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## 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

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 [×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 1 2A 2B 2C 2D 2E 2F 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 4K 8A 8B 8C 8D 8E 8F 8G 8H size 1 1 2 4 4 4 8 1 1 2 4 4 4 4 4 4 4 8 8 8 8 8 8 8 8 8 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 1 1 -1 -1 -1 1 -1 -1 -1 -1 1 1 -1 1 1 1 -1 -1 1 -1 1 1 -1 1 -1 linear of order 2 ρ3 1 1 1 -1 -1 -1 1 -1 -1 -1 -1 1 1 -1 1 1 1 -1 1 -1 1 -1 -1 1 -1 1 linear of order 2 ρ4 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 -1 -1 linear of order 2 ρ5 1 1 1 -1 -1 -1 -1 -1 -1 -1 1 1 -1 1 1 1 -1 1 1 1 1 1 -1 -1 -1 -1 linear of order 2 ρ6 1 1 1 1 1 1 -1 1 1 1 -1 1 -1 -1 1 1 -1 -1 -1 1 -1 1 -1 1 -1 1 linear of order 2 ρ7 1 1 1 1 1 1 -1 1 1 1 -1 1 -1 -1 1 1 -1 -1 1 -1 1 -1 1 -1 1 -1 linear of order 2 ρ8 1 1 1 -1 -1 -1 -1 -1 -1 -1 1 1 -1 1 1 1 -1 1 -1 -1 -1 -1 1 1 1 1 linear of order 2 ρ9 1 1 1 -1 1 -1 -1 1 1 1 1 1 1 1 -1 -1 1 -1 -i i i -i i -i -i i linear of order 4 ρ10 1 1 1 1 -1 1 -1 -1 -1 -1 -1 1 1 -1 -1 -1 1 1 i i -i -i i i -i -i linear of order 4 ρ11 1 1 1 -1 1 -1 -1 1 1 1 1 1 1 1 -1 -1 1 -1 i -i -i i -i i i -i linear of order 4 ρ12 1 1 1 1 -1 1 -1 -1 -1 -1 -1 1 1 -1 -1 -1 1 1 -i -i i i -i -i i i linear of order 4 ρ13 1 1 1 1 -1 1 1 -1 -1 -1 1 1 -1 1 -1 -1 -1 -1 -i i i -i -i i i -i linear of order 4 ρ14 1 1 1 -1 1 -1 1 1 1 1 -1 1 -1 -1 -1 -1 -1 1 i i -i -i -i -i i i linear of order 4 ρ15 1 1 1 1 -1 1 1 -1 -1 -1 1 1 -1 1 -1 -1 -1 -1 i -i -i i i -i -i i linear of order 4 ρ16 1 1 1 -1 1 -1 1 1 1 1 -1 1 -1 -1 -1 -1 -1 1 -i -i i i i i -i -i linear of order 4 ρ17 2 2 2 -2 2 2 0 -2 -2 -2 0 -2 0 0 2 -2 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ18 2 2 2 2 -2 -2 0 2 2 2 0 -2 0 0 2 -2 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ19 2 2 2 2 2 -2 0 -2 -2 -2 0 -2 0 0 -2 2 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ20 2 2 2 -2 -2 2 0 2 2 2 0 -2 0 0 -2 2 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ21 4 4 -4 0 0 0 0 4 4 -4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from C23⋊C4 ρ22 4 4 -4 0 0 0 0 -4 -4 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from C23⋊C4 ρ23 4 -4 0 0 0 0 0 -4i 4i 0 2 0 -2i -2 0 0 2i 0 0 0 0 0 0 0 0 0 complex faithful ρ24 4 -4 0 0 0 0 0 -4i 4i 0 -2 0 2i 2 0 0 -2i 0 0 0 0 0 0 0 0 0 complex faithful ρ25 4 -4 0 0 0 0 0 4i -4i 0 2 0 2i -2 0 0 -2i 0 0 0 0 0 0 0 0 0 complex faithful ρ26 4 -4 0 0 0 0 0 4i -4i 0 -2 0 -2i 2 0 0 2i 0 0 0 0 0 0 0 0 0 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

 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 0 0 3 2 0 0 0 0 2 0 0 0 0 2 0
,
 0 2 0 0 0 0 4 0 0 0 0 3 1 0 0 0
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

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