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G = C42.41C23order 128 = 27

41st non-split extension by C42 of C23 acting faithfully

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

Aliases: C42.41C23, C4.512+ 1+4, C89D49C2, D46(C4○D4), D46D49C2, D45D49C2, C82D423C2, C88D447C2, C4⋊C832C22, C4⋊C4.153D4, C4⋊Q822C22, C22⋊D830C2, D8⋊C421C2, D4⋊D441C2, D42Q817C2, (C2×D4).313D4, (C4×D4)⋊25C22, (C2×C8).94C23, C8⋊C421C22, C4.Q824C22, D4.2D440C2, C4⋊D416C22, C4⋊C4.231C23, C22⋊C828C22, (C2×C4).498C24, C22⋊C4.163D4, (C2×D8).83C22, C23.473(C2×D4), D4⋊C454C22, C2.72(D4○SD16), Q8⋊C442C22, (C2×SD16)⋊52C22, (C2×D4).228C23, C22.8(C8⋊C22), (C2×Q8).213C23, C2.134(D45D4), C22⋊Q8.77C22, C23.36D413C2, C23.46D414C2, C23.47D414C2, (C2×M4(2))⋊26C22, (C22×C8).361C22, C4.4D4.62C22, C22.758(C22×D4), (C22×C4).1142C23, (C22×D4).409C22, C42.28C2212C2, (C2×C4⋊C4)⋊57C22, C4.223(C2×C4○D4), (C2×C4).595(C2×D4), C2.74(C2×C8⋊C22), (C2×D4⋊C4)⋊42C2, (C2×C4○D4).204C22, SmallGroup(128,2038)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C42.41C23
C1C2C4C2×C4C22×C4C22×D4D45D4 — C42.41C23
C1C2C2×C4 — C42.41C23
C1C22C4×D4 — C42.41C23
C1C2C2C2×C4 — C42.41C23

Generators and relations for C42.41C23
 G = < a,b,c,d,e | a4=b4=d2=e2=1, c2=a2, ab=ba, cac-1=eae=a-1, dad=ab2, cbc-1=dbd=b-1, be=eb, dcd=bc, ece=a2c, ede=b2d >

Subgroups: 504 in 221 conjugacy classes, 88 normal (84 characteristic)
C1, C2 [×3], C2 [×7], C4 [×2], C4 [×9], C22, C22 [×2], C22 [×19], C8 [×4], C2×C4 [×5], C2×C4 [×18], D4 [×2], D4 [×16], Q8 [×4], C23 [×2], C23 [×8], C42, C22⋊C4 [×2], C22⋊C4 [×8], C4⋊C4 [×5], C4⋊C4 [×4], C2×C8 [×4], C2×C8, M4(2), D8 [×3], SD16, C22×C4 [×2], C22×C4 [×5], C2×D4 [×4], C2×D4 [×8], C2×Q8, C2×Q8, C4○D4 [×7], C24, C8⋊C4, C22⋊C8 [×2], D4⋊C4 [×7], Q8⋊C4 [×3], C4⋊C8, C4.Q8 [×3], C2×C22⋊C4, C2×C4⋊C4 [×2], C4×D4 [×3], C22≀C2, C4⋊D4 [×3], C4⋊D4, C22⋊Q8, C22⋊Q8, C22.D4 [×3], C4.4D4, C4⋊Q8, C22×C8, C2×M4(2), C2×D8 [×2], C2×SD16, C22×D4, C2×C4○D4, C2×C4○D4, C2×D4⋊C4, C23.36D4, C89D4, D8⋊C4, C22⋊D8, D4⋊D4, D4.2D4, C88D4, C82D4, D42Q8, C23.46D4, C23.47D4, C42.28C22, D45D4, D46D4, C42.41C23
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, C42.41C23

Character table of C42.41C23

 class 12A2B2C2D2E2F2G2H2I2J4A4B4C4D4E4F4G4H4I4J4K4L8A8B8C8D8E8F
 size 11112244488224444448888444488
ρ111111111111111111111111111111    trivial
ρ21111-1-111-1-1-111-11-1-11-11-111-1-1111-1    linear of order 2
ρ311111111-11-1111-11-1-11-1-1-111111-1-1    linear of order 2
ρ41111-1-1111-1111-1-1-11-1-1-11-11-1-111-11    linear of order 2
ρ51111-1-1-1-1-1111111-1-111-11-1-1-1-1111-1    linear of order 2
ρ6111111-1-11-1-111-11111-1-1-1-1-1111111    linear of order 2
ρ71111-1-1-1-111-1111-1-11-111-11-1-1-111-11    linear of order 2
ρ8111111-1-1-1-1111-1-11-1-1-1111-11111-1-1    linear of order 2
ρ91111111111-111-11111-1-111-1-1-1-1-1-1-1    linear of order 2
ρ101111-1-111-1-111111-1-111-1-11-111-1-1-11    linear of order 2
ρ1111111111-11111-1-11-1-1-11-1-1-1-1-1-1-111    linear of order 2
ρ121111-1-1111-1-1111-1-11-1111-1-111-1-11-1    linear of order 2
ρ131111-1-1-1-1-11-111-11-1-11-111-1111-1-1-11    linear of order 2
ρ14111111-1-11-11111111111-1-11-1-1-1-1-1-1    linear of order 2
ρ151111-1-1-1-111111-1-1-11-1-1-1-11111-1-11-1    linear of order 2
ρ16111111-1-1-1-1-1111-11-1-11-1111-1-1-1-111    linear of order 2
ρ172222-2-200200-2-20-22-2200000000000    orthogonal lifted from D4
ρ182222-2-200-200-2-20222-200000000000    orthogonal lifted from D4
ρ1922222200200-2-202-2-2-200000000000    orthogonal lifted from D4
ρ2022222200-200-2-20-2-22200000000000    orthogonal lifted from D4
ρ212-22-2002-2000-222i0000-2i0000-2i2i0000    complex lifted from C4○D4
ρ222-22-2002-2000-22-2i00002i00002i-2i0000    complex lifted from C4○D4
ρ232-22-200-22000-222i0000-2i00002i-2i0000    complex lifted from C4○D4
ρ242-22-200-22000-22-2i00002i0000-2i2i0000    complex lifted from C4○D4
ρ254-4-444-400000000000000000000000    orthogonal lifted from C8⋊C22
ρ264-44-400000004-40000000000000000    orthogonal lifted from 2+ 1+4
ρ274-4-44-4400000000000000000000000    orthogonal lifted from C8⋊C22
ρ2844-4-40000000000000000000002-2-2-200    complex lifted from D4○SD16
ρ2944-4-4000000000000000000000-2-22-200    complex lifted from D4○SD16

Smallest permutation representation of C42.41C23
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 19 23 25)(2 20 24 26)(3 17 21 27)(4 18 22 28)(5 15 9 32)(6 16 10 29)(7 13 11 30)(8 14 12 31)
(1 10 3 12)(2 9 4 11)(5 22 7 24)(6 21 8 23)(13 20 15 18)(14 19 16 17)(25 29 27 31)(26 32 28 30)
(1 3)(2 22)(4 24)(5 13)(6 31)(7 15)(8 29)(9 30)(10 14)(11 32)(12 16)(17 25)(18 20)(19 27)(21 23)(26 28)
(1 4)(2 3)(5 6)(7 8)(9 10)(11 12)(13 14)(15 16)(17 20)(18 19)(21 24)(22 23)(25 28)(26 27)(29 32)(30 31)

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

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

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

Matrix representation of C42.41C23 in GL6(𝔽17)

010000
1600000
000010
000001
0016000
0001600
,
100000
010000
000100
0016000
000001
0000160
,
1300000
040000
0012500
005500
0000512
00001212
,
100000
010000
0016000
000100
000010
0000016
,
010000
100000
000010
000001
001000
000100

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

C42.41C23 in GAP, Magma, Sage, TeX

C_4^2._{41}C_2^3
% in TeX

G:=Group("C4^2.41C2^3");
// GroupNames label

G:=SmallGroup(128,2038);
// by ID

G=gap.SmallGroup(128,2038);
# by ID

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

Export

Character table of C42.41C23 in TeX

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