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G = C42.53C23order 128 = 27

53rd non-split extension by C42 of C23 acting faithfully

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

Aliases: C42.53C23, C4.632+ 1+4, D429C2, C4⋊D838C2, C87D439C2, C82D425C2, C89D420C2, C4⋊C836C22, C4⋊C4.159D4, D4.Q837C2, D8⋊C423C2, C22⋊D831C2, (C2×D4).319D4, C2.49(D4○D8), (C4×D4)⋊26C22, (C2×D8)⋊10C22, C22⋊C4.52D4, C8⋊C425C22, C2.D813C22, C4.Q826C22, D4.26(C4○D4), C4⋊C4.237C23, C4⋊D417C22, C22⋊C832C22, (C2×C8).100C23, (C2×C4).510C24, (C22×C8)⋊32C22, C23.327(C2×D4), D4⋊C441C22, (C2×D4).236C23, C41D4.89C22, C2.146(D45D4), C42.C210C22, C42⋊C224C22, C23.46D417C2, C23.37D414C2, C23.19D435C2, C22.12(C8⋊C22), (C2×M4(2))⋊29C22, C22.770(C22×D4), C22.47C245C2, (C22×C4).1154C23, (C22×D4).413C22, C42.29C2211C2, (C2×C4⋊C4)⋊60C22, C4.235(C2×C4○D4), (C2×C4).607(C2×D4), C2.77(C2×C8⋊C22), (C2×D4⋊C4)⋊32C2, SmallGroup(128,2050)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C42.53C23
C1C2C4C2×C4C22×C4C22×D4D42 — C42.53C23
C1C2C2×C4 — C42.53C23
C1C22C4×D4 — C42.53C23
C1C2C2C2×C4 — C42.53C23

Generators and relations for C42.53C23
 G = < a,b,c,d,e | a4=b4=d2=1, c2=a2, 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: 576 in 231 conjugacy classes, 88 normal (84 characteristic)
C1, C2 [×3], C2 [×8], C4 [×2], C4 [×8], C22, C22 [×2], C22 [×26], C8 [×4], C2×C4 [×5], C2×C4 [×12], D4 [×2], D4 [×23], C23 [×2], C23 [×14], C42, C42, C22⋊C4 [×2], C22⋊C4 [×7], C4⋊C4 [×5], C4⋊C4 [×3], C2×C8 [×4], C2×C8, M4(2), D8 [×4], C22×C4 [×2], C22×C4 [×3], C2×D4 [×5], C2×D4 [×18], C24 [×2], C8⋊C4, C22⋊C8 [×2], D4⋊C4 [×10], C4⋊C8, C4.Q8 [×2], C2.D8, C2×C4⋊C4, C42⋊C2, C4×D4 [×3], C4×D4, C22≀C2 [×2], C4⋊D4 [×4], C4⋊D4 [×2], C22.D4, C42.C2, C422C2, C41D4, C22×C8, C2×M4(2), C2×D8 [×3], C22×D4 [×2], C22×D4, C2×D4⋊C4, C23.37D4, C89D4, D8⋊C4, C22⋊D8 [×2], C4⋊D8, C87D4, C82D4, D4.Q8, C23.46D4, C23.19D4, C42.29C22, D42, C22.47C24, C42.53C23
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, C42.53C23

Character table of C42.53C23

 class 12A2B2C2D2E2F2G2H2I2J2K4A4B4C4D4E4F4G4H4I4J4K8A8B8C8D8E8F
 size 11112244488822444444888444488
ρ111111111111111111111111111111    trivial
ρ21111-1-1-1-111-1111-11-1-1-1-111-1-11-11-11    linear of order 2
ρ31111-1-1111-11-111-11-11-11-1-11-11-11-11    linear of order 2
ρ4111111-1-11-1-1-111111-11-1-1-1-1111111    linear of order 2
ρ51111-1-111-11-1-1111-1-11111-1-1-11-111-1    linear of order 2
ρ6111111-1-1-111-111-1-11-1-1-11-111111-1-1    linear of order 2
ρ71111-1-1-1-1-1-111111-1-1-11-1-111-11-111-1    linear of order 2
ρ811111111-1-1-1111-1-111-11-11-11111-1-1    linear of order 2
ρ91111-1-111-1-1-1-1111-1-1-11-11111-11-1-11    linear of order 2
ρ10111111-1-1-1-11-111-1-111-1111-1-1-1-1-111    linear of order 2
ρ111111-1-1-1-1-1111111-1-1111-1-1-11-11-1-11    linear of order 2
ρ1211111111-11-1111-1-11-1-1-1-1-11-1-1-1-111    linear of order 2
ρ131111-1-1-1-11-1-1111-11-11-111-111-11-11-1    linear of order 2
ρ14111111111-11111111-11-11-1-1-1-1-1-1-1-1    linear of order 2
ρ151111-1-111111-111-11-1-1-1-1-11-11-11-11-1    linear of order 2
ρ16111111-1-111-1-111111111-111-1-1-1-1-1-1    linear of order 2
ρ17222222002000-2-2-2-2-2020000000000    orthogonal lifted from D4
ρ1822222200-2000-2-222-20-20000000000    orthogonal lifted from D4
ρ192222-2-200-2000-2-2-222020000000000    orthogonal lifted from D4
ρ202222-2-2002000-2-22-220-20000000000    orthogonal lifted from D4
ρ212-22-200-2200002-20002i0-2i0000-2i02i00    complex lifted from C4○D4
ρ222-22-200-2200002-2000-2i02i00002i0-2i00    complex lifted from C4○D4
ρ232-22-2002-200002-20002i0-2i00002i0-2i00    complex lifted from C4○D4
ρ242-22-2002-200002-2000-2i02i0000-2i02i00    complex lifted from C4○D4
ρ254-4-444-400000000000000000000000    orthogonal lifted from C8⋊C22
ρ264-44-400000000-44000000000000000    orthogonal lifted from 2+ 1+4
ρ274-4-44-4400000000000000000000000    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 C42.53C23
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 29 3 31)(2 9 4 11)(5 19 7 17)(6 27 8 25)(10 22 12 24)(13 26 15 28)(14 20 16 18)(21 32 23 30)
(1 3)(2 23)(4 21)(5 10)(6 30)(7 12)(8 32)(9 14)(11 16)(13 31)(15 29)(17 28)(18 20)(19 26)(22 24)(25 27)
(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), (1,29,3,31)(2,9,4,11)(5,19,7,17)(6,27,8,25)(10,22,12,24)(13,26,15,28)(14,20,16,18)(21,32,23,30), (1,3)(2,23)(4,21)(5,10)(6,30)(7,12)(8,32)(9,14)(11,16)(13,31)(15,29)(17,28)(18,20)(19,26)(22,24)(25,27), (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), (1,29,3,31)(2,9,4,11)(5,19,7,17)(6,27,8,25)(10,22,12,24)(13,26,15,28)(14,20,16,18)(21,32,23,30), (1,3)(2,23)(4,21)(5,10)(6,30)(7,12)(8,32)(9,14)(11,16)(13,31)(15,29)(17,28)(18,20)(19,26)(22,24)(25,27), (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)], [(1,29,3,31),(2,9,4,11),(5,19,7,17),(6,27,8,25),(10,22,12,24),(13,26,15,28),(14,20,16,18),(21,32,23,30)], [(1,3),(2,23),(4,21),(5,10),(6,30),(7,12),(8,32),(9,14),(11,16),(13,31),(15,29),(17,28),(18,20),(19,26),(22,24),(25,27)], [(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 C42.53C23 in GL6(𝔽17)

1620000
1610000
00160150
000011
000010
0001160
,
100000
010000
00161500
001100
000101
001616160
,
1300000
1340000
006600
00141100
001114143
003333
,
100000
010000
001000
00161600
00160160
000001
,
1620000
010000
00160150
000011
001010
001616160

G:=sub<GL(6,GF(17))| [16,16,0,0,0,0,2,1,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,1,0,0,15,1,1,16,0,0,0,1,0,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,1,0,16,0,0,15,1,1,16,0,0,0,0,0,16,0,0,0,0,1,0],[13,13,0,0,0,0,0,4,0,0,0,0,0,0,6,14,11,3,0,0,6,11,14,3,0,0,0,0,14,3,0,0,0,0,3,3],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,16,16,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,1],[16,0,0,0,0,0,2,1,0,0,0,0,0,0,16,0,1,16,0,0,0,0,0,16,0,0,15,1,1,16,0,0,0,1,0,0] >;

C42.53C23 in GAP, Magma, Sage, TeX

C_4^2._{53}C_2^3
% in TeX

G:=Group("C4^2.53C2^3");
// GroupNames label

G:=SmallGroup(128,2050);
// by ID

G=gap.SmallGroup(128,2050);
# by ID

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

G:=Group<a,b,c,d,e|a^4=b^4=d^2=1,c^2=a^2,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 C42.53C23 in TeX

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