Copied to
clipboard

G = C42.45C23order 128 = 27

45th non-split extension by C42 of C23 acting faithfully

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

Aliases: C42.45C23, C4.552+ 1+4, D42.5C2, C8⋊D436C2, C89D413C2, C88D449C2, C4⋊C833C22, C4⋊C4.155D4, D4.Q835C2, C4⋊SD1621C2, (C2×D4).315D4, (C2×C8).96C23, C22⋊C4.48D4, (C4×Q8)⋊26C22, C8⋊C422C22, C2.D835C22, C4.Q839C22, D4.22(C4○D4), C22⋊SD1620C2, C4⋊C4.233C23, C22⋊C829C22, (C2×C4).502C24, (C22×C8)⋊42C22, C23.321(C2×D4), C22⋊Q817C22, SD16⋊C433C2, C42.C29C22, D4⋊C440C22, C2.74(D4○SD16), Q8⋊C456C22, (C2×SD16)⋊53C22, (C4×D4).155C22, (C2×D4).422C23, C41D4.87C22, C22.D827C2, C4⋊D4.81C22, C22.9(C8⋊C22), (C2×Q8).215C23, C2.138(D45D4), C42⋊C222C22, C23.19D431C2, C23.37D412C2, (C2×M4(2))⋊27C22, C22.762(C22×D4), C42.29C229C2, (C22×C4).1146C23, C22.46C243C2, (C22×D4).410C22, (C2×C4⋊C4)⋊58C22, C4.227(C2×C4○D4), (C2×C4).599(C2×D4), C2.76(C2×C8⋊C22), (C2×D4⋊C4)⋊43C2, SmallGroup(128,2042)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C42.45C23
C1C2C4C2×C4C22×C4C22×D4D42 — C42.45C23
C1C2C2×C4 — C42.45C23
C1C22C4×D4 — C42.45C23
C1C2C2C2×C4 — C42.45C23

Generators and relations for C42.45C23
 G = < a,b,c,d,e | a4=b4=d2=1, c2=a2b2, e2=b2, ab=ba, cac-1=eae-1=a-1b2, dad=ab2, cbc-1=dbd=b-1, be=eb, dcd=bc, ece-1=a2b2c, ede-1=b2d >

Subgroups: 536 in 225 conjugacy classes, 88 normal (84 characteristic)
C1, C2 [×3], C2 [×7], C4 [×2], C4 [×9], C22, C22 [×2], C22 [×23], C8 [×4], C2×C4 [×5], C2×C4 [×13], D4 [×2], D4 [×18], Q8 [×2], C23 [×2], C23 [×13], C42, C42 [×2], C22⋊C4 [×2], C22⋊C4 [×6], C4⋊C4 [×5], C4⋊C4 [×6], C2×C8 [×4], C2×C8, M4(2), SD16 [×4], C22×C4 [×2], C22×C4 [×2], C2×D4 [×4], C2×D4 [×16], C2×Q8, C24 [×2], C8⋊C4, C22⋊C8 [×2], D4⋊C4 [×9], Q8⋊C4, C4⋊C8, C4.Q8, C2.D8 [×2], C2×C4⋊C4, C42⋊C2, C42⋊C2, C4×D4 [×2], C4×Q8, C22≀C2 [×2], C4⋊D4 [×2], C4⋊D4, C22⋊Q8 [×2], C22.D4, C42.C2, C42.C2, C422C2, C41D4, C22×C8, C2×M4(2), C2×SD16 [×3], C22×D4 [×2], C22×D4, C2×D4⋊C4, C23.37D4, C89D4, SD16⋊C4, C22⋊SD16 [×2], C4⋊SD16, C88D4, C8⋊D4, D4.Q8, C22.D8, C23.19D4, C42.29C22, D42, C22.46C24, C42.45C23
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], C2×D4 [×6], C4○D4 [×2], C24, C8⋊C22 [×2], C22×D4, C2×C4○D4, 2+ 1+4, D45D4, C2×C8⋊C22, D4○SD16, C42.45C23

Character table of C42.45C23

 class 12A2B2C2D2E2F2G2H2I2J4A4B4C4D4E4F4G4H4I4J4K4L8A8B8C8D8E8F
 size 11112244488224444448888444488
ρ111111111111111111111111111111    trivial
ρ21111-1-1-1-111-11111-1-11-11-11-11-11-1-11    linear of order 2
ρ311111111-11-111-1-1-11-1-11-1-11-1-1-1-111    linear of order 2
ρ41111-1-1-1-1-11111-1-11-1-1111-1-1-11-11-11    linear of order 2
ρ51111-1-1111-111111-1-11-11-1-1-1-11-111-1    linear of order 2
ρ6111111-1-11-1-11111111111-11-1-1-1-1-1-1    linear of order 2
ρ71111-1-111-1-1-111-1-11-1-11111-11-11-11-1    linear of order 2
ρ8111111-1-1-1-1111-1-1-11-1-11-1111111-1-1    linear of order 2
ρ9111111-1-1-1-11111-1-111-1-111-1-1-1-1-111    linear of order 2
ρ101111-1-111-1-1-1111-11-111-1-111-11-11-11    linear of order 2
ρ11111111-1-11-1-111-1111-11-1-1-1-1111111    linear of order 2
ρ121111-1-1111-1111-11-1-1-1-1-11-111-11-1-11    linear of order 2
ρ131111-1-1-1-1-111111-11-111-1-1-111-11-11-1    linear of order 2
ρ1411111111-11-1111-1-111-1-11-1-11111-1-1    linear of order 2
ρ151111-1-1-1-111-111-11-1-1-1-1-1111-11-111-1    linear of order 2
ρ161111111111111-1111-11-1-11-1-1-1-1-1-1-1    linear of order 2
ρ1722222200-200-2-2022-20-20000000000    orthogonal lifted from D4
ρ1822222200200-2-20-2-2-2020000000000    orthogonal lifted from D4
ρ192222-2-200-200-2-202-22020000000000    orthogonal lifted from D4
ρ202222-2-200200-2-20-2220-20000000000    orthogonal lifted from D4
ρ212-22-2002-20002-22i000-2i000000-2i02i00    complex lifted from C4○D4
ρ222-22-200-220002-2-2i0002i000000-2i02i00    complex lifted from C4○D4
ρ232-22-200-220002-22i000-2i0000002i0-2i00    complex lifted from C4○D4
ρ242-22-2002-20002-2-2i0002i0000002i0-2i00    complex lifted from C4○D4
ρ254-4-444-400000000000000000000000    orthogonal lifted from C8⋊C22
ρ264-44-40000000-440000000000000000    orthogonal lifted from 2+ 1+4
ρ274-4-44-4400000000000000000000000    orthogonal lifted from C8⋊C22
ρ2844-4-40000000000000000000-2-202-2000    complex lifted from D4○SD16
ρ2944-4-400000000000000000002-20-2-2000    complex lifted from D4○SD16

Smallest permutation representation of C42.45C23
On 32 points
Generators in S32
(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 28 24 19)(2 25 21 20)(3 26 22 17)(4 27 23 18)(5 12 15 31)(6 9 16 32)(7 10 13 29)(8 11 14 30)
(1 10 22 31)(2 32 23 11)(3 12 24 29)(4 30 21 9)(5 19 13 26)(6 27 14 20)(7 17 15 28)(8 25 16 18)
(1 22)(2 4)(3 24)(5 29)(6 11)(7 31)(8 9)(10 15)(12 13)(14 32)(16 30)(17 19)(18 25)(20 27)(21 23)(26 28)
(1 23 24 4)(2 3 21 22)(5 14 15 8)(6 7 16 13)(9 10 32 29)(11 12 30 31)(17 25 26 20)(18 19 27 28)

G:=sub<Sym(32)| (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), (1,28,24,19)(2,25,21,20)(3,26,22,17)(4,27,23,18)(5,12,15,31)(6,9,16,32)(7,10,13,29)(8,11,14,30), (1,10,22,31)(2,32,23,11)(3,12,24,29)(4,30,21,9)(5,19,13,26)(6,27,14,20)(7,17,15,28)(8,25,16,18), (1,22)(2,4)(3,24)(5,29)(6,11)(7,31)(8,9)(10,15)(12,13)(14,32)(16,30)(17,19)(18,25)(20,27)(21,23)(26,28), (1,23,24,4)(2,3,21,22)(5,14,15,8)(6,7,16,13)(9,10,32,29)(11,12,30,31)(17,25,26,20)(18,19,27,28)>;

G:=Group( (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), (1,28,24,19)(2,25,21,20)(3,26,22,17)(4,27,23,18)(5,12,15,31)(6,9,16,32)(7,10,13,29)(8,11,14,30), (1,10,22,31)(2,32,23,11)(3,12,24,29)(4,30,21,9)(5,19,13,26)(6,27,14,20)(7,17,15,28)(8,25,16,18), (1,22)(2,4)(3,24)(5,29)(6,11)(7,31)(8,9)(10,15)(12,13)(14,32)(16,30)(17,19)(18,25)(20,27)(21,23)(26,28), (1,23,24,4)(2,3,21,22)(5,14,15,8)(6,7,16,13)(9,10,32,29)(11,12,30,31)(17,25,26,20)(18,19,27,28) );

G=PermutationGroup([(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)], [(1,28,24,19),(2,25,21,20),(3,26,22,17),(4,27,23,18),(5,12,15,31),(6,9,16,32),(7,10,13,29),(8,11,14,30)], [(1,10,22,31),(2,32,23,11),(3,12,24,29),(4,30,21,9),(5,19,13,26),(6,27,14,20),(7,17,15,28),(8,25,16,18)], [(1,22),(2,4),(3,24),(5,29),(6,11),(7,31),(8,9),(10,15),(12,13),(14,32),(16,30),(17,19),(18,25),(20,27),(21,23),(26,28)], [(1,23,24,4),(2,3,21,22),(5,14,15,8),(6,7,16,13),(9,10,32,29),(11,12,30,31),(17,25,26,20),(18,19,27,28)])

Matrix representation of C42.45C23 in GL6(𝔽17)

1150000
1160000
000010
000001
001000
000100
,
100000
010000
0011500
0011600
0000115
0000116
,
1300000
1340000
0001000
005000
000007
0000120
,
100000
010000
0016000
0016100
000010
0000116
,
1150000
0160000
000010
000001
0016000
0001600

G:=sub<GL(6,GF(17))| [1,1,0,0,0,0,15,16,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,1,0,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,1,0,0,0,0,15,16,0,0,0,0,0,0,1,1,0,0,0,0,15,16],[13,13,0,0,0,0,0,4,0,0,0,0,0,0,0,5,0,0,0,0,10,0,0,0,0,0,0,0,0,12,0,0,0,0,7,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,16,0,0,0,0,0,1,0,0,0,0,0,0,1,1,0,0,0,0,0,16],[1,0,0,0,0,0,15,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,1,0,0,0,0,0,0,1,0,0] >;

C42.45C23 in GAP, Magma, Sage, TeX

C_4^2._{45}C_2^3
% in TeX

G:=Group("C4^2.45C2^3");
// GroupNames label

G:=SmallGroup(128,2042);
// by ID

G=gap.SmallGroup(128,2042);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,112,253,456,758,723,346,4037,1027,124]);
// Polycyclic

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

Export

Character table of C42.45C23 in TeX

׿
×
𝔽