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G = C24.36D4  order 128 = 27

36th non-split extension by C24 of D4 acting faithfully

p-group, metabelian, nilpotent (class 4), monomial

Aliases: C24.36D4, C2≀C4⋊5C2, C24⋊3(C2×C4), C22≀C2⋊2C4, (C22×D4)⋊9C4, C23.4(C2×D4), (C2×D4).126D4, (C22×C4).91D4, (C2×D4).15C23, C23⋊3D4.7C2, C22.D4⋊2C4, C23.D4⋊5C2, C23⋊C4.8C22, C22≀C2.1C22, C22.8(C23⋊C4), C23.54(C22×C4), C4.D4.9C22, C23.84(C22⋊C4), (C22×D4).100C22, C22.D4.1C22, (C2×C4).4(C2×D4), C22⋊C4⋊3(C2×C4), (C2×C22⋊C4)⋊9C4, (C22×C4)⋊3(C2×C4), (C2×C23⋊C4)⋊12C2, C2.33(C2×C23⋊C4), (C2×D4).124(C2×C4), (C2×C4.D4)⋊27C2, (C2×C4).25(C22⋊C4), C22.57(C2×C22⋊C4), SmallGroup(128,853)

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

Derived series
C1 — C23 — C24.36D4
Chief series
C1 — C2 — C22 — C23 — C2×D4 — C22×D4 — C23⋊3D4 — C24.36D4
Lower central
C1 — C2 — C22 — C23 — C24.36D4
Upper central
C1 — C2 — C23 — C22×D4 — C24.36D4
Jennings
C1 — C2 — C22 — C2×D4 — C24.36D4

Generators and relations for C24.36D4
 G = < a,b,c,d,e,f | a2=b2=c2=d2=1, e4=d, f2=b, ab=ba, ac=ca, eae-1=ad=da, af=fa, bc=cb, bd=db, ebe-1=bcd, bf=fb, ece-1=fcf-1=cd=dc, de=ed, df=fd, fef-1=bde3 >

Subgroups: 412 in 140 conjugacy classes, 42 normal (34 characteristic)
C1, C2, C2, C4, C22, C22, C8, C2×C4, C2×C4, D4, C23, C23, C22⋊C4, C22⋊C4, C4⋊C4, C2×C8, M4(2), C22×C4, C22×C4, C2×D4, C2×D4, C24, C23⋊C4, C23⋊C4, C4.D4, C4.D4, C2×C22⋊C4, C2×C22⋊C4, C22≀C2, C22≀C2, C4⋊D4, C22.D4, C22.D4, C2×M4(2), C22×D4, C2≀C4, C23.D4, C2×C23⋊C4, C2×C4.D4, C23⋊3D4, C24.36D4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22⋊C4, C22×C4, C2×D4, C23⋊C4, C2×C22⋊C4, C2×C23⋊C4, C24.36D4

Character table of C24.36D4

 class 12A2B2C2D2E2F2G2H2I4A4B4C4D4E4F4G4H4I8A8B8C8D
 size 11222444484488888888888
ρ111111111111111111111111    trivial
ρ2111111111111-1-111-1-11-1-1-1-1    linear of order 2
ρ311-11-1-111-111-11-11-11-1-1-11-11    linear of order 2
ρ411-11-1-111-111-1-111-1-11-11-11-1    linear of order 2
ρ511-11-1-111-1-11-1-11-11-111-11-11    linear of order 2
ρ611-11-1-111-1-11-11-1-111-111-11-1    linear of order 2
ρ7111111111-111-1-1-1-1-1-1-11111    linear of order 2
ρ8111111111-11111-1-111-1-1-1-1-1    linear of order 2
ρ911-11-1-1-1111-11i-i-11-ii-1ii-i-i    linear of order 4
ρ1011-11-1-1-1111-11-ii-11i-i-1-i-iii    linear of order 4
ρ11111111-11-11-1-1ii-1-1-i-i1-iii-i    linear of order 4
ρ12111111-11-11-1-1-i-i-1-1ii1i-i-ii    linear of order 4
ρ13111111-11-1-1-1-1-i-i11ii-1-iii-i    linear of order 4
ρ14111111-11-1-1-1-1ii11-i-i-1i-i-ii    linear of order 4
ρ1511-11-1-1-111-1-11-ii1-1i-i1ii-i-i    linear of order 4
ρ1611-11-1-1-111-1-11i-i1-1-ii1-i-iii    linear of order 4
ρ1722222-22-220-2-200000000000    orthogonal lifted from D4
ρ1822222-2-2-2-202200000000000    orthogonal lifted from D4
ρ1922-22-222-2-20-2200000000000    orthogonal lifted from D4
ρ2022-22-22-2-2202-200000000000    orthogonal lifted from D4
ρ21444-4-4000000000000000000    orthogonal lifted from C23⋊C4
ρ2244-4-44000000000000000000    orthogonal lifted from C23⋊C4
ρ238-8000000000000000000000    orthogonal faithful

Permutation representations of C24.36D4
►On 16 points - transitive group 16T219
Generators in S16
(1 10)(2 15)(3 12)(4 9)(5 14)(6 11)(7 16)(8 13)

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

(2 6)(4 8)(9 13)(11 15)

(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 16 15)(2 10 13 7)(3 6 14 9)(4 12 11 5)
 
G:=sub<Sym(16)| (1,10)(2,15)(3,12)(4,9)(5,14)(6,11)(7,16)(8,13), (1,16)(2,13)(3,14)(4,11)(5,12)(6,9)(7,10)(8,15), (2,6)(4,8)(9,13)(11,15), (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), (1,8,16,15)(2,10,13,7)(3,6,14,9)(4,12,11,5)>;
 
G:=Group( (1,10)(2,15)(3,12)(4,9)(5,14)(6,11)(7,16)(8,13), (1,16)(2,13)(3,14)(4,11)(5,12)(6,9)(7,10)(8,15), (2,6)(4,8)(9,13)(11,15), (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), (1,8,16,15)(2,10,13,7)(3,6,14,9)(4,12,11,5) );
 
G=PermutationGroup([[(1,10),(2,15),(3,12),(4,9),(5,14),(6,11),(7,16),(8,13)], [(1,16),(2,13),(3,14),(4,11),(5,12),(6,9),(7,10),(8,15)], [(2,6),(4,8),(9,13),(11,15)], [(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)], [(1,8,16,15),(2,10,13,7),(3,6,14,9),(4,12,11,5)]])
 
G:=TransitiveGroup(16,219);
 

►On 16 points - transitive group 16T236
Generators in S16
(1 13)(2 10)(3 15)(4 12)(5 9)(6 14)(7 11)(8 16)

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

(1 5)(3 7)(9 13)(11 15)

(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 16 13 8)(2 7 14 15)(3 10 11 6)(4 5 12 9)
 
G:=sub<Sym(16)| (1,13)(2,10)(3,15)(4,12)(5,9)(6,14)(7,11)(8,16), (1,13)(2,14)(3,11)(4,12)(5,9)(6,10)(7,15)(8,16), (1,5)(3,7)(9,13)(11,15), (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), (1,16,13,8)(2,7,14,15)(3,10,11,6)(4,5,12,9)>;
 
G:=Group( (1,13)(2,10)(3,15)(4,12)(5,9)(6,14)(7,11)(8,16), (1,13)(2,14)(3,11)(4,12)(5,9)(6,10)(7,15)(8,16), (1,5)(3,7)(9,13)(11,15), (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), (1,16,13,8)(2,7,14,15)(3,10,11,6)(4,5,12,9) );
 
G=PermutationGroup([[(1,13),(2,10),(3,15),(4,12),(5,9),(6,14),(7,11),(8,16)], [(1,13),(2,14),(3,11),(4,12),(5,9),(6,10),(7,15),(8,16)], [(1,5),(3,7),(9,13),(11,15)], [(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)], [(1,16,13,8),(2,7,14,15),(3,10,11,6),(4,5,12,9)]])
 
G:=TransitiveGroup(16,236);
 

►On 16 points - transitive group 16T266
Generators in S16
(2 6)(4 8)(9 13)(11 15)

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

(2 6)(4 8)(10 14)(12 16)

(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 14 7 16)(2 11 4 13)(3 12 5 10)(6 15 8 9)
 
G:=sub<Sym(16)| (2,6)(4,8)(9,13)(11,15), (1,7)(2,4)(3,5)(6,8)(9,15)(10,12)(11,13)(14,16), (2,6)(4,8)(10,14)(12,16), (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), (1,14,7,16)(2,11,4,13)(3,12,5,10)(6,15,8,9)>;
 
G:=Group( (2,6)(4,8)(9,13)(11,15), (1,7)(2,4)(3,5)(6,8)(9,15)(10,12)(11,13)(14,16), (2,6)(4,8)(10,14)(12,16), (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), (1,14,7,16)(2,11,4,13)(3,12,5,10)(6,15,8,9) );
 
G=PermutationGroup([[(2,6),(4,8),(9,13),(11,15)], [(1,7),(2,4),(3,5),(6,8),(9,15),(10,12),(11,13),(14,16)], [(2,6),(4,8),(10,14),(12,16)], [(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)], [(1,14,7,16),(2,11,4,13),(3,12,5,10),(6,15,8,9)]])
 
G:=TransitiveGroup(16,266);
 

►On 16 points - transitive group 16T287
Generators in S16
(1 10)(2 15)(3 12)(4 9)(5 14)(6 11)(7 16)(8 13)

(2 6)(3 7)(11 15)(12 16)

(2 6)(4 8)(9 13)(11 15)

(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)(2 3 6 7)(4 5)(9 14)(10 13)(11 16 15 12)
 
G:=sub<Sym(16)| (1,10)(2,15)(3,12)(4,9)(5,14)(6,11)(7,16)(8,13), (2,6)(3,7)(11,15)(12,16), (2,6)(4,8)(9,13)(11,15), (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)(2,3,6,7)(4,5)(9,14)(10,13)(11,16,15,12)>;
 
G:=Group( (1,10)(2,15)(3,12)(4,9)(5,14)(6,11)(7,16)(8,13), (2,6)(3,7)(11,15)(12,16), (2,6)(4,8)(9,13)(11,15), (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)(2,3,6,7)(4,5)(9,14)(10,13)(11,16,15,12) );
 
G=PermutationGroup([[(1,10),(2,15),(3,12),(4,9),(5,14),(6,11),(7,16),(8,13)], [(2,6),(3,7),(11,15),(12,16)], [(2,6),(4,8),(9,13),(11,15)], [(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),(2,3,6,7),(4,5),(9,14),(10,13),(11,16,15,12)]])
 
G:=TransitiveGroup(16,287);
 

►On 16 points - transitive group 16T317
Generators in S16
(1 15)(2 12)(3 9)(4 14)(5 11)(6 16)(7 13)(8 10)

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

(2 6)(4 8)(10 14)(12 16)

(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 6 7 8)(2 3 4 5)(9 14 11 12)(10 15 16 13)
 
G:=sub<Sym(16)| (1,15)(2,12)(3,9)(4,14)(5,11)(6,16)(7,13)(8,10), (1,7)(2,4)(3,5)(6,8)(9,11)(10,16)(12,14)(13,15), (2,6)(4,8)(10,14)(12,16), (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), (1,6,7,8)(2,3,4,5)(9,14,11,12)(10,15,16,13)>;
 
G:=Group( (1,15)(2,12)(3,9)(4,14)(5,11)(6,16)(7,13)(8,10), (1,7)(2,4)(3,5)(6,8)(9,11)(10,16)(12,14)(13,15), (2,6)(4,8)(10,14)(12,16), (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), (1,6,7,8)(2,3,4,5)(9,14,11,12)(10,15,16,13) );
 
G=PermutationGroup([[(1,15),(2,12),(3,9),(4,14),(5,11),(6,16),(7,13),(8,10)], [(1,7),(2,4),(3,5),(6,8),(9,11),(10,16),(12,14),(13,15)], [(2,6),(4,8),(10,14),(12,16)], [(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)], [(1,6,7,8),(2,3,4,5),(9,14,11,12),(10,15,16,13)]])
 
G:=TransitiveGroup(16,317);
 

►On 16 points - transitive group 16T319
Generators in S16
(2 6)(4 8)(9 13)(11 15)

(2 9)(3 7)(4 15)(6 13)(8 11)(10 14)

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

(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)

(2 11 9 8)(3 14 7 10)(4 6 15 13)(12 16)
 
G:=sub<Sym(16)| (2,6)(4,8)(9,13)(11,15), (2,9)(3,7)(4,15)(6,13)(8,11)(10,14), (1,12)(2,9)(3,14)(4,11)(5,16)(6,13)(7,10)(8,15), (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), (2,11,9,8)(3,14,7,10)(4,6,15,13)(12,16)>;
 
G:=Group( (2,6)(4,8)(9,13)(11,15), (2,9)(3,7)(4,15)(6,13)(8,11)(10,14), (1,12)(2,9)(3,14)(4,11)(5,16)(6,13)(7,10)(8,15), (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), (2,11,9,8)(3,14,7,10)(4,6,15,13)(12,16) );
 
G=PermutationGroup([[(2,6),(4,8),(9,13),(11,15)], [(2,9),(3,7),(4,15),(6,13),(8,11),(10,14)], [(1,12),(2,9),(3,14),(4,11),(5,16),(6,13),(7,10),(8,15)], [(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)], [(2,11,9,8),(3,14,7,10),(4,6,15,13),(12,16)]])
 
G:=TransitiveGroup(16,319);
 

►On 16 points - transitive group 16T324
Generators in S16
(2 6)(4 8)(10 14)(12 16)

(2 6)(3 7)(11 15)(12 16)

(2 6)(4 8)(9 13)(11 15)

(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 13)(2 16 6 12)(3 11 7 15)(4 10)(5 9)(8 14)
 
G:=sub<Sym(16)| (2,6)(4,8)(10,14)(12,16), (2,6)(3,7)(11,15)(12,16), (2,6)(4,8)(9,13)(11,15), (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), (1,13)(2,16,6,12)(3,11,7,15)(4,10)(5,9)(8,14)>;
 
G:=Group( (2,6)(4,8)(10,14)(12,16), (2,6)(3,7)(11,15)(12,16), (2,6)(4,8)(9,13)(11,15), (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), (1,13)(2,16,6,12)(3,11,7,15)(4,10)(5,9)(8,14) );
 
G=PermutationGroup([[(2,6),(4,8),(10,14),(12,16)], [(2,6),(3,7),(11,15),(12,16)], [(2,6),(4,8),(9,13),(11,15)], [(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)], [(1,13),(2,16,6,12),(3,11,7,15),(4,10),(5,9),(8,14)]])
 
G:=TransitiveGroup(16,324);
 

Matrix representation of C24.36D4 ►in GL8(ℤ)

00100000
000-10000
10000000
0-1000000
00000010
0000000-1
00001000
00000-100
,
01000000
10000000
000-10000
00-100000
00000100
00001000
0000000-1
000000-10
,
10000000
01000000
00100000
00010000
0000-1000
00000-100
000000-10
0000000-1
,
-10000000
0-1000000
00-100000
000-10000
0000-1000
00000-100
000000-10
0000000-1
,
00000100
00001000
00000001
00000010
00-100000
000-10000
10000000
01000000
,
00001000
00000100
00000010
00000001
01000000
10000000
000-10000
00-100000

G:=sub<GL(8,Integers())| [0,0,1,0,0,0,0,0,0,0,0,-1,0,0,0,0,1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,-1,0,0,0,0,1,0,0,0,0,0,0,0,0,-1,0,0],[0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1],[-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1],[0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0],[0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0] >;
 

C24.36D4 in GAP, Magma, Sage, TeX 

C_2^4._{36}D_4
 
% in TeX
 
G:=Group("C2^4.36D4");
 
// GroupNames label
 
G:=SmallGroup(128,853);
 
// by ID
 
G=gap.SmallGroup(128,853);
 
# by ID
 
G:=PCGroup([7,-2,2,2,-2,2,-2,-2,112,141,723,1123,851,375,4037]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=1,e^4=d,f^2=b,a*b=b*a,a*c=c*a,e*a*e^-1=a*d=d*a,a*f=f*a,b*c=c*b,b*d=d*b,e*b*e^-1=b*c*d,b*f=f*b,e*c*e^-1=f*c*f^-1=c*d=d*c,d*e=e*d,d*f=f*d,f*e*f^-1=b*d*e^3>;
 
// generators/relations
 

Export

Character table of C24.36D4 in TeX

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