Copied to
clipboard

G = C24.4C4order 64 = 26

2nd non-split extension by C24 of C4 acting via C4/C2=C2

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

Aliases: C24.4C4, C222M4(2), (C2×C8)⋊7C22, C4.67(C2×D4), C22⋊C811C2, (C2×C4).117D4, (C23×C4).7C2, (C2×M4(2))⋊6C2, (C22×C4).13C4, C23.29(C2×C4), C2.6(C2×M4(2)), C4.23(C22⋊C4), (C2×C4).145C23, (C22×C4).92C22, C22.41(C22×C4), C22.16(C22⋊C4), (C2×C4).71(C2×C4), C2.10(C2×C22⋊C4), SmallGroup(64,88)

Series: Derived Chief Lower central Upper central Jennings

C1C22 — C24.4C4
C1C2C4C2×C4C22×C4C23×C4 — C24.4C4
C1C22 — C24.4C4
C1C2×C4 — C24.4C4
C1C2C2C2×C4 — C24.4C4

Generators and relations for C24.4C4
 G = < a,b,c,d,e | a2=b2=c2=d2=1, e4=d, ab=ba, eae-1=ac=ca, ad=da, bc=cb, ebe-1=bd=db, cd=dc, ce=ec, de=ed >

Subgroups: 145 in 95 conjugacy classes, 45 normal (11 characteristic)
C1, C2, C2 [×2], C2 [×6], C4 [×4], C4 [×2], C22, C22 [×6], C22 [×14], C8 [×4], C2×C4 [×2], C2×C4 [×6], C2×C4 [×10], C23, C23 [×2], C23 [×6], C2×C8 [×4], M4(2) [×4], C22×C4 [×2], C22×C4 [×4], C22×C4 [×4], C24, C22⋊C8 [×4], C2×M4(2) [×2], C23×C4, C24.4C4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×4], C23, C22⋊C4 [×4], M4(2) [×4], C22×C4, C2×D4 [×2], C2×C22⋊C4, C2×M4(2) [×2], C24.4C4

Character table of C24.4C4

 class 12A2B2C2D2E2F2G2H2I4A4B4C4D4E4F4G4H4I4J8A8B8C8D8E8F8G8H
 size 1111222222111122222244444444
ρ11111111111111111111111111111    trivial
ρ21111-1-11-11-1111111-1-1-1-11-1-111-1-11    linear of order 2
ρ31111-1-11-11-1111111-1-1-1-1-111-1-111-1    linear of order 2
ρ411111111111111111111-1-1-1-1-1-1-1-1    linear of order 2
ρ51111-11-11-1-11111-1-1-1-111-11-11-11-11    linear of order 2
ρ611111-1-1-1-111111-1-111-1-1-1-111-1-111    linear of order 2
ρ71111-11-11-1-11111-1-1-1-1111-11-11-11-1    linear of order 2
ρ811111-1-1-1-111111-1-111-1-111-1-111-1-1    linear of order 2
ρ91111-1-11-11-1-1-1-1-1-1-11111-iii-ii-i-ii    linear of order 4
ρ101111111111-1-1-1-1-1-1-1-1-1-1-i-i-i-iiiii    linear of order 4
ρ111111-1-11-11-1-1-1-1-1-1-11111i-i-ii-iii-i    linear of order 4
ρ121111111111-1-1-1-1-1-1-1-1-1-1iiii-i-i-i-i    linear of order 4
ρ131111-11-11-1-1-1-1-1-11111-1-1i-ii-i-ii-ii    linear of order 4
ρ1411111-1-1-1-11-1-1-1-111-1-111ii-i-i-i-iii    linear of order 4
ρ151111-11-11-1-1-1-1-1-11111-1-1-ii-iii-ii-i    linear of order 4
ρ1611111-1-1-1-11-1-1-1-111-1-111-i-iiiii-i-i    linear of order 4
ρ172-2-2220000-2-222-200-220000000000    orthogonal lifted from D4
ρ182-2-2220000-22-2-22002-20000000000    orthogonal lifted from D4
ρ192-2-22-200002-222-2002-20000000000    orthogonal lifted from D4
ρ202-2-22-2000022-2-2200-220000000000    orthogonal lifted from D4
ρ212-22-200-20202i2i-2i-2i-2i2i000000000000    complex lifted from M4(2)
ρ222-22-20020-202i2i-2i-2i2i-2i000000000000    complex lifted from M4(2)
ρ2322-2-20-20200-2i2i-2i2i0000-2i2i00000000    complex lifted from M4(2)
ρ242-22-200-2020-2i-2i2i2i2i-2i000000000000    complex lifted from M4(2)
ρ252-22-20020-20-2i-2i2i2i-2i2i000000000000    complex lifted from M4(2)
ρ2622-2-20-202002i-2i2i-2i00002i-2i00000000    complex lifted from M4(2)
ρ2722-2-2020-2002i-2i2i-2i0000-2i2i00000000    complex lifted from M4(2)
ρ2822-2-2020-200-2i2i-2i2i00002i-2i00000000    complex lifted from M4(2)

Permutation representations of C24.4C4
On 16 points - transitive group 16T95
Generators in S16
(1 5)(2 16)(3 7)(4 10)(6 12)(8 14)(9 13)(11 15)
(1 15)(2 12)(3 9)(4 14)(5 11)(6 16)(7 13)(8 10)
(1 11)(2 12)(3 13)(4 14)(5 15)(6 16)(7 9)(8 10)
(1 5)(2 6)(3 7)(4 8)(9 13)(10 14)(11 15)(12 16)
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)

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

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

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

G:=TransitiveGroup(16,95);

C24.4C4 is a maximal subgroup of
C24.46D4  C24.2Q8  C24.48D4  C24.3Q8  C23.C42  C23.8C42  C24.C8  C25.3C4  C23⋊C8⋊C2  C24.(C2×C4)  C24.45(C2×C4)  C24.150D4  C24.54D4  C24.55D4  C24.56D4  C24.57D4  C24.58D4  C25.C4  C4.C22≀C2  (C23×C4).C4  C24.66D4  C24.19Q8  C24.9Q8  (C22×C4).275D4  (C22×C4).276D4  C24.70D4  C24.10Q8  (C2×C8)⋊D4  (C2×C4)≀C2  C427D4  M4(2)⋊20D4  M4(2).45D4  C422D4  C24.11Q8  C24.73(C2×C4)  D4○(C22⋊C8)  C42.259C23  C42.262C23  C42.264C23  C42.265C23  D4×M4(2)  C233M4(2)  C42.297C23  C42.299C23  C24.177D4  C24.178D4  C24.104D4  C24.105D4  C24.106D4  C24.183D4  C24.116D4  C24.117D4  C24.118D4  C24.125D4  C24.126D4  C24.127D4  C24.128D4  C24.129D4  C24.130D4  A4⋊M4(2)
 D2p⋊M4(2): D47M4(2)  D6⋊M4(2)  D66M4(2)  D107M4(2)  D108M4(2)  D109M4(2)  D1010M4(2)  D14⋊M4(2) ...
 (C2×C2p)⋊M4(2): (C2×C4)⋊M4(2)  C23⋊M4(2)  C42.677C23  C42.693C23  C24.6Dic3  C24.4Dic5  C24.4F5  C24.4Dic7 ...
C24.4C4 is a maximal quotient of
C25.3C4  C42.42D4  C42.43D4  C42.44D4  Q8⋊M4(2)  C42.374D4  Q85M4(2)  C42.378D4  C243C8  C42.425D4  C23.32M4(2)  C42.109D4  C42.120D4
 D2p⋊M4(2): D4⋊M4(2)  D44M4(2)  D45M4(2)  D6⋊M4(2)  D66M4(2)  D107M4(2)  D108M4(2)  D109M4(2) ...
 (C2×C2p)⋊M4(2): (C2×C4)⋊M4(2)  C23⋊M4(2)  C23.28C42  C232M4(2)  C24.6Dic3  C24.4Dic5  C24.4F5  C24.4Dic7 ...

Matrix representation of C24.4C4 in GL4(𝔽17) generated by

1000
131600
0010
0001
,
1000
131600
00160
0001
,
16000
01600
0010
0001
,
16000
01600
00160
00016
,
131500
10400
0001
0040
G:=sub<GL(4,GF(17))| [1,13,0,0,0,16,0,0,0,0,1,0,0,0,0,1],[1,13,0,0,0,16,0,0,0,0,16,0,0,0,0,1],[16,0,0,0,0,16,0,0,0,0,1,0,0,0,0,1],[16,0,0,0,0,16,0,0,0,0,16,0,0,0,0,16],[13,10,0,0,15,4,0,0,0,0,0,4,0,0,1,0] >;

C24.4C4 in GAP, Magma, Sage, TeX

C_2^4._4C_4
% in TeX

G:=Group("C2^4.4C4");
// GroupNames label

G:=SmallGroup(64,88);
// by ID

G=gap.SmallGroup(64,88);
# by ID

G:=PCGroup([6,-2,2,2,-2,2,-2,96,121,650,88]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^2=c^2=d^2=1,e^4=d,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 C24.4C4 in TeX

׿
×
𝔽