Copied to
clipboard

G = C42.471C23order 128 = 27

332nd non-split extension by C42 of C23 acting via C23/C2=C22

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

Aliases: C42.471C23, C4.672+ 1+4, (C4×D8)⋊42C2, C82D427C2, C86D410C2, C8⋊D442C2, C4⋊C838C22, C4⋊C4.161D4, (C4×C8)⋊37C22, D4.Q838C2, D4⋊D447C2, C22⋊D833C2, D45D411C2, (C2×D4).175D4, C2.51(D4○D8), (C4×D4)⋊28C22, C2.D839C22, C4.Q828C22, D4.28(C4○D4), D4.2D444C2, C4⋊C4.414C23, C4⋊D419C22, C22⋊C834C22, (C2×C8).102C23, (C2×C4).514C24, C22⋊C4.171D4, C23.331(C2×D4), D4⋊C496C22, Q8⋊C410C22, (C2×SD16)⋊34C22, (C2×D8).140C22, (C2×D4).240C23, C22.D829C2, (C2×Q8).225C23, C2.150(D45D4), C42.C211C22, C22⋊Q8.88C22, C23.36D421C2, C23.37D416C2, C23.20D439C2, (C2×M4(2))⋊30C22, (C22×C4).327C23, C4.4D4.70C22, C22.774(C22×D4), C2.90(D8⋊C22), C22.47C246C2, (C22×D4).415C22, C42⋊C2.194C22, C42.78C2211C2, (C2×C4⋊C4)⋊61C22, C4.239(C2×C4○D4), (C2×C4).927(C2×D4), (C2×C4○D4).216C22, SmallGroup(128,2054)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C42.471C23
C1C2C4C2×C4C22×C4C22×D4D45D4 — C42.471C23
C1C2C2×C4 — C42.471C23
C1C22C4×D4 — C42.471C23
C1C2C2C2×C4 — C42.471C23

Generators and relations for C42.471C23
 G = < a,b,c,d,e | a4=b4=d2=e2=1, c2=a2, ab=ba, cac-1=a-1b2, dad=ab2, eae=a-1, cbc-1=dbd=b-1, be=eb, dcd=bc, ece=a2c, ede=b2d >

Subgroups: 472 in 207 conjugacy classes, 86 normal (84 characteristic)
C1, C2 [×3], C2 [×6], C4 [×2], C4 [×9], C22, C22 [×20], C8 [×4], C2×C4 [×5], C2×C4 [×14], D4 [×2], D4 [×14], Q8 [×2], C23 [×2], C23 [×8], C42, C42, C22⋊C4 [×2], C22⋊C4 [×9], C4⋊C4 [×5], C4⋊C4 [×4], C2×C8 [×4], M4(2) [×2], D8 [×3], SD16, C22×C4 [×2], C22×C4 [×4], C2×D4 [×4], C2×D4 [×8], C2×Q8, C4○D4 [×3], C24, C4×C8, C22⋊C8 [×2], D4⋊C4 [×7], Q8⋊C4 [×3], C4⋊C8, C4.Q8, C2.D8 [×2], C2×C22⋊C4, C2×C4⋊C4, C42⋊C2, C4×D4 [×3], C4×D4, C22≀C2, C4⋊D4 [×3], C4⋊D4 [×2], C22⋊Q8, C22.D4 [×2], C4.4D4, C42.C2, C422C2, C2×M4(2) [×2], C2×D8 [×2], C2×SD16, C22×D4, C2×C4○D4, C23.36D4, C23.37D4, C86D4, C4×D8, C22⋊D8, D4⋊D4, D4.2D4, C8⋊D4, C82D4, D4.Q8, C22.D8, C23.20D4, C42.78C22, D45D4, C22.47C24, C42.471C23
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], C2×D4 [×6], C4○D4 [×2], C24, C22×D4, C2×C4○D4, 2+ 1+4, D45D4, D8⋊C22, D4○D8, C42.471C23

Character table of C42.471C23

 class 12A2B2C2D2E2F2G2H2I4A4B4C4D4E4F4G4H4I4J4K4L4M8A8B8C8D8E8F
 size 11114444882222444448888444488
ρ111111111111111111111111111111    trivial
ρ2111111-1-11-111111111111-1-1-1-1-1-1-1-1    linear of order 2
ρ31111-1-1111-111111-11-11-1-1-111111-1-1    linear of order 2
ρ41111-1-1-1-11111111-11-11-1-11-1-1-1-1-111    linear of order 2
ρ511111-111-1-1-111-1-1-11111-11-1-111-1-11    linear of order 2
ρ611111-1-1-1-11-111-1-1-11111-1-111-1-111-1    linear of order 2
ρ71111-1111-11-111-1-111-11-11-1-1-111-11-1    linear of order 2
ρ81111-11-1-1-1-1-111-1-111-11-11111-1-11-11    linear of order 2
ρ91111-1-1-1-1-1111111-1-1-1-1111-11111-1-1    linear of order 2
ρ101111-1-111-1-111111-1-1-1-111-11-1-1-1-111    linear of order 2
ρ11111111-1-1-1-1111111-11-1-1-1-1-1111111    linear of order 2
ρ1211111111-11111111-11-1-1-111-1-1-1-1-1-1    linear of order 2
ρ131111-11-1-11-1-111-1-11-1-1-11-111-111-11-1    linear of order 2
ρ141111-111111-111-1-11-1-1-11-1-1-11-1-11-11    linear of order 2
ρ1511111-1-1-111-111-1-1-1-11-1-11-11-111-1-11    linear of order 2
ρ1611111-1111-1-111-1-1-1-11-1-111-11-1-111-1    linear of order 2
ρ1722222-200002-2-22-220-200000000000    orthogonal lifted from D4
ρ182222-2200002-2-22-2-20200000000000    orthogonal lifted from D4
ρ192222-2-20000-2-2-2-2220200000000000    orthogonal lifted from D4
ρ202222220000-2-2-2-22-20-200000000000    orthogonal lifted from D4
ρ212-22-2002-2000-220002i0-2i000002i-2i000    complex lifted from C4○D4
ρ222-22-200-22000-22000-2i02i000002i-2i000    complex lifted from C4○D4
ρ232-22-200-22000-220002i0-2i00000-2i2i000    complex lifted from C4○D4
ρ242-22-2002-2000-22000-2i02i00000-2i2i000    complex lifted from C4○D4
ρ254-44-400000004-40000000000000000    orthogonal lifted from 2+ 1+4
ρ2644-4-400000000000000000002200-2200    orthogonal lifted from D4○D8
ρ2744-4-40000000000000000000-22002200    orthogonal lifted from D4○D8
ρ284-4-440000004i00-4i000000000000000    complex lifted from D8⋊C22
ρ294-4-44000000-4i004i000000000000000    complex lifted from D8⋊C22

Smallest permutation representation of C42.471C23
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 18 25 24)(2 19 26 21)(3 20 27 22)(4 17 28 23)(5 15 29 12)(6 16 30 9)(7 13 31 10)(8 14 32 11)
(1 6 3 8)(2 29 4 31)(5 28 7 26)(9 20 11 18)(10 21 12 23)(13 19 15 17)(14 24 16 22)(25 30 27 32)
(1 3)(2 28)(4 26)(5 13)(6 11)(7 15)(8 9)(10 29)(12 31)(14 30)(16 32)(17 19)(18 22)(20 24)(21 23)(25 27)
(1 4)(2 3)(5 30)(6 29)(7 32)(8 31)(9 15)(10 14)(11 13)(12 16)(17 18)(19 20)(21 22)(23 24)(25 28)(26 27)

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

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

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

Matrix representation of C42.471C23 in GL6(𝔽17)

16150000
110000
0001300
004000
000004
0000130
,
100000
010000
000100
0016000
0000016
000010
,
400000
13130000
000010
000001
001000
000100
,
100000
010000
001000
0001600
0000016
0000160
,
16150000
010000
0001300
004000
0000013
000040

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

C42.471C23 in GAP, Magma, Sage, TeX

C_4^2._{471}C_2^3
% in TeX

G:=Group("C4^2.471C2^3");
// GroupNames label

G:=SmallGroup(128,2054);
// by ID

G=gap.SmallGroup(128,2054);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,112,253,758,723,2019,346,4037,1027,124]);
// Polycyclic

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

Export

Character table of C42.471C23 in TeX

׿
×
𝔽