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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)])

C2.D8 is a maximal subgroup of
C2.D16  C2.Q32  C23.25D4  M4(2)⋊C4  C4×D8  C4×Q16  SD16⋊C4  C87D4  C8.18D4  C8⋊D4  C4.Q16  D4.Q8  Q8.Q8  C22.D8  C23.19D4  C23.48D4  C23.20D4  C8.5Q8  C8⋊Q8  C62.7D4  C4.2PSU3(𝔽2)  C3⋊S3.4D8
 C8p⋊C4: C163C4  C164C4  C241C4  C405C4  D5.D8  C561C4  C44.5Q8  C1045C4 ...
 C2p.D8: D4⋊Q8  C82Q8  C6.Q16  C10.D8  C28.Q8  C44.Q8  C26.D8 ...
C2.D8 is a maximal quotient of
C22.4Q16  C62.7D4  C4.2PSU3(𝔽2)  C3⋊S3.4D8
 C8p⋊C4: C163C4  C164C4  C241C4  C405C4  D5.D8  C561C4  C44.5Q8  C1045C4 ...
 C2p.D8: C81C8  C8.4Q8  C6.Q16  C10.D8  C28.Q8  C44.Q8  C26.D8 ...

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

Export

Subgroup lattice of C2.D8 in TeX
Character table of C2.D8 in TeX

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