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G = (C2×D4).135D4  order 128 = 27

97th non-split extension by C2×D4 of D4 acting via D4/C2=C22

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

Aliases: (C2×D4).135D4, C4⋊1D4.16C4, (C2×C42).25C4, C42.27(C2×C4), (C22×C4).99D4, C4.31(C23⋊C4), C4○(C42.3C4), C4○(C42.C4), C42.C4⋊8C2, C42.3C4⋊8C2, C4.4D4.14C4, C4⋊Q8.253C22, (C2×Q8).12C23, C23.25(C22⋊C4), C4.10D4.6C22, C4.4D4.123C22, C22.26C24.24C2, M4(2).8C22⋊16C2, (C2×C4).9(C2×D4), (C2×C4○D4).9C4, (C2×D4).40(C2×C4), C2.44(C2×C23⋊C4), (C2×Q8).39(C2×C4), (C22×C4).84(C2×C4), (C2×C4).101(C22×C4), (C2×C4○D4).77C22, C22.68(C2×C22⋊C4), (C2×C4).365(C22⋊C4), SmallGroup(128,864)

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

Derived series
C1 — C2×C4 — (C2×D4).135D4
Chief series
C1 — C2 — C22 — C2×C4 — C2×Q8 — C2×C4○D4 — C22.26C24 — (C2×D4).135D4
Lower central
C1 — C2 — C22 — C2×C4 — (C2×D4).135D4
Upper central
C1 — C4 — C2×C4 — C2×C4○D4 — (C2×D4).135D4
Jennings
C1 — C2 — C22 — C2×Q8 — (C2×D4).135D4

Generators and relations for (C2×D4).135D4
 G = < a,b,c,d,e | a2=b4=c2=1, d4=b2, e2=ab-1, dbd-1=ab=ba, ece-1=ac=ca, dad-1=eae-1=ab2, cbc=ebe-1=b-1, dcd-1=ab2c, ede-1=abd3 >

Subgroups: 292 in 125 conjugacy classes, 42 normal (24 characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, C23, C42, C42, C22⋊C4, C4⋊C4, C2×C8, M4(2), C22×C4, C22×C4, C22×C4, C2×D4, C2×D4, C2×D4, C2×Q8, C4○D4, C4.D4, C4.10D4, C2×C42, C4×D4, C4⋊D4, C4.4D4, C4⋊1D4, C4⋊Q8, C2×M4(2), C2×C4○D4, C42.C4, C42.3C4, M4(2).8C22, C22.26C24, (C2×D4).135D4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22⋊C4, C22×C4, C2×D4, C23⋊C4, C2×C22⋊C4, C2×C23⋊C4, (C2×D4).135D4

Character table of (C2×D4).135D4

 class 12A2B2C2D2E2F4A4B4C4D4E4F4G4H4I4J4K8A8B8C8D8E8F8G8H
 size 11244481124444444888888888
ρ111111111111111111111111111    trivial
ρ2111-1-1-11-1-1-1-111-1111-1-11-111-11-1    linear of order 2
ρ3111-1-1-11-1-1-1-111-1111-11-11-1-11-11    linear of order 2
ρ4111111111111111111-1-1-1-1-1-1-1-1    linear of order 2
ρ5111-1-1-1-1-1-1-111-1111-111111-1-1-1-1    linear of order 2
ρ6111111-1111-11-1-111-1-1-11-11-11-11    linear of order 2
ρ7111111-1111-11-1-111-1-11-11-11-11-1    linear of order 2
ρ8111-1-1-1-1-1-1-111-1111-11-1-1-1-11111    linear of order 2
ρ9111-11-1-11111111-1-11-1-iii-ii-i-ii    linear of order 4
ρ101111-11-1-1-1-1-111-1-1-111ii-i-iii-i-i    linear of order 4
ρ11111-11-1-11111111-1-11-1i-i-ii-iii-i    linear of order 4
ρ121111-11-1-1-1-1-111-1-1-111-i-iii-i-iii    linear of order 4
ρ131111-111-1-1-111-11-1-1-1-1-iii-i-iii-i    linear of order 4
ρ14111-11-11111-11-1-1-1-1-11ii-i-i-i-iii    linear of order 4
ρ151111-111-1-1-111-11-1-1-1-1i-i-iii-i-ii    linear of order 4
ρ16111-11-11111-11-1-1-1-1-11-i-iiiii-i-i    linear of order 4
ρ17222-2220-2-2-20-2002-20000000000    orthogonal lifted from D4
ρ182222-2-202220-2002-20000000000    orthogonal lifted from D4
ρ1922222-20-2-2-20-200-220000000000    orthogonal lifted from D4
ρ20222-2-2202220-200-220000000000    orthogonal lifted from D4
ρ2144-4000044-40000000000000000    orthogonal lifted from C23⋊C4
ρ2244-40000-4-440000000000000000    orthogonal lifted from C23⋊C4
ρ234-400000-4i4i020-2i-2002i000000000    complex faithful
ρ244-400000-4i4i0-202i200-2i000000000    complex faithful
ρ254-4000004i-4i0202i-200-2i000000000    complex faithful
ρ264-4000004i-4i0-20-2i2002i000000000    complex faithful

Permutation representations of (C2×D4).135D4
►On 16 points - transitive group 16T300
Generators in S16
(1 5)(3 7)(10 14)(12 16)

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

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

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

(1 13 14 8 5 9 10 4)(2 7 15 16 6 3 11 12)
 
G:=sub<Sym(16)| (1,5)(3,7)(10,14)(12,16), (1,14,5,10)(2,11,6,15)(3,12,7,16)(4,9,8,13), (1,10)(2,11)(3,16)(4,9)(5,14)(6,15)(7,12)(8,13), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16), (1,13,14,8,5,9,10,4)(2,7,15,16,6,3,11,12)>;
 
G:=Group( (1,5)(3,7)(10,14)(12,16), (1,14,5,10)(2,11,6,15)(3,12,7,16)(4,9,8,13), (1,10)(2,11)(3,16)(4,9)(5,14)(6,15)(7,12)(8,13), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16), (1,13,14,8,5,9,10,4)(2,7,15,16,6,3,11,12) );
 
G=PermutationGroup([[(1,5),(3,7),(10,14),(12,16)], [(1,14,5,10),(2,11,6,15),(3,12,7,16),(4,9,8,13)], [(1,10),(2,11),(3,16),(4,9),(5,14),(6,15),(7,12),(8,13)], [(1,2,3,4,5,6,7,8),(9,10,11,12,13,14,15,16)], [(1,13,14,8,5,9,10,4),(2,7,15,16,6,3,11,12)]])
 
G:=TransitiveGroup(16,300);
 

►On 16 points - transitive group 16T327
Generators in S16
(1 5)(3 7)(10 14)(12 16)

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

(3 7)(4 8)(9 13)(12 16)

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

(1 4 16 15 5 8 12 11)(2 14 9 7 6 10 13 3)
 
G:=sub<Sym(16)| (1,5)(3,7)(10,14)(12,16), (1,16,5,12)(2,13,6,9)(3,14,7,10)(4,11,8,15), (3,7)(4,8)(9,13)(12,16), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16), (1,4,16,15,5,8,12,11)(2,14,9,7,6,10,13,3)>;
 
G:=Group( (1,5)(3,7)(10,14)(12,16), (1,16,5,12)(2,13,6,9)(3,14,7,10)(4,11,8,15), (3,7)(4,8)(9,13)(12,16), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16), (1,4,16,15,5,8,12,11)(2,14,9,7,6,10,13,3) );
 
G=PermutationGroup([[(1,5),(3,7),(10,14),(12,16)], [(1,16,5,12),(2,13,6,9),(3,14,7,10),(4,11,8,15)], [(3,7),(4,8),(9,13),(12,16)], [(1,2,3,4,5,6,7,8),(9,10,11,12,13,14,15,16)], [(1,4,16,15,5,8,12,11),(2,14,9,7,6,10,13,3)]])
 
G:=TransitiveGroup(16,327);
 

Matrix representation of (C2×D4).135D4 ►in GL4(𝔽5) generated by

1000
0400
0010
0004
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0020
0002
2000
0200
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4000
0100
0010
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0003
2000
0200
0020
,
0200
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0003
1000
G:=sub<GL(4,GF(5))| [1,0,0,0,0,4,0,0,0,0,1,0,0,0,0,4],[0,0,2,0,0,0,0,2,2,0,0,0,0,2,0,0],[4,0,0,0,0,1,0,0,0,0,1,0,0,0,0,4],[0,2,0,0,0,0,2,0,0,0,0,2,3,0,0,0],[0,0,0,1,2,0,0,0,0,4,0,0,0,0,3,0] >;
 

(C2×D4).135D4 in GAP, Magma, Sage, TeX 

(C_2\times D_4)._{135}D_4
 
% in TeX
 
G:=Group("(C2xD4).135D4");
 
// GroupNames label
 
G:=SmallGroup(128,864);
 
// by ID
 
G=gap.SmallGroup(128,864);
 
# by ID
 
G:=PCGroup([7,-2,2,2,-2,2,-2,-2,112,141,352,1123,1018,248,1971,375,172,4037]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^2=b^4=c^2=1,d^4=b^2,e^2=a*b^-1,d*b*d^-1=a*b=b*a,e*c*e^-1=a*c=c*a,d*a*d^-1=e*a*e^-1=a*b^2,c*b*c=e*b*e^-1=b^-1,d*c*d^-1=a*b^2*c,e*d*e^-1=a*b*d^3>;
 
// generators/relations
 

Export

Character table of (C2×D4).135D4 in TeX

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