Copied to
clipboard

## G = (C2×D4).137D4order 128 = 27

### 99th 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).137D4
 Chief series C1 — C2 — C22 — C2×C4 — C2×Q8 — C2×C4○D4 — C23.38C23 — (C2×D4).137D4
 Lower central C1 — C2 — C22 — C2×C4 — (C2×D4).137D4
 Upper central C1 — C2 — C2×C4 — C2×C4○D4 — (C2×D4).137D4
 Jennings C1 — C2 — C22 — C2×Q8 — (C2×D4).137D4

Generators and relations for (C2×D4).137D4
G = < a,b,c,d,e | a2=b4=c2=1, d4=b2, e2=ab-1, dbd-1=ab=ba, dcd-1=ece-1=ac=ca, dad-1=eae-1=ab2, cbc=ebe-1=b-1, ede-1=ab-1d3 >

Subgroups: 252 in 115 conjugacy classes, 42 normal (14 characteristic)
C1, C2, C2 [×4], C4 [×2], C4 [×7], C22, C22 [×6], C8 [×4], C2×C4 [×2], C2×C4 [×2], C2×C4 [×12], D4 [×3], Q8 [×5], C23, C23 [×2], C42 [×2], C22⋊C4 [×5], C4⋊C4 [×5], C2×C8 [×2], M4(2) [×6], C22×C4, C22×C4 [×2], C22×C4, C2×D4, C2×D4 [×2], C2×Q8, C2×Q8 [×2], C2×Q8 [×3], C4○D4 [×2], C4.D4 [×2], C4.10D4 [×4], C42⋊C2, C22⋊Q8 [×2], C22.D4 [×2], C4.4D4 [×2], C4⋊Q8 [×2], C2×M4(2) [×2], C22×Q8, C2×C4○D4, C42.3C4 [×4], M4(2).8C22 [×2], C23.38C23, (C2×D4).137D4
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).137D4

Character table of (C2×D4).137D4

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 4D 4E 4F 4G 4H 4I 8A 8B 8C 8D 8E 8F 8G 8H size 1 1 2 4 4 4 2 2 4 4 4 8 8 8 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 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 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 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 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 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 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 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 linear of order 2 ρ9 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 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 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 -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 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 -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 -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 i i -i i -i -i i -i linear of order 4 ρ17 2 2 2 2 -2 -2 2 2 2 -2 -2 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ18 2 2 2 -2 -2 2 2 2 -2 -2 2 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ19 2 2 2 2 2 -2 -2 -2 -2 -2 2 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ20 2 2 2 -2 2 2 -2 -2 2 -2 -2 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ21 4 4 -4 0 0 0 4 -4 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 -4 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from C23⋊C4 ρ23 8 -8 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 symplectic faithful, Schur index 2

Smallest permutation representation of (C2×D4).137D4
On 32 points
Generators in S32
(1 5)(3 7)(9 13)(11 15)(17 21)(19 23)(26 30)(28 32)
(1 28 5 32)(2 25 6 29)(3 26 7 30)(4 31 8 27)(9 17 13 21)(10 22 14 18)(11 23 15 19)(12 20 16 24)
(1 7)(2 4)(3 5)(6 8)(9 11)(10 16)(12 14)(13 15)(17 19)(18 24)(20 22)(21 23)(25 27)(26 28)(29 31)(30 32)
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)(17 18 19 20 21 22 23 24)(25 26 27 28 29 30 31 32)
(1 14 28 22 5 10 32 18)(2 17 29 13 6 21 25 9)(3 12 26 24 7 16 30 20)(4 19 27 11 8 23 31 15)

G:=sub<Sym(32)| (1,5)(3,7)(9,13)(11,15)(17,21)(19,23)(26,30)(28,32), (1,28,5,32)(2,25,6,29)(3,26,7,30)(4,31,8,27)(9,17,13,21)(10,22,14,18)(11,23,15,19)(12,20,16,24), (1,7)(2,4)(3,5)(6,8)(9,11)(10,16)(12,14)(13,15)(17,19)(18,24)(20,22)(21,23)(25,27)(26,28)(29,31)(30,32), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32), (1,14,28,22,5,10,32,18)(2,17,29,13,6,21,25,9)(3,12,26,24,7,16,30,20)(4,19,27,11,8,23,31,15)>;

G:=Group( (1,5)(3,7)(9,13)(11,15)(17,21)(19,23)(26,30)(28,32), (1,28,5,32)(2,25,6,29)(3,26,7,30)(4,31,8,27)(9,17,13,21)(10,22,14,18)(11,23,15,19)(12,20,16,24), (1,7)(2,4)(3,5)(6,8)(9,11)(10,16)(12,14)(13,15)(17,19)(18,24)(20,22)(21,23)(25,27)(26,28)(29,31)(30,32), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32), (1,14,28,22,5,10,32,18)(2,17,29,13,6,21,25,9)(3,12,26,24,7,16,30,20)(4,19,27,11,8,23,31,15) );

G=PermutationGroup([(1,5),(3,7),(9,13),(11,15),(17,21),(19,23),(26,30),(28,32)], [(1,28,5,32),(2,25,6,29),(3,26,7,30),(4,31,8,27),(9,17,13,21),(10,22,14,18),(11,23,15,19),(12,20,16,24)], [(1,7),(2,4),(3,5),(6,8),(9,11),(10,16),(12,14),(13,15),(17,19),(18,24),(20,22),(21,23),(25,27),(26,28),(29,31),(30,32)], [(1,2,3,4,5,6,7,8),(9,10,11,12,13,14,15,16),(17,18,19,20,21,22,23,24),(25,26,27,28,29,30,31,32)], [(1,14,28,22,5,10,32,18),(2,17,29,13,6,21,25,9),(3,12,26,24,7,16,30,20),(4,19,27,11,8,23,31,15)])

Matrix representation of (C2×D4).137D4 in GL8(𝔽17)

 16 0 0 0 0 0 0 0 0 16 0 0 0 0 0 0 0 0 16 0 0 0 0 0 0 0 0 16 0 0 0 0 0 0 0 0 1 0 0 0 13 16 14 0 0 1 0 0 13 12 1 0 0 0 1 0 0 4 13 0 0 0 0 1
,
 0 1 0 0 0 0 0 0 16 0 0 0 0 0 0 0 16 16 1 2 0 0 0 0 0 1 16 16 0 0 0 0 13 16 14 0 16 2 0 0 13 0 0 3 16 1 0 0 13 4 13 16 16 1 0 1 0 0 0 4 0 1 16 0
,
 16 16 1 2 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 16 1 1 0 0 0 0 0 4 13 0 16 0 0 2 13 16 1 4 16 0 1 1 13 16 1 4 16 1 0 1 0 4 13 0 0 0 0 1
,
 0 0 0 0 13 0 0 0 16 4 12 0 4 9 0 0 16 3 13 0 4 0 9 0 0 1 16 0 0 13 4 13 13 13 4 8 0 0 0 0 13 13 4 4 0 13 5 0 13 13 0 4 0 14 4 0 13 13 4 4 0 16 1 0
,
 0 0 0 0 1 0 0 0 4 1 3 0 1 15 0 0 4 5 16 0 1 0 15 0 0 0 0 0 0 16 1 1 0 1 0 0 0 0 0 0 0 0 0 0 4 16 14 0 0 1 16 16 4 12 1 0 0 0 0 0 0 4 13 0

G:=sub<GL(8,GF(17))| [16,0,0,0,0,13,13,0,0,16,0,0,0,16,12,4,0,0,16,0,0,14,1,13,0,0,0,16,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[0,16,16,0,13,13,13,0,1,0,16,1,16,0,4,0,0,0,1,16,14,0,13,0,0,0,2,16,0,3,16,4,0,0,0,0,16,16,16,0,0,0,0,0,2,1,1,1,0,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0],[16,0,0,0,0,13,13,0,16,0,1,16,4,16,16,4,1,1,0,1,13,1,1,13,2,0,0,1,0,4,4,0,0,0,0,0,16,16,16,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,2,1,1,1],[0,16,16,0,13,13,13,13,0,4,3,1,13,13,13,13,0,12,13,16,4,4,0,4,0,0,0,0,8,4,4,4,13,4,4,0,0,0,0,0,0,9,0,13,0,13,14,16,0,0,9,4,0,5,4,1,0,0,0,13,0,0,0,0],[0,4,4,0,0,0,0,0,0,1,5,0,1,0,1,0,0,3,16,0,0,0,16,0,0,0,0,0,0,0,16,0,1,1,1,0,0,4,4,0,0,15,0,16,0,16,12,4,0,0,15,1,0,14,1,13,0,0,0,1,0,0,0,0] >;

(C2×D4).137D4 in GAP, Magma, Sage, TeX

(C_2\times D_4)._{137}D_4
% in TeX

G:=Group("(C2xD4).137D4");
// GroupNames label

G:=SmallGroup(128,867);
// by ID

G=gap.SmallGroup(128,867);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,-2,112,141,456,723,352,1123,1018,248,1971,375,172,4037]);
// 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,d*c*d^-1=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,e*d*e^-1=a*b^-1*d^3>;
// generators/relations

Export

׿
×
𝔽