Copied to
clipboard

## G = (C2×Q8).211D4order 128 = 27

### 19th non-split extension by C2×Q8 of D4 acting via D4/C22=C2

p-group, metabelian, nilpotent (class 3), monomial

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C23 — (C2×Q8).211D4
 Chief series C1 — C2 — C4 — C2×C4 — C22×C4 — C42⋊C2 — C23.32C23 — (C2×Q8).211D4
 Lower central C1 — C2 — C23 — (C2×Q8).211D4
 Upper central C1 — C2 — C22×C4 — (C2×Q8).211D4
 Jennings C1 — C2 — C2 — C22×C4 — (C2×Q8).211D4

Generators and relations for (C2×Q8).211D4
G = < a,b,c,d,e | a2=b4=1, c2=d4=b2, e2=ab2c, ab=ba, ac=ca, dad-1=eae-1=ab2, cbc-1=b-1, dbd-1=ebe-1=ab-1, dcd-1=ece-1=b2c, ede-1=b2cd3 >

Subgroups: 204 in 118 conjugacy classes, 50 normal (8 characteristic)
C1, C2, C2 [×3], C4 [×4], C4 [×8], C22, C22 [×2], C22, C8 [×4], C2×C4 [×10], C2×C4 [×8], Q8 [×8], C23, C42 [×6], C22⋊C4 [×4], C4⋊C4 [×6], C2×C8 [×4], M4(2) [×8], C22×C4, C22×C4 [×2], C2×Q8 [×4], C2×Q8 [×4], C4.10D4 [×4], C42⋊C2 [×6], C4×Q8 [×4], C2×M4(2) [×4], C22×Q8, M4(2)⋊4C4 [×4], C2×C4.10D4 [×2], C23.32C23, (C2×Q8).211D4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×4], C23, C22⋊C4 [×4], C22×C4, C2×D4 [×2], C4○D4 [×4], C2×C22⋊C4, C42⋊C2 [×2], C22.D4 [×4], C23.34D4, (C2×Q8).211D4

Character table of (C2×Q8).211D4

 class 1 2A 2B 2C 2D 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 4K 4L 4M 4N 4O 4P 8A 8B 8C 8D 8E 8F 8G 8H size 1 1 2 2 2 2 2 2 2 4 4 4 4 4 4 4 4 4 4 4 4 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 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 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 -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 -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 -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 -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 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 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 -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 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 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 -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 -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 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 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 -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 0 0 0 0 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 0 0 0 0 -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 0 0 0 0 -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 0 0 0 0 2 2 -2 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ21 2 2 2 -2 -2 2 -2 2 -2 0 -2i 0 0 0 0 0 0 2i 2i -2i 0 0 0 0 0 0 0 0 0 complex lifted from C4○D4 ρ22 2 2 -2 2 -2 2 2 -2 -2 0 0 2i -2i -2i 0 0 0 0 0 0 2i 0 0 0 0 0 0 0 0 complex lifted from C4○D4 ρ23 2 2 -2 2 -2 -2 -2 2 2 0 0 -2i -2i 2i 0 0 0 0 0 0 2i 0 0 0 0 0 0 0 0 complex lifted from C4○D4 ρ24 2 2 2 -2 -2 2 -2 2 -2 0 2i 0 0 0 0 0 0 -2i -2i 2i 0 0 0 0 0 0 0 0 0 complex lifted from C4○D4 ρ25 2 2 -2 2 -2 -2 -2 2 2 0 0 2i 2i -2i 0 0 0 0 0 0 -2i 0 0 0 0 0 0 0 0 complex lifted from C4○D4 ρ26 2 2 2 -2 -2 -2 2 -2 2 0 2i 0 0 0 0 0 0 2i -2i -2i 0 0 0 0 0 0 0 0 0 complex lifted from C4○D4 ρ27 2 2 -2 2 -2 2 2 -2 -2 0 0 -2i 2i 2i 0 0 0 0 0 0 -2i 0 0 0 0 0 0 0 0 complex lifted from C4○D4 ρ28 2 2 2 -2 -2 -2 2 -2 2 0 -2i 0 0 0 0 0 0 -2i 2i 2i 0 0 0 0 0 0 0 0 0 complex lifted from C4○D4 ρ29 8 -8 0 0 0 0 0 0 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×Q8).211D4
On 32 points
Generators in S32
(2 6)(4 8)(10 14)(12 16)(18 22)(20 24)(26 30)(28 32)
(1 19 5 23)(2 24 6 20)(3 17 7 21)(4 22 8 18)(9 27 13 31)(10 32 14 28)(11 25 15 29)(12 30 16 26)
(1 29 5 25)(2 26 6 30)(3 31 7 27)(4 28 8 32)(9 17 13 21)(10 22 14 18)(11 19 15 23)(12 24 16 20)
(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 10 25 22 5 14 29 18)(2 21 26 13 6 17 30 9)(3 16 27 20 7 12 31 24)(4 19 28 11 8 23 32 15)

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

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

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

Matrix representation of (C2×Q8).211D4 in GL8(𝔽17)

 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 16 0 0 0 13 4 0 0 0 16 0 0 0 0 0 0 0 0 16 0 13 4 0 0 0 0 0 16
,
 7 5 5 7 0 0 0 0 0 5 5 0 0 0 0 0 0 5 12 0 0 0 0 0 5 0 12 10 0 0 0 0 14 3 0 0 12 0 5 7 3 0 0 3 12 5 0 12 14 3 0 0 5 7 12 0 3 0 0 3 0 12 12 5
,
 0 16 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 16 16 2 0 0 0 0 1 0 16 1 0 0 0 0 13 4 0 0 1 15 0 0 0 4 0 0 1 16 0 0 13 4 0 0 0 0 1 15 0 4 0 0 0 0 1 16
,
 14 3 0 0 12 0 5 7 3 14 0 0 12 10 12 0 3 14 0 0 12 10 5 0 6 11 0 0 12 10 12 10 0 5 5 7 0 0 0 0 12 5 5 12 0 3 14 3 0 12 12 0 0 0 0 0 7 5 0 12 0 3 14 3
,
 0 0 0 0 1 0 0 0 13 4 0 0 1 15 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 16 0 1 0 1 0 0 0 0 0 0 1 0 0 0 0 13 0 0 16 1 1 15 0 0 0 0 0 0 1 16 0 13 0 0

G:=sub<GL(8,GF(17))| [1,0,0,0,0,13,0,13,0,1,0,0,0,4,0,4,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16],[7,0,0,5,14,3,14,3,5,5,5,0,3,0,3,0,5,5,12,12,0,0,0,0,7,0,0,10,0,3,0,3,0,0,0,0,12,12,5,0,0,0,0,0,0,5,7,12,0,0,0,0,5,0,12,12,0,0,0,0,7,12,0,5],[0,1,1,1,13,0,13,0,16,0,16,0,4,4,4,4,0,0,16,16,0,0,0,0,0,0,2,1,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,15,16,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,15,16],[14,3,3,6,0,12,0,7,3,14,14,11,5,5,12,5,0,0,0,0,5,5,12,0,0,0,0,0,7,12,0,12,12,12,12,12,0,0,0,0,0,10,10,10,0,3,0,3,5,12,5,12,0,14,0,14,7,0,0,10,0,3,0,3],[0,13,0,0,0,1,16,0,0,4,0,0,1,0,1,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,15,16,1,1,0,0,0,0,0,0,0,15,0,16,0,13,0,13,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0] >;

(C2×Q8).211D4 in GAP, Magma, Sage, TeX

(C_2\times Q_8)._{211}D_4
% in TeX

G:=Group("(C2xQ8).211D4");
// GroupNames label

G:=SmallGroup(128,562);
// by ID

G=gap.SmallGroup(128,562);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,456,422,58,2019,718,2028,124]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^4=1,c^2=d^4=b^2,e^2=a*b^2*c,a*b=b*a,a*c=c*a,d*a*d^-1=e*a*e^-1=a*b^2,c*b*c^-1=b^-1,d*b*d^-1=e*b*e^-1=a*b^-1,d*c*d^-1=e*c*e^-1=b^2*c,e*d*e^-1=b^2*c*d^3>;
// generators/relations

Export

׿
×
𝔽