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G = C24order 16 = 24

Elementary abelian group of type [2,2,2,2]

direct product, p-group, elementary abelian, monomial, rational

Aliases: C24, SmallGroup(16,14)

Series: Derived Chief Lower central Upper central Jennings

C1 — C24
C1C2C22C23 — C24
C1 — C24
C1 — C24
C1 — C24

Generators and relations for C24
 G = < a,b,c,d | a2=b2=c2=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, cd=dc >

Subgroups: 67, all normal (2 characteristic)
C1, C2 [×15], C22 [×35], C23 [×15], C24
Quotients: C1, C2 [×15], C22 [×35], C23 [×15], C24

Character table of C24

 class 12A2B2C2D2E2F2G2H2I2J2K2L2M2N2O
 size 1111111111111111
ρ11111111111111111    trivial
ρ21-1111111-1-1-1-1-1-1-11    linear of order 2
ρ31-111-1-1-1-11111-1-1-11    linear of order 2
ρ41111-1-1-1-1-1-1-1-11111    linear of order 2
ρ51-1-1-111-1-111-1-111-11    linear of order 2
ρ611-1-111-1-1-1-111-1-111    linear of order 2
ρ711-1-1-1-11111-1-1-1-111    linear of order 2
ρ81-1-1-1-1-111-1-11111-11    linear of order 2
ρ91-11-11-11-11-11-11-11-1    linear of order 2
ρ10111-11-11-1-11-11-11-1-1    linear of order 2
ρ11111-1-11-111-11-1-11-1-1    linear of order 2
ρ121-11-1-11-11-11-111-11-1    linear of order 2
ρ1311-111-1-111-1-111-1-1-1    linear of order 2
ρ141-1-111-1-11-111-1-111-1    linear of order 2
ρ151-1-11-111-11-1-11-111-1    linear of order 2
ρ1611-11-111-1-111-11-1-1-1    linear of order 2

Permutation representations of C24
Regular action on 16 points - transitive group 16T3
Generators in S16
(1 2)(3 4)(5 6)(7 8)(9 10)(11 12)(13 14)(15 16)
(1 5)(2 6)(3 11)(4 12)(7 10)(8 9)(13 15)(14 16)
(1 11)(2 12)(3 5)(4 6)(7 13)(8 14)(9 16)(10 15)
(1 13)(2 14)(3 10)(4 9)(5 15)(6 16)(7 11)(8 12)

G:=sub<Sym(16)| (1,2)(3,4)(5,6)(7,8)(9,10)(11,12)(13,14)(15,16), (1,5)(2,6)(3,11)(4,12)(7,10)(8,9)(13,15)(14,16), (1,11)(2,12)(3,5)(4,6)(7,13)(8,14)(9,16)(10,15), (1,13)(2,14)(3,10)(4,9)(5,15)(6,16)(7,11)(8,12)>;

G:=Group( (1,2)(3,4)(5,6)(7,8)(9,10)(11,12)(13,14)(15,16), (1,5)(2,6)(3,11)(4,12)(7,10)(8,9)(13,15)(14,16), (1,11)(2,12)(3,5)(4,6)(7,13)(8,14)(9,16)(10,15), (1,13)(2,14)(3,10)(4,9)(5,15)(6,16)(7,11)(8,12) );

G=PermutationGroup([(1,2),(3,4),(5,6),(7,8),(9,10),(11,12),(13,14),(15,16)], [(1,5),(2,6),(3,11),(4,12),(7,10),(8,9),(13,15),(14,16)], [(1,11),(2,12),(3,5),(4,6),(7,13),(8,14),(9,16),(10,15)], [(1,13),(2,14),(3,10),(4,9),(5,15),(6,16),(7,11),(8,12)])

G:=TransitiveGroup(16,3);

Matrix representation of C24 in GL4(ℤ) generated by

1000
0-100
0010
000-1
,
-1000
0100
00-10
0001
,
1000
0100
00-10
0001
,
1000
0100
0010
000-1
G:=sub<GL(4,Integers())| [1,0,0,0,0,-1,0,0,0,0,1,0,0,0,0,-1],[-1,0,0,0,0,1,0,0,0,0,-1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,-1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,1,0,0,0,0,-1] >;

C24 in GAP, Magma, Sage, TeX

C_2^4
% in TeX

G:=Group("C2^4");
// GroupNames label

G:=SmallGroup(16,14);
// by ID

G=gap.SmallGroup(16,14);
# by ID

G:=PCGroup([4,-2,2,2,2]);
// Polycyclic

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

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