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G = C42.46C23order 128 = 27

46th non-split extension by C42 of C23 acting faithfully

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

Aliases: C42.46C23, C4.562+ 1+4, C88D450C2, C89D414C2, C4⋊C834C22, C4⋊C4.365D4, C4⋊Q823C22, D42Q818C2, (C2×D4).169D4, D45D4.3C2, C8.D423C2, (C4×Q8)⋊27C22, C4.Q825C22, C8⋊C423C22, C2.D836C22, D4.23(C4○D4), C22⋊SD1621C2, D4.7D442C2, C4⋊C4.408C23, C22⋊C830C22, (C2×C4).503C24, (C2×C8).353C23, Q8.D440C2, C22⋊C4.165D4, (C22×C8)⋊43C22, (C2×Q16)⋊32C22, C23.322(C2×D4), C22⋊Q818C22, SD16⋊C434C2, C2.75(D4○SD16), Q8⋊C443C22, (C4×D4).156C22, (C2×D4).232C23, C4⋊D4.82C22, (C2×Q8).216C23, C2.139(D45D4), C42⋊C223C22, C23.19D432C2, C23.24D431C2, C23.37D413C2, C23.20D432C2, (C2×SD16).54C22, C4.4D4.64C22, C22.763(C22×D4), D4⋊C4.187C22, C2.84(D8⋊C22), C22.50C244C2, (C22×C4).1147C23, (C22×D4).411C22, C42.28C2214C2, (C2×M4(2)).111C22, C4.228(C2×C4○D4), (C2×C4).600(C2×D4), (C2×C4○D4).208C22, SmallGroup(128,2043)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C42.46C23
C1C2C4C2×C4C22×C4C2×C4○D4D45D4 — C42.46C23
C1C2C2×C4 — C42.46C23
C1C22C4×D4 — C42.46C23
C1C2C2C2×C4 — C42.46C23

Generators and relations for C42.46C23
 G = < a,b,c,d,e | a4=b4=d2=1, c2=a2, e2=b2, ab=ba, cac-1=eae-1=a-1, dad=ab2, cbc-1=dbd=b-1, be=eb, dcd=bc, ece-1=a2c, ede-1=b2d >

Subgroups: 432 in 201 conjugacy classes, 86 normal (84 characteristic)
C1, C2 [×3], C2 [×5], C4 [×2], C4 [×10], C22, C22 [×17], C8 [×4], C2×C4 [×5], C2×C4 [×13], D4 [×2], D4 [×9], Q8 [×6], C23 [×2], C23 [×7], C42, C42 [×3], C22⋊C4 [×2], C22⋊C4 [×9], C4⋊C4 [×5], C4⋊C4 [×5], C2×C8 [×4], C2×C8, M4(2), SD16 [×3], Q16, C22×C4 [×2], C22×C4 [×2], C2×D4 [×3], C2×D4 [×6], C2×Q8 [×2], C2×Q8, C4○D4 [×3], C24, C8⋊C4, C22⋊C8 [×2], D4⋊C4 [×6], Q8⋊C4 [×4], C4⋊C8, C4.Q8 [×2], C2.D8, C2×C22⋊C4, C42⋊C2 [×2], C4×D4 [×2], C4×Q8, C4×Q8, C22≀C2, C4⋊D4, C4⋊D4, C22⋊Q8 [×3], C22.D4, C4.4D4, C4.4D4, C422C2 [×2], C4⋊Q8, C22×C8, C2×M4(2), C2×SD16 [×2], C2×Q16, C22×D4, C2×C4○D4, C23.24D4, C23.37D4, C89D4, SD16⋊C4, C22⋊SD16, D4.7D4, Q8.D4, C88D4, C8.D4, D42Q8, C23.19D4, C23.20D4, C42.28C22, D45D4, C22.50C24, C42.46C23
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○SD16, C42.46C23

Character table of C42.46C23

 class 12A2B2C2D2E2F2G2H4A4B4C4D4E4F4G4H4I4J4K4L4M4N8A8B8C8D8E8F
 size 11114444822224444488888444488
ρ111111111111111111111111111111    trivial
ρ211111-11111-1-111-1-1-1-1-11-11-11-11-1-11    linear of order 2
ρ3111111111111111-11-11-1-1-11-1-1-1-1-1-1    linear of order 2
ρ411111-11111-1-111-11-11-1-11-1-1-11-111-1    linear of order 2
ρ51111-1-1-1-111-1-11-1111111-1-1-11-11-11-1    linear of order 2
ρ61111-11-1-111111-1-1-1-1-1-111-111111-1-1    linear of order 2
ρ71111-1-1-1-111-1-11-11-11-11-111-1-11-11-11    linear of order 2
ρ81111-11-1-111111-1-11-11-1-1-111-1-1-1-111    linear of order 2
ρ91111-11-11-1111111-11-1-1-1-1-1-1111111    linear of order 2
ρ101111-1-1-11-11-1-111-11-111-11-111-11-1-11    linear of order 2
ρ111111-11-11-1111111111-1111-1-1-1-1-1-1-1    linear of order 2
ρ121111-1-1-11-11-1-111-1-1-1-111-111-11-111-1    linear of order 2
ρ1311111-11-1-11-1-11-11-11-1-1-11111-11-11-1    linear of order 2
ρ141111111-1-11111-1-11-111-1-11-11111-1-1    linear of order 2
ρ1511111-11-1-11-1-11-11111-11-1-11-11-11-11    linear of order 2
ρ161111111-1-11111-1-1-1-1-1111-1-1-1-1-1-111    linear of order 2
ρ172222020-20-2-2-2-2220-2000000000000    orthogonal lifted from D4
ρ18222202020-2-2-2-2-2-202000000000000    orthogonal lifted from D4
ρ1922220-20-20-222-22-202000000000000    orthogonal lifted from D4
ρ2022220-2020-222-2-220-2000000000000    orthogonal lifted from D4
ρ212-22-220-200-2002002i0-2i0000002i0-2i00    complex lifted from C4○D4
ρ222-22-2-20200-200200-2i02i0000002i0-2i00    complex lifted from C4○D4
ρ232-22-2-20200-2002002i0-2i000000-2i02i00    complex lifted from C4○D4
ρ242-22-220-200-200200-2i02i000000-2i02i00    complex lifted from C4○D4
ρ254-44-400000400-40000000000000000    orthogonal lifted from 2+ 1+4
ρ264-4-440000004i-4i00000000000000000    complex lifted from D8⋊C22
ρ274-4-44000000-4i4i00000000000000000    complex lifted from D8⋊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 C42.46C23
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 25 18 22)(2 26 19 23)(3 27 20 24)(4 28 17 21)(5 10 29 16)(6 11 30 13)(7 12 31 14)(8 9 32 15)
(1 9 3 11)(2 12 4 10)(5 26 7 28)(6 25 8 27)(13 18 15 20)(14 17 16 19)(21 29 23 31)(22 32 24 30)
(1 3)(2 17)(4 19)(5 14)(6 9)(7 16)(8 11)(10 31)(12 29)(13 32)(15 30)(18 20)(21 23)(22 27)(24 25)(26 28)
(1 22 18 25)(2 21 19 28)(3 24 20 27)(4 23 17 26)(5 10 29 16)(6 9 30 15)(7 12 31 14)(8 11 32 13)

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

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

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

Matrix representation of C42.46C23 in GL6(𝔽17)

1300000
240000
000400
0013000
000004
0000130
,
100000
010000
000100
0016000
0000016
000010
,
140000
8160000
000010
000001
001000
000100
,
100000
010000
001000
0001600
0000016
0000160
,
140000
0160000
0001600
001000
0000016
000010

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

C42.46C23 in GAP, Magma, Sage, TeX

C_4^2._{46}C_2^3
% in TeX

G:=Group("C4^2.46C2^3");
// GroupNames label

G:=SmallGroup(128,2043);
// by ID

G=gap.SmallGroup(128,2043);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,112,253,456,758,723,346,248,4037,1027,124]);
// 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,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*c,e*d*e^-1=b^2*d>;
// generators/relations

Export

Character table of C42.46C23 in TeX

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