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G = C2.D8  order 32 = 25

2nd central extension by C2 of D8

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

Aliases: C8⋊1C4, 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

Derived series
C1 — C4 — C2.D8
Chief series
C1 — C2 — C22 — C2×C4 — C2×C8 — C2.D8
Lower central
C1 — C2 — C4 — C2.D8
Upper central
C1 — C22 — C2×C4 — C2.D8
Jennings
C1 — C2 — C2 — C2×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 >

C1
C2
C2
C2
C4
C22
C4
4C4
4C4
C8
C8
C2×C4
2C2×C4
2C2×C4
C4⋊C4
C2×C8
C4⋊C4
C2.D8

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-√2√2√2    orthogonal lifted from D8
ρ112-2-22000000√2√2-√2-√2    orthogonal lifted from D8
ρ1222-2-2000000-√2√2√2-√2    symplectic lifted from Q16, Schur index 2
ρ1322-2-2000000√2-√2-√2√2    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  C8⋊7D4  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: C16⋊3C4  C16⋊4C4  C24⋊1C4  C40⋊5C4  D5.D8  C56⋊1C4  C44.5Q8  C104⋊5C4 ...
 C2p.D8: D4⋊Q8  C8⋊2Q8  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: C16⋊3C4  C16⋊4C4  C24⋊1C4  C40⋊5C4  D5.D8  C56⋊1C4  C44.5Q8  C104⋊5C4 ...
 C2p.D8: C8⋊1C8  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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