρ141-11-1-111-1-11-1-111-ii-iiii-i-i    linear of order 4
ρ151-11-1-11-111-1-1-111i-ii-iii-i-i    linear of order 4
ρ161-11-1-111-1-11-1-111i-ii-i-i-iii    linear of order 4
ρ172222-2-20000-222-200000000    orthogonal lifted from D4
ρ182222220000-2-2-2-200000000    orthogonal lifted from D4
ρ192-22-2-22000022-2-200000000    orthogonal lifted from D4
ρ202-22-22-200002-22-200000000    orthogonal lifted from D4
ρ2144-4-4000000000000000000    orthogonal lifted from C8⋊C22
ρ224-4-44000000000000000000    orthogonal lifted from C8⋊C22

Permutation representations of C23.37D4
►On 16 points - transitive group 16T75
Generators in S16
(1 9)(2 14)(3 11)(4 16)(5 13)(6 10)(7 15)(8 12)

(1 13)(2 14)(3 15)(4 16)(5 9)(6 10)(7 11)(8 12)

(1 5)(2 6)(3 7)(4 8)(9 13)(10 14)(11 15)(12 16)

(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)

(1 8 13 12)(2 11 14 7)(3 6 15 10)(4 9 16 5)
 
G:=sub<Sym(16)| (1,9)(2,14)(3,11)(4,16)(5,13)(6,10)(7,15)(8,12), (1,13)(2,14)(3,15)(4,16)(5,9)(6,10)(7,11)(8,12), (1,5)(2,6)(3,7)(4,8)(9,13)(10,14)(11,15)(12,16), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16), (1,8,13,12)(2,11,14,7)(3,6,15,10)(4,9,16,5)>;
 
G:=Group( (1,9)(2,14)(3,11)(4,16)(5,13)(6,10)(7,15)(8,12), (1,13)(2,14)(3,15)(4,16)(5,9)(6,10)(7,11)(8,12), (1,5)(2,6)(3,7)(4,8)(9,13)(10,14)(11,15)(12,16), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16), (1,8,13,12)(2,11,14,7)(3,6,15,10)(4,9,16,5) );
 
G=PermutationGroup([[(1,9),(2,14),(3,11),(4,16),(5,13),(6,10),(7,15),(8,12)], [(1,13),(2,14),(3,15),(4,16),(5,9),(6,10),(7,11),(8,12)], [(1,5),(2,6),(3,7),(4,8),(9,13),(10,14),(11,15),(12,16)], [(1,2,3,4,5,6,7,8),(9,10,11,12,13,14,15,16)], [(1,8,13,12),(2,11,14,7),(3,6,15,10),(4,9,16,5)]])
 
G:=TransitiveGroup(16,75);
 

C23.37D4 is a maximal subgroup of
 C4○D4.D4  C8⋊C22⋊C4  C4.4D4⋊13C4  C42.5D4  C42.129D4  (C2×D4)⋊2Q8  M4(2).8D4  C24.98D4  2+ 1+4⋊5C4  C4×C8⋊C22  C42.277C23  C42.278C23  C24.177D4  C24.104D4  C4○D4⋊D4  D4.(C2×D4)  C42.444D4  C42.446D4  C42.14C23  C42.16C23  M4(2)⋊14D4  C42.219D4  C42.20C23  C24.117D4  (C2×D4).301D4  C42.366C23  C42.240D4  C42.242D4  C42.45C23  C42.46C23  C42.53C23  C42.54C23  C42.471C23  C42.472C23  C42.473C23  C42.474C23  (D4×C10)⋊C4
 C4⋊C4⋊D2p: C24.24D4  C42⋊9D4  C4⋊C4⋊19D6  C4⋊C4⋊36D6  (D4×D5)⋊C4  C4⋊C4⋊36D10  (D4×D7)⋊C4  C4⋊C4⋊36D14 ...
 (C2×C8)⋊D2p: (C2×C8)⋊4D4  M4(2)⋊5D4  (C2×C8)⋊11D4  (C2×C8)⋊12D4  C23.53D12  C23.48D20  C23.48D28 ...
 (C2×C4p).D4: M4(2).47D4  C4.D4⋊3C4  M4(2).31D4  M4(2).32D4  C4⋊C4.96D4  C4⋊C4.97D4  M4(2).12D4  (C2×C8).D4 ...
C23.37D4 is a maximal quotient of
 C42.398D4  C42.400D4  D4⋊5M4(2)  C42.53D4  C42.55D4  C24.54D4  C24.56D4  C42.59D4  C42.413D4  C42.416D4  C42.417D4  C42.82D4  C42.83D4  C42.84D4  C24.152D4  D4⋊C42  C24.157D4  C24.76D4  C42.112D4  C42.121D4  C42.125D4  (D4×C10)⋊C4
 C23.D4p: C23.35D8  C23.53D12  C23.48D20  C23.48D28 ...
 C4⋊C4⋊D2p: C24.74D4  C42.118D4  C4⋊C4⋊19D6  C4⋊C4⋊36D6  (D4×D5)⋊C4  C4⋊C4⋊36D10  (D4×D7)⋊C4  C4⋊C4⋊36D14 ...
 (C2×D4).D2p: C42.100D4  (C6×D4)⋊6C4  (D4×C10)⋊18C4  (D4×C14)⋊6C4 ...

Matrix representation of C23.37D4 ►in GL6(𝔽17)

100000
010000
0016000
0001600
000010
000001
,
1600000
0160000
0016000
0001600
0000160
0000016
,
100000
010000
0016000
0001600
0000160
0000016
,
040000
1300000
0000160
000001
0001600
0016000
,
040000
400000
000010
000001
0016000
0001600

G:=sub<GL(6,GF(17))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[0,13,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,16,0,0,0,0,16,0,0,0,16,0,0,0,0,0,0,1,0,0],[0,4,0,0,0,0,4,0,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,1,0,0,0,0,0,0,1,0,0] >;
 

C23.37D4 in GAP, Magma, Sage, TeX 

C_2^3._{37}D_4
 
% in TeX
 
G:=Group("C2^3.37D4");
 
// GroupNames label
 
G:=SmallGroup(64,99);
 
// by ID
 
G=gap.SmallGroup(64,99);
 
# by ID
 
G:=PCGroup([6,-2,2,2,-2,2,-2,96,121,332,158,963,489,117]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^2=b^2=c^2=1,d^4=c,e^2=b,a*b=b*a,d*a*d^-1=e*a*e^-1=a*c=c*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*c*d^3>;
 
// generators/relations
 

Export

Character table of C23.37D4 in TeX

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