Copied to
clipboard

G = (C2×D4).135D4order 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, C41D4.16C4, (C2×C42).25C4, C42.27(C2×C4), (C22×C4).99D4, C4.31(C23⋊C4), C4(C42.3C4), C4(C42.C4), C42.C48C2, C42.3C48C2, C4.4D4.14C4, C4⋊Q8.253C22, (C2×Q8).12C23, C23.25(C22⋊C4), C4.10D4.6C22, C4.4D4.123C22, C22.26C24.24C2, M4(2).8C2216C2, (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

C1C2×C4 — (C2×D4).135D4
C1C2C22C2×C4C2×Q8C2×C4○D4C22.26C24 — (C2×D4).135D4
C1C2C22C2×C4 — (C2×D4).135D4
C1C4C2×C4C2×C4○D4 — (C2×D4).135D4
C1C2C22C2×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 [×5], C4 [×2], C4 [×7], C22, C22 [×9], C8 [×4], C2×C4 [×2], C2×C4 [×2], C2×C4 [×13], D4 [×10], Q8 [×2], C23, C23 [×2], C23, C42 [×2], C42, C22⋊C4 [×4], C4⋊C4 [×2], C2×C8 [×2], M4(2) [×6], C22×C4, C22×C4 [×2], C22×C4 [×2], C2×D4 [×2], C2×D4 [×2], C2×D4 [×3], C2×Q8 [×2], C4○D4 [×4], C4.D4 [×2], C4.10D4 [×4], C2×C42, C4×D4 [×2], C4⋊D4 [×2], C4.4D4 [×2], C41D4, C4⋊Q8, C2×M4(2) [×2], C2×C4○D4 [×2], C42.C4 [×2], C42.3C4 [×2], M4(2).8C22 [×2], C22.26C24, (C2×D4).135D4
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, (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 16 5 12)(2 13 6 9)(3 14 7 10)(4 11 8 15)
(1 12)(2 13)(3 10)(4 11)(5 16)(6 9)(7 14)(8 15)
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)
(1 15 16 8 5 11 12 4)(2 7 9 10 6 3 13 14)

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

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

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

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
,
0020
0002
2000
0200
,
4000
0100
0010
0004
,
0003
2000
0200
0020
,
0200
0040
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

׿
×
𝔽