Copied to
clipboard

G = C2.D8order 32 = 25

2nd central extension by C2 of D8

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

Aliases: C81C4, C2.2D8, C4.2Q8, C2.2Q16, C22.11D4, C4⋊C4.3C2, (C2×C8).3C2, C4.7(C2×C4), C2.4(C4⋊C4), (C2×C4).18C22, SmallGroup(32,14)

Series: Derived Chief Lower central Upper central Jennings

C1C4 — C2.D8
C1C2C22C2×C4C2×C8 — C2.D8
C1C2C4 — C2.D8
C1C22C2×C4 — C2.D8
C1C2C2C2×C4 — C2.D8

Generators and relations for C2.D8
 G = < a,b,c | a2=b8=1, c2=a, ab=ba, ac=ca, cbc-1=b-1 >

4C4
4C4
2C2×C4
2C2×C4

Character table of C2.D8

 class 12A2B2C4A4B4C4D4E4F8A8B8C8D
 size 11112244442222
ρ111111111111111    trivial
ρ2111111-111-1-1-1-1-1    linear of order 2
ρ31111111-1-11-1-1-1-1    linear of order 2
ρ4111111-1-1-1-11111    linear of order 2
ρ51-11-11-1i-ii-i-11-11    linear of order 4
ρ61-11-11-1-i-iii1-11-1    linear of order 4
ρ71-11-11-1ii-i-i1-11-1    linear of order 4
ρ81-11-11-1-ii-ii-11-11    linear of order 4
ρ92222-2-200000000    orthogonal lifted from D4
ρ102-2-22000000-2-222    orthogonal lifted from D8
ρ112-2-2200000022-2-2    orthogonal lifted from D8
ρ1222-2-2000000-222-2    symplectic lifted from Q16, Schur index 2
ρ1322-2-20000002-2-22    symplectic lifted from Q16, Schur index 2
ρ142-22-2-2200000000    symplectic lifted from Q8, Schur index 2

Smallest permutation representation of C2.D8
Regular action on 32 points
Generators in S32
(1 23)(2 24)(3 17)(4 18)(5 19)(6 20)(7 21)(8 22)(9 31)(10 32)(11 25)(12 26)(13 27)(14 28)(15 29)(16 30)
(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 9 23 31)(2 16 24 30)(3 15 17 29)(4 14 18 28)(5 13 19 27)(6 12 20 26)(7 11 21 25)(8 10 22 32)

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

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

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

Matrix representation of C2.D8 in GL3(𝔽17) generated by

1600
010
001
,
1600
090
002
,
400
002
090
G:=sub<GL(3,GF(17))| [16,0,0,0,1,0,0,0,1],[16,0,0,0,9,0,0,0,2],[4,0,0,0,0,9,0,2,0] >;

C2.D8 in GAP, Magma, Sage, TeX

C_2.D_8
% in TeX

G:=Group("C2.D8");
// GroupNames label

G:=SmallGroup(32,14);
// by ID

G=gap.SmallGroup(32,14);
# by ID

G:=PCGroup([5,-2,2,-2,2,-2,40,61,106,302,72]);
// Polycyclic

G:=Group<a,b,c|a^2=b^8=1,c^2=a,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations

׿
×
𝔽