Copied to
clipboard

G = C243C4order 64 = 26

1st semidirect product of C24 and C4 acting via C4/C2=C2

p-group, metabelian, nilpotent (class 2), monomial

Aliases: C243C4, C25.1C2, C23.33D4, C23.55C23, C24.23C22, C2.1C22≀C2, C23.24(C2×C4), (C22×C4)⋊1C22, C22.29(C2×D4), C222(C22⋊C4), C22.28(C22×C4), (C2×C22⋊C4)⋊1C2, C2.4(C2×C22⋊C4), SmallGroup(64,60)

Series: Derived Chief Lower central Upper central Jennings

C1C22 — C243C4
C1C2C22C23C24C25 — C243C4
C1C22 — C243C4
C1C23 — C243C4
C1C23 — C243C4

Generators and relations for C243C4
 G = < a,b,c,d,e | a2=b2=c2=d2=e4=1, ab=ba, eae-1=ac=ca, ad=da, bc=cb, ebe-1=bd=db, cd=dc, ce=ec, de=ed >

Subgroups: 449 in 253 conjugacy classes, 65 normal (5 characteristic)
C1, C2 [×7], C2 [×12], C4 [×4], C22, C22 [×18], C22 [×68], C2×C4 [×12], C23, C23 [×18], C23 [×68], C22⋊C4 [×12], C22×C4 [×4], C24 [×7], C24 [×12], C2×C22⋊C4 [×6], C25, C243C4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×12], C23, C22⋊C4 [×12], C22×C4, C2×D4 [×6], C2×C22⋊C4 [×3], C22≀C2 [×4], C243C4

Character table of C243C4

 class 12A2B2C2D2E2F2G2H2I2J2K2L2M2N2O2P2Q2R2S4A4B4C4D4E4F4G4H
 size 1111111122222222222244444444
ρ11111111111111111111111111111    trivial
ρ211111111-1-11-1-1-1-1-1-1111-1-1111-1-11    linear of order 2
ρ311111111-11-1-1-1-1111-1-1-1-11-111-11-1    linear of order 2
ρ4111111111-1-1111-1-1-1-1-1-11-1-1111-1-1    linear of order 2
ρ511111111-11-1-1-1-1111-1-1-11-11-1-11-11    linear of order 2
ρ611111111-1-11-1-1-1-1-1-111111-1-1-111-1    linear of order 2
ρ7111111111-1-1111-1-1-1-1-1-1-111-1-1-111    linear of order 2
ρ811111111111111111111-1-1-1-1-1-1-1-1    linear of order 2
ρ91-11-11-11-1-1-111-111-11-11-1-i-iii-iii-i    linear of order 4
ρ101-11-11-11-1-11-11-11-11-11-11-ii-ii-ii-ii    linear of order 4
ρ111-11-11-11-11-1-1-11-11-111-11i-i-ii-i-iii    linear of order 4
ρ121-11-11-11-1111-11-1-11-1-11-1iiii-i-i-i-i    linear of order 4
ρ131-11-11-11-1-1-111-111-11-11-1ii-i-ii-i-ii    linear of order 4
ρ141-11-11-11-1-11-11-11-11-11-11i-ii-ii-ii-i    linear of order 4
ρ151-11-11-11-11-1-1-11-11-111-11-iii-iii-i-i    linear of order 4
ρ161-11-11-11-1111-11-1-11-1-11-1-i-i-i-iiiii    linear of order 4
ρ17222-2-22-2-2200-2-2200000000000000    orthogonal lifted from D4
ρ1822-222-2-2-2020000-2-2200000000000    orthogonal lifted from D4
ρ192-222-2-2-222002-2-200000000000000    orthogonal lifted from D4
ρ202-222-2-2-22-200-22200000000000000    orthogonal lifted from D4
ρ2122-2-2-2-2220020000002-2-200000000    orthogonal lifted from D4
ρ222-2-22-222-200-200000022-200000000    orthogonal lifted from D4
ρ2322-222-2-2-20-2000022-200000000000    orthogonal lifted from D4
ρ24222-2-22-2-2-20022-200000000000000    orthogonal lifted from D4
ρ252-2-2-222-220200002-2-200000000000    orthogonal lifted from D4
ρ262-2-22-222-2002000000-2-2200000000    orthogonal lifted from D4
ρ272-2-2-222-220-20000-22200000000000    orthogonal lifted from D4
ρ2822-2-2-2-22200-2000000-22200000000    orthogonal lifted from D4

Permutation representations of C243C4
On 16 points - transitive group 16T79
Generators in S16
(1 9)(2 16)(3 11)(4 14)(5 12)(6 15)(7 10)(8 13)
(1 6)(2 10)(3 8)(4 12)(5 14)(7 16)(9 15)(11 13)
(1 6)(2 7)(3 8)(4 5)(9 15)(10 16)(11 13)(12 14)
(1 15)(2 16)(3 13)(4 14)(5 12)(6 9)(7 10)(8 11)
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)

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

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

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

G:=TransitiveGroup(16,79);

C243C4 is a maximal subgroup of
C24⋊C8  C24⋊D4  C242Q8  C25.85C22  C4×C22≀C2  C23.191C24  C23.194C24  C23.203C24  D4×C22⋊C4  C23.224C24  C23.257C24  C247D4  C23.308C24  C248D4  C23.311C24  C23.318C24  C23.335C24  C244Q8  C23.372C24  C23.380C24  C23.382C24  C23.434C24  C23.461C24  C249D4  C245Q8  C23.568C24  C23.570C24  C23.578C24  C23.584C24  C23.585C24  C23.597C24  C23.635C24  C23.636C24  C2411D4  C246Q8  C24.15Q8  C23≀C2  C248Q8  C24⋊C12  C244Dic3  C244F5  C25.D5
 C24.D2p: C24.4Q8  C25⋊C4  C24.68D4  C25.C22  C24.78D4  C24.90D4  C24.94D4  C24.96D4 ...
C243C4 is a maximal quotient of
C24.636C23  C243C8  C24.51(C2×C4)  C24.165C23  C25.C4  C4.C22≀C2  (C23×C4).C4  2+ 1+4.2C4  2- 1+42C4  2+ 1+44C4  C4○D4.D4  (C22×Q8)⋊C4  C244F5
 C24.D2p: C24.17Q8  C24.50D4  C25⋊C4  C23.35D8  C24.155D4  C24.65D4  C24.66D4  2+ 1+42C4 ...

Matrix representation of C243C4 in GL5(𝔽5)

40000
04000
00100
00040
00004
,
10000
04000
00100
00010
00044
,
10000
04000
00400
00010
00001
,
10000
04000
00400
00040
00004
,
30000
00100
04000
00043
00001

G:=sub<GL(5,GF(5))| [4,0,0,0,0,0,4,0,0,0,0,0,1,0,0,0,0,0,4,0,0,0,0,0,4],[1,0,0,0,0,0,4,0,0,0,0,0,1,0,0,0,0,0,1,4,0,0,0,0,4],[1,0,0,0,0,0,4,0,0,0,0,0,4,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,4,0,0,0,0,0,4,0,0,0,0,0,4,0,0,0,0,0,4],[3,0,0,0,0,0,0,4,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,3,1] >;

C243C4 in GAP, Magma, Sage, TeX

C_2^4\rtimes_3C_4
% in TeX

G:=Group("C2^4:3C4");
// GroupNames label

G:=SmallGroup(64,60);
// by ID

G=gap.SmallGroup(64,60);
# by ID

G:=PCGroup([6,-2,2,2,-2,2,2,192,121,362]);
// Polycyclic

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

Export

Character table of C243C4 in TeX

׿
×
𝔽