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G = C42.15D4  order 128 = 27

15th non-split extension by C42 of D4 acting faithfully

p-group, non-abelian, nilpotent (class 4), monomial, rational

Aliases: C42.15D4, 2- 1+4⋊1C22, (C2×D4).36D4, C2.26C2≀C22, D4.8D4⋊1C2, (C2×Q8).1C23, C42.C4⋊1C2, C24⋊C22⋊3C2, C22.50C22≀C2, C4.10D4⋊1C22, C4.4D4.21C22, (C2×C4).19(C2×D4), 2-Sylow(PSL(3,4).C2), SmallGroup(128,934)

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

Derived series
C1 — C2 — C2×Q8 — C42.15D4
Chief series
C1 — C2 — C22 — C2×C4 — C2×Q8 — C4.4D4 — C24⋊C22 — C42.15D4
Lower central
C1 — C2 — C22 — C2×Q8 — C42.15D4
Upper central
C1 — C2 — C22 — C2×Q8 — C42.15D4
Jennings
C1 — C2 — C22 — C2×Q8 — C42.15D4

Generators and relations for C42.15D4
 G = < a,b,c,d | a4=b4=d2=1, c4=b2, ab=ba, cac-1=a-1b, dad=a-1b-1, cbc-1=a2b-1, bd=db, dcd=b2c3 >

Subgroups: 376 in 124 conjugacy classes, 28 normal (7 characteristic)
C1, C2, C2, C4, C22, C22, C8, C2×C4, C2×C4, D4, Q8, C23, C42, C22⋊C4, M4(2), D8, SD16, C2×D4, C2×D4, C2×Q8, C2×Q8, C4○D4, C24, C4.10D4, C4≀C2, C22≀C2, C4.4D4, C4.4D4, C8⋊C22, 2- 1+4, C42.C4, D4.8D4, C24⋊C22, C42.15D4
Quotients: C1, C2, C22, D4, C23, C2×D4, C22≀C2, C2≀C22, C42.15D4

Character table of C42.15D4

 class 12A2B2C2D2E2F4A4B4C4D4E4F4G8A8B8C
 size 11288884448888161616
ρ111111111111111111    trivial
ρ2111-1-1-11111-11-1-11-11    linear of order 2
ρ3111-1-11-1111-1-1-11-111    linear of order 2
ρ411111-1-11111-11-1-1-11    linear of order 2
ρ51111-1-1-1111-1-11-111-1    linear of order 2
ρ6111-111-11111-1-111-1-1    linear of order 2
ρ71111-111111-1111-1-1-1    linear of order 2
ρ8111-11-1111111-1-1-11-1    linear of order 2
ρ9222-2000-2-220020000    orthogonal lifted from D4
ρ102222000-2-2200-20000    orthogonal lifted from D4
ρ1122200-202-2-20002000    orthogonal lifted from D4
ρ12222000-2-22-20200000    orthogonal lifted from D4
ρ1322200202-2-2000-2000    orthogonal lifted from D4
ρ142220002-22-20-200000    orthogonal lifted from D4
ρ1544-40200000-2000000    orthogonal lifted from C2≀C22
ρ1644-40-2000002000000    orthogonal lifted from C2≀C22
ρ178-8000000000000000    orthogonal faithful

Permutation representations of C42.15D4
►On 16 points - transitive group 16T349
Generators in S16
(1 14)(2 4 6 8)(3 12)(5 10)(7 16)(9 15 13 11)

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

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

(1 2)(3 8)(4 7)(5 6)(9 12)(10 11)(13 16)(14 15)
 
G:=sub<Sym(16)| (1,14)(2,4,6,8)(3,12)(5,10)(7,16)(9,15,13,11), (1,12,5,16)(2,9,6,13)(3,10,7,14)(4,15,8,11), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16), (1,2)(3,8)(4,7)(5,6)(9,12)(10,11)(13,16)(14,15)>;
 
G:=Group( (1,14)(2,4,6,8)(3,12)(5,10)(7,16)(9,15,13,11), (1,12,5,16)(2,9,6,13)(3,10,7,14)(4,15,8,11), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16), (1,2)(3,8)(4,7)(5,6)(9,12)(10,11)(13,16)(14,15) );
 
G=PermutationGroup([[(1,14),(2,4,6,8),(3,12),(5,10),(7,16),(9,15,13,11)], [(1,12,5,16),(2,9,6,13),(3,10,7,14),(4,15,8,11)], [(1,2,3,4,5,6,7,8),(9,10,11,12,13,14,15,16)], [(1,2),(3,8),(4,7),(5,6),(9,12),(10,11),(13,16),(14,15)]])
 
G:=TransitiveGroup(16,349);
 

►On 16 points - transitive group 16T357
Generators in S16
(2 11 6 15)(3 7)(4 13 8 9)(12 16)

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

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

(1 15)(2 14)(3 13)(4 12)(5 11)(6 10)(7 9)(8 16)
 
G:=sub<Sym(16)| (2,11,6,15)(3,7)(4,13,8,9)(12,16), (1,10,5,14)(2,15,6,11)(3,16,7,12)(4,13,8,9), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16), (1,15)(2,14)(3,13)(4,12)(5,11)(6,10)(7,9)(8,16)>;
 
G:=Group( (2,11,6,15)(3,7)(4,13,8,9)(12,16), (1,10,5,14)(2,15,6,11)(3,16,7,12)(4,13,8,9), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16), (1,15)(2,14)(3,13)(4,12)(5,11)(6,10)(7,9)(8,16) );
 
G=PermutationGroup([[(2,11,6,15),(3,7),(4,13,8,9),(12,16)], [(1,10,5,14),(2,15,6,11),(3,16,7,12),(4,13,8,9)], [(1,2,3,4,5,6,7,8),(9,10,11,12,13,14,15,16)], [(1,15),(2,14),(3,13),(4,12),(5,11),(6,10),(7,9),(8,16)]])
 
G:=TransitiveGroup(16,357);
 

►On 16 points - transitive group 16T404
Generators in S16
(1 14 5 10)(2 6)(3 16 7 12)(11 15)

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

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

(1 8)(2 7)(3 6)(4 5)(9 14)(10 13)(11 12)(15 16)
 
G:=sub<Sym(16)| (1,14,5,10)(2,6)(3,16,7,12)(11,15), (1,10,5,14)(2,11,6,15)(3,16,7,12)(4,9,8,13), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16), (1,8)(2,7)(3,6)(4,5)(9,14)(10,13)(11,12)(15,16)>;
 
G:=Group( (1,14,5,10)(2,6)(3,16,7,12)(11,15), (1,10,5,14)(2,11,6,15)(3,16,7,12)(4,9,8,13), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16), (1,8)(2,7)(3,6)(4,5)(9,14)(10,13)(11,12)(15,16) );
 
G=PermutationGroup([[(1,14,5,10),(2,6),(3,16,7,12),(11,15)], [(1,10,5,14),(2,11,6,15),(3,16,7,12),(4,9,8,13)], [(1,2,3,4,5,6,7,8),(9,10,11,12,13,14,15,16)], [(1,8),(2,7),(3,6),(4,5),(9,14),(10,13),(11,12),(15,16)]])
 
G:=TransitiveGroup(16,404);
 

Matrix representation of C42.15D4 ►in GL8(ℤ)

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

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

C42.15D4 in GAP, Magma, Sage, TeX 

C_4^2._{15}D_4
 
% in TeX
 
G:=Group("C4^2.15D4");
 
// GroupNames label
 
G:=SmallGroup(128,934);
 
// by ID
 
G=gap.SmallGroup(128,934);
 
# by ID
 
G:=PCGroup([7,-2,2,2,-2,2,-2,-2,141,422,2019,1018,297,248,2804,1971,718,375,172,4037]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^4=b^4=d^2=1,c^4=b^2,a*b=b*a,c*a*c^-1=a^-1*b,d*a*d=a^-1*b^-1,c*b*c^-1=a^2*b^-1,b*d=d*b,d*c*d=b^2*c^3>;
 
// generators/relations
 

Export

Character table of C42.15D4 in TeX

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