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G = C24.39D4order 128 = 27

39th non-split extension by C24 of D4 acting faithfully

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

Aliases: C24.39D4, C41D45C4, C422(C2×C4), C4.4D45C4, C423C43C2, C42⋊C29C4, C42⋊C43C2, (C2×D4).132D4, C23.10(C2×D4), (C2×D4).21C23, C41D4.53C22, C23⋊C4.13C22, C22.9(C23⋊C4), C23.24(C22⋊C4), C22.29C24.7C2, C4.4D4.14C22, (C22×D4).103C22, (C2×C4○D4)⋊8C4, (C2×D4)⋊4(C2×C4), (C2×Q8)⋊4(C2×C4), (C2×C23⋊C4)⋊14C2, C2.39(C2×C23⋊C4), (C2×C4).96(C22×C4), (C22×C4).31(C2×C4), (C2×C4).51(C22⋊C4), C22.63(C2×C22⋊C4), SmallGroup(128,859)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C24.39D4
C1C2C22C23C2×D4C22×D4C22.29C24 — C24.39D4
C1C2C22C2×C4 — C24.39D4
C1C2C23C22×D4 — C24.39D4
C1C2C22C2×D4 — C24.39D4

Generators and relations for C24.39D4
 G = < a,b,c,d,e,f | a2=b2=c2=d2=e4=1, f2=b, ab=ba, ac=ca, eae-1=ad=da, af=fa, ebe-1=bc=cb, bd=db, bf=fb, ece-1=fcf-1=cd=dc, de=ed, df=fd, fef-1=be-1 >

Subgroups: 428 in 143 conjugacy classes, 42 normal (22 characteristic)
C1, C2, C2 [×8], C4 [×9], C22, C22 [×2], C22 [×18], C2×C4 [×2], C2×C4 [×15], D4 [×11], Q8, C23, C23 [×4], C23 [×7], C42 [×2], C22⋊C4 [×11], C4⋊C4, C22×C4, C22×C4 [×3], C2×D4 [×3], C2×D4 [×2], C2×D4 [×7], C2×Q8, C4○D4 [×2], C24 [×2], C23⋊C4 [×4], C23⋊C4 [×2], C2×C22⋊C4 [×2], C42⋊C2, C22≀C2 [×2], C4⋊D4 [×2], C4.4D4 [×2], C41D4 [×2], C22×D4, C2×C4○D4, C42⋊C4 [×2], C423C4 [×2], C2×C23⋊C4 [×2], C22.29C24, C24.39D4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×4], C23, C22⋊C4 [×4], C22×C4, C2×D4 [×2], C23⋊C4 [×2], C2×C22⋊C4, C2×C23⋊C4, C24.39D4

Character table of C24.39D4

 class 12A2B2C2D2E2F2G2H2I4A4B4C4D4E4F4G4H4I4J4K4L4M
 size 11222444484488888888888
ρ111111111111111111111111    trivial
ρ211-11-1-1-11111-1-1-11-1-11-111-11    linear of order 2
ρ3111111111-1111-1-1-111-1-1-1-11    linear of order 2
ρ411-11-1-1-111-11-1-11-11-111-1-111    linear of order 2
ρ511-11-1-1-11111-1111-11-11-1-1-1-1    linear of order 2
ρ6111111111111-1-111-1-1-1-1-11-1    linear of order 2
ρ711-11-1-1-111-11-11-1-111-1-1111-1    linear of order 2
ρ8111111111-111-11-1-1-1-1111-1-1    linear of order 2
ρ911-11-111-1-111-1-ii-11ii-ii-i-1-i    linear of order 4
ρ1011111-1-1-1-1111i-i-1-1-iiii-i1-i    linear of order 4
ρ1111-11-111-1-1-11-1-i-i1-1iii-ii1-i    linear of order 4
ρ1211111-1-1-1-1-111ii11-ii-i-ii-1-i    linear of order 4
ρ1311111-1-1-1-1111-ii-1-1i-i-i-ii1i    linear of order 4
ρ1411-11-111-1-111-1i-i-11-i-ii-ii-1i    linear of order 4
ρ1511111-1-1-1-1-111-i-i11i-iii-i-1i    linear of order 4
ρ1611-11-111-1-1-11-1ii1-1-i-i-ii-i1i    linear of order 4
ρ1722222-22-220-2-200000000000    orthogonal lifted from D4
ρ1822-22-2-222-20-2200000000000    orthogonal lifted from D4
ρ19222222-22-20-2-200000000000    orthogonal lifted from D4
ρ2022-22-22-2-220-2200000000000    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 C24.39D4
On 16 points - transitive group 16T234
Generators in S16
(1 8)(2 11)(3 6)(4 9)(5 15)(7 13)(10 14)(12 16)
(1 8)(2 5)(3 12)(4 9)(6 16)(7 13)(10 14)(11 15)
(2 15)(4 13)(5 11)(7 9)
(1 14)(2 15)(3 16)(4 13)(5 11)(6 12)(7 9)(8 10)
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)
(1 9 8 4)(2 3 5 12)(6 15 16 11)(7 10 13 14)

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

G:=Group( (1,8)(2,11)(3,6)(4,9)(5,15)(7,13)(10,14)(12,16), (1,8)(2,5)(3,12)(4,9)(6,16)(7,13)(10,14)(11,15), (2,15)(4,13)(5,11)(7,9), (1,14)(2,15)(3,16)(4,13)(5,11)(6,12)(7,9)(8,10), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16), (1,9,8,4)(2,3,5,12)(6,15,16,11)(7,10,13,14) );

G=PermutationGroup([(1,8),(2,11),(3,6),(4,9),(5,15),(7,13),(10,14),(12,16)], [(1,8),(2,5),(3,12),(4,9),(6,16),(7,13),(10,14),(11,15)], [(2,15),(4,13),(5,11),(7,9)], [(1,14),(2,15),(3,16),(4,13),(5,11),(6,12),(7,9),(8,10)], [(1,2,3,4),(5,6,7,8),(9,10,11,12),(13,14,15,16)], [(1,9,8,4),(2,3,5,12),(6,15,16,11),(7,10,13,14)])

G:=TransitiveGroup(16,234);

On 16 points - transitive group 16T250
Generators in S16
(1 4)(2 8)(3 5)(6 7)(9 15)(10 14)(11 13)(12 16)
(1 9)(2 10)(3 16)(4 15)(5 12)(6 11)(7 13)(8 14)
(2 5)(3 8)(10 12)(14 16)
(1 6)(2 5)(3 8)(4 7)(9 11)(10 12)(13 15)(14 16)
(1 2)(3 4)(5 6)(7 8)(9 10 11 12)(13 14 15 16)
(1 10 9 2)(3 7 16 13)(4 14 15 8)(5 6 12 11)

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

G:=Group( (1,4)(2,8)(3,5)(6,7)(9,15)(10,14)(11,13)(12,16), (1,9)(2,10)(3,16)(4,15)(5,12)(6,11)(7,13)(8,14), (2,5)(3,8)(10,12)(14,16), (1,6)(2,5)(3,8)(4,7)(9,11)(10,12)(13,15)(14,16), (1,2)(3,4)(5,6)(7,8)(9,10,11,12)(13,14,15,16), (1,10,9,2)(3,7,16,13)(4,14,15,8)(5,6,12,11) );

G=PermutationGroup([(1,4),(2,8),(3,5),(6,7),(9,15),(10,14),(11,13),(12,16)], [(1,9),(2,10),(3,16),(4,15),(5,12),(6,11),(7,13),(8,14)], [(2,5),(3,8),(10,12),(14,16)], [(1,6),(2,5),(3,8),(4,7),(9,11),(10,12),(13,15),(14,16)], [(1,2),(3,4),(5,6),(7,8),(9,10,11,12),(13,14,15,16)], [(1,10,9,2),(3,7,16,13),(4,14,15,8),(5,6,12,11)])

G:=TransitiveGroup(16,250);

On 16 points - transitive group 16T297
Generators in S16
(1 4)(2 3)(5 6)(7 8)(9 13)(10 12)(11 15)(14 16)
(1 5)(2 7)(3 8)(4 6)(9 10)(11 14)(12 13)(15 16)
(1 3)(2 4)(5 8)(6 7)(9 13)(10 12)(11 15)(14 16)
(1 2)(3 4)(5 7)(6 8)(9 15)(10 16)(11 13)(12 14)
(3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)
(1 16 5 15)(2 10 7 9)(3 12 8 13)(4 14 6 11)

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

G:=Group( (1,4)(2,3)(5,6)(7,8)(9,13)(10,12)(11,15)(14,16), (1,5)(2,7)(3,8)(4,6)(9,10)(11,14)(12,13)(15,16), (1,3)(2,4)(5,8)(6,7)(9,13)(10,12)(11,15)(14,16), (1,2)(3,4)(5,7)(6,8)(9,15)(10,16)(11,13)(12,14), (3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16), (1,16,5,15)(2,10,7,9)(3,12,8,13)(4,14,6,11) );

G=PermutationGroup([(1,4),(2,3),(5,6),(7,8),(9,13),(10,12),(11,15),(14,16)], [(1,5),(2,7),(3,8),(4,6),(9,10),(11,14),(12,13),(15,16)], [(1,3),(2,4),(5,8),(6,7),(9,13),(10,12),(11,15),(14,16)], [(1,2),(3,4),(5,7),(6,8),(9,15),(10,16),(11,13),(12,14)], [(3,4),(5,6,7,8),(9,10,11,12),(13,14,15,16)], [(1,16,5,15),(2,10,7,9),(3,12,8,13),(4,14,6,11)])

G:=TransitiveGroup(16,297);

On 16 points - transitive group 16T310
Generators in S16
(1 5)(4 8)(9 11)(14 16)
(1 11)(2 12)(3 15)(4 16)(5 9)(6 10)(7 13)(8 14)
(2 6)(4 8)(10 12)(14 16)
(1 5)(2 6)(3 7)(4 8)(9 11)(10 12)(13 15)(14 16)
(1 2)(3 4)(5 6)(7 8)(9 10 11 12)(13 14 15 16)
(1 14 11 8)(2 7 12 13)(3 10 15 6)(4 5 16 9)

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

G:=Group( (1,5)(4,8)(9,11)(14,16), (1,11)(2,12)(3,15)(4,16)(5,9)(6,10)(7,13)(8,14), (2,6)(4,8)(10,12)(14,16), (1,5)(2,6)(3,7)(4,8)(9,11)(10,12)(13,15)(14,16), (1,2)(3,4)(5,6)(7,8)(9,10,11,12)(13,14,15,16), (1,14,11,8)(2,7,12,13)(3,10,15,6)(4,5,16,9) );

G=PermutationGroup([(1,5),(4,8),(9,11),(14,16)], [(1,11),(2,12),(3,15),(4,16),(5,9),(6,10),(7,13),(8,14)], [(2,6),(4,8),(10,12),(14,16)], [(1,5),(2,6),(3,7),(4,8),(9,11),(10,12),(13,15),(14,16)], [(1,2),(3,4),(5,6),(7,8),(9,10,11,12),(13,14,15,16)], [(1,14,11,8),(2,7,12,13),(3,10,15,6),(4,5,16,9)])

G:=TransitiveGroup(16,310);

Matrix representation of C24.39D4 in GL8(ℤ)

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

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,-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],[0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,-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],[0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0],[-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,-1,0,0,0,0,0,0,0,0,0,-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,-1,0,0,0,0,0,1,0,0,0,0,0,0,-1,0,0,0],[0,0,0,0,1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,1,0,0,0,0,0,0,-1,0,0,0,0,0,0,0] >;

C24.39D4 in GAP, Magma, Sage, TeX

C_2^4._{39}D_4
% in TeX

G:=Group("C2^4.39D4");
// GroupNames label

G:=SmallGroup(128,859);
// by ID

G=gap.SmallGroup(128,859);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,-2,112,141,723,1123,1018,248,1971,375,4037]);
// Polycyclic

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

Export

Character table of C24.39D4 in TeX

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