Copied to
clipboard

G = C42.474C23order 128 = 27

335th non-split extension by C42 of C23 acting via C23/C2=C22

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

Aliases: C42.474C23, C4.702+ 1+4, D4210C2, (C4×D8)⋊43C2, C4⋊D839C2, C82D428C2, C86D413C2, C4⋊C841C22, (C4×C8)⋊38C22, C4⋊C4.372D4, C4⋊Q827C22, C22⋊D834C2, D42Q819C2, (C2×D4).322D4, C2.52(D4○D8), C4.4D821C2, (C2×D8)⋊32C22, (C4×D4)⋊29C22, C22⋊C4.55D4, C2.D861C22, C4.Q830C22, D4.31(C4○D4), D4⋊C47C22, C4⋊C4.417C23, C4⋊D420C22, C4.47(C8⋊C22), C22⋊C837C22, (C2×C8).191C23, (C2×C4).517C24, C23.334(C2×D4), (C2×D4).241C23, C41D4.91C22, C2.153(D45D4), C42⋊C227C22, C23.37D419C2, C23.19D439C2, (C2×M4(2))⋊33C22, (C22×C4).330C23, C22.777(C22×D4), C22.49C247C2, (C22×D4).418C22, C4.242(C2×C4○D4), (C2×C4).612(C2×D4), C2.79(C2×C8⋊C22), SmallGroup(128,2057)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C42.474C23
C1C2C4C2×C4C22×C4C22×D4D42 — C42.474C23
C1C2C2×C4 — C42.474C23
C1C22C4×D4 — C42.474C23
C1C2C2C2×C4 — C42.474C23

Generators and relations for C42.474C23
 G = < a,b,c,d,e | a4=b4=d2=1, c2=a2, e2=b2, ab=ba, cac-1=a-1, dad=ab2, eae-1=a-1b2, cbc-1=dbd=b-1, be=eb, dcd=bc, ece-1=a2b2c, ede-1=b2d >

Subgroups: 576 in 230 conjugacy classes, 88 normal (38 characteristic)
C1, C2 [×3], C2 [×7], C4 [×2], C4 [×2], C4 [×7], C22, C22 [×27], C8 [×4], C2×C4 [×3], C2×C4 [×2], C2×C4 [×12], D4 [×2], D4 [×22], Q8, C23 [×2], C23 [×14], C42, C42 [×2], C22⋊C4 [×2], C22⋊C4 [×8], C4⋊C4 [×3], C4⋊C4 [×2], C4⋊C4, C2×C8 [×2], C2×C8 [×2], M4(2) [×2], D8 [×4], C22×C4 [×2], C22×C4 [×2], C2×D4 [×3], C2×D4 [×2], C2×D4 [×18], C2×Q8, C24 [×2], C4×C8, C22⋊C8 [×2], D4⋊C4 [×2], D4⋊C4 [×8], C4⋊C8, C4.Q8 [×2], C2.D8, C42⋊C2 [×2], C42⋊C2, C4×D4 [×3], C22≀C2 [×2], C4⋊D4 [×4], C4⋊D4 [×2], C4.4D4 [×2], C41D4, C4⋊Q8, C2×M4(2) [×2], C2×D8, C2×D8 [×2], C22×D4 [×2], C22×D4, C23.37D4 [×2], C86D4, C4×D8, C22⋊D8 [×2], C4⋊D8, C82D4 [×2], D42Q8, C23.19D4 [×2], C4.4D8, D42, C22.49C24, C42.474C23
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○D8, C42.474C23

Character table of C42.474C23

 class 12A2B2C2D2E2F2G2H2I2J4A4B4C4D4E4F4G4H4I4J4K4L8A8B8C8D8E8F
 size 11114444888222244444888444488
ρ111111111111111111111111111111    trivial
ρ21111-11-11-11-11111111111-11-1-1-1-1-1-1    linear of order 2
ρ3111111111-111111111-1-1-11-1-1-1-1-1-1-1    linear of order 2
ρ41111-11-11-1-1-11111111-1-1-1-1-1111111    linear of order 2
ρ51111111-1-111-111-1-1-11-1-1-1-111-11-1-11    linear of order 2
ρ61111-11-1-111-1-111-1-1-11-1-1-111-11-111-1    linear of order 2
ρ71111111-1-1-11-111-1-1-11111-1-1-11-111-1    linear of order 2
ρ81111-11-1-11-1-1-111-1-1-111111-11-11-1-11    linear of order 2
ρ91111-1-1-11-1-11-111-1-11-111-1111-11-11-1    linear of order 2
ρ1011111-1111-1-1-111-1-11-111-1-11-11-11-11    linear of order 2
ρ111111-1-1-11-111-111-1-11-1-1-111-1-11-11-11    linear of order 2
ρ1211111-11111-1-111-1-11-1-1-11-1-11-11-11-1    linear of order 2
ρ131111-1-1-1-11-1111111-1-1-1-11-111111-1-1    linear of order 2
ρ1411111-11-1-1-1-111111-1-1-1-1111-1-1-1-111    linear of order 2
ρ151111-1-1-1-111111111-1-111-1-1-1-1-1-1-111    linear of order 2
ρ1611111-11-1-11-111111-1-111-11-11111-1-1    linear of order 2
ρ172222020-2000-2-2-2-222-200000000000    orthogonal lifted from D4
ρ18222202020002-2-22-2-2-200000000000    orthogonal lifted from D4
ρ1922220-202000-2-2-2-22-2200000000000    orthogonal lifted from D4
ρ2022220-20-20002-2-22-22200000000000    orthogonal lifted from D4
ρ212-22-220-2000002-200002i-2i00002i0-2i00    complex lifted from C4○D4
ρ222-22-2-202000002-20000-2i2i00002i0-2i00    complex lifted from C4○D4
ρ232-22-220-2000002-20000-2i2i0000-2i02i00    complex lifted from C4○D4
ρ242-22-2-202000002-200002i-2i0000-2i02i00    complex lifted from C4○D4
ρ254-4-440000000-400400000000000000    orthogonal lifted from C8⋊C22
ρ264-44-400000000-44000000000000000    orthogonal lifted from 2+ 1+4
ρ274-4-440000000400-400000000000000    orthogonal lifted from C8⋊C22
ρ2844-4-40000000000000000000220-22000    orthogonal lifted from D4○D8
ρ2944-4-40000000000000000000-22022000    orthogonal lifted from D4○D8

Smallest permutation representation of C42.474C23
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 21 25 18)(2 22 26 19)(3 23 27 20)(4 24 28 17)(5 12 30 15)(6 9 31 16)(7 10 32 13)(8 11 29 14)
(1 32 3 30)(2 31 4 29)(5 25 7 27)(6 28 8 26)(9 24 11 22)(10 23 12 21)(13 20 15 18)(14 19 16 17)
(1 3)(2 28)(4 26)(5 13)(6 11)(7 15)(8 9)(10 30)(12 32)(14 31)(16 29)(17 19)(18 23)(20 21)(22 24)(25 27)
(1 28 25 4)(2 3 26 27)(5 8 30 29)(6 32 31 7)(9 13 16 10)(11 15 14 12)(17 18 24 21)(19 20 22 23)

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

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

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

Matrix representation of C42.474C23 in GL6(𝔽17)

1150000
1160000
00161500
001100
000012
00001616
,
100000
010000
00161500
001100
000012
00001616
,
1380000
040000
000010
000001
0016000
0001600
,
100000
010000
0016000
001100
00001615
000001
,
1600000
1610000
00161500
001100
00001615
000011

G:=sub<GL(6,GF(17))| [1,1,0,0,0,0,15,16,0,0,0,0,0,0,16,1,0,0,0,0,15,1,0,0,0,0,0,0,1,16,0,0,0,0,2,16],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,1,0,0,0,0,15,1,0,0,0,0,0,0,1,16,0,0,0,0,2,16],[13,0,0,0,0,0,8,4,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,1,0,0,0,0,0,0,1,0,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,1,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,15,1],[16,16,0,0,0,0,0,1,0,0,0,0,0,0,16,1,0,0,0,0,15,1,0,0,0,0,0,0,16,1,0,0,0,0,15,1] >;

C42.474C23 in GAP, Magma, Sage, TeX

C_4^2._{474}C_2^3
% in TeX

G:=Group("C4^2.474C2^3");
// GroupNames label

G:=SmallGroup(128,2057);
// by ID

G=gap.SmallGroup(128,2057);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,560,253,456,758,723,2019,346,4037,1027,124]);
// Polycyclic

G:=Group<a,b,c,d,e|a^4=b^4=d^2=1,c^2=a^2,e^2=b^2,a*b=b*a,c*a*c^-1=a^-1,d*a*d=a*b^2,e*a*e^-1=a^-1*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 C42.474C23 in TeX

׿
×
𝔽