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G = C42.47C23order 128 = 27

47th non-split extension by C42 of C23 acting faithfully

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

Aliases: C42.47C23, C4.572+ 1+4, (D4×Q8).6C2, C4⋊C4.156D4, C89D4.1C2, Q8.Q835C2, C42Q1638C2, (C2×D4).316D4, C8.D424C2, (C2×C8).97C23, C22⋊C4.49D4, C2.48(Q8○D8), Q16⋊C421C2, C8.18D437C2, C4⋊C4.234C23, C4⋊C8.101C22, (C2×C4).504C24, Q8.22(C4○D4), C22⋊Q1630C2, C23.323(C2×D4), C4⋊Q8.149C22, C8⋊C4.42C22, C4.Q8.55C22, C2.D8.59C22, (C4×D4).157C22, C22⋊C8.79C22, (C4×Q8).156C22, (C2×Q16).83C22, (C2×Q8).217C23, C2.140(D45D4), C22⋊Q8.80C22, C23.38D412C2, C23.47D416C2, C23.20D433C2, (C22×C8).307C22, Q8⋊C4.70C22, C22.764(C22×D4), C22.4(C8.C22), C42.C2.39C22, C42.30C229C2, (C22×C4).1148C23, (C22×Q8).342C22, C42⋊C2.188C22, (C2×M4(2)).112C22, C22.46C24.1C2, C4.229(C2×C4○D4), (C2×C4).601(C2×D4), (C2×Q8⋊C4)⋊32C2, C2.74(C2×C8.C22), (C2×C4⋊C4).667C22, SmallGroup(128,2044)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C42.47C23
C1C2C4C2×C4C22×C4C22×Q8D4×Q8 — C42.47C23
C1C2C2×C4 — C42.47C23
C1C22C4×D4 — C42.47C23
C1C2C2C2×C4 — C42.47C23

Generators and relations for C42.47C23
 G = < a,b,c,d,e | a4=b4=1, c2=a2b2, d2=e2=b2, ab=ba, cac-1=eae-1=a-1b2, dad-1=ab2, cbc-1=dbd-1=b-1, be=eb, dcd-1=bc, ece-1=a2b2c, ede-1=b2d >

Subgroups: 336 in 188 conjugacy classes, 88 normal (84 characteristic)
C1, C2 [×3], C2 [×3], C4 [×2], C4 [×13], C22, C22 [×2], C22 [×5], C8 [×4], C2×C4 [×5], C2×C4 [×18], D4 [×2], Q8 [×2], Q8 [×11], C23 [×2], C42, C42 [×3], C22⋊C4 [×2], C22⋊C4 [×5], C4⋊C4 [×5], C4⋊C4 [×11], C2×C8 [×4], C2×C8, M4(2), Q16 [×4], C22×C4 [×2], C22×C4 [×3], C2×D4, C2×Q8 [×4], C2×Q8 [×8], C8⋊C4, C22⋊C8 [×2], Q8⋊C4 [×10], C4⋊C8, C4.Q8 [×2], C2.D8, C2×C4⋊C4, C42⋊C2, C42⋊C2, C4×D4, C4×D4, C4×Q8 [×2], C22⋊Q8 [×4], C22⋊Q8 [×2], C22.D4, C42.C2, C42.C2, C422C2, C4⋊Q8, C4⋊Q8, C22×C8, C2×M4(2), C2×Q16 [×3], C22×Q8 [×2], C2×Q8⋊C4, C23.38D4, C89D4, Q16⋊C4, C22⋊Q16 [×2], C42Q16, C8.18D4, C8.D4, Q8.Q8, C23.47D4, C23.20D4, C42.30C22, D4×Q8, C22.46C24, C42.47C23
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, Q8○D8, C42.47C23

Character table of C42.47C23

 class 12A2B2C2D2E2F4A4B4C4D4E4F4G4H4I4J4K4L4M4N4O4P8A8B8C8D8E8F
 size 11112242244444444888888444488
ρ111111111111111111111111111111    trivial
ρ21111-1-11111-1-1-1-1-1-1-11-11-11111-1-11-1    linear of order 2
ρ31111-1-1-111-1-1-1111-11-111-1-11-1-1111-1    linear of order 2
ρ4111111-111-111-1-1-11-1-1-111-11-1-1-1-111    linear of order 2
ρ51111-1-11111-11-1-1-11-1111-1-1-1-1-111-11    linear of order 2
ρ611111111111-1111-111-111-1-1-1-1-1-1-1-1    linear of order 2
ρ7111111-111-11-1-1-1-1-1-1-11111-11111-1-1    linear of order 2
ρ81111-1-1-111-1-1111111-1-11-11-111-1-1-11    linear of order 2
ρ91111-1-11111-1-11-11-1-1-1-1-1111-1-111-11    linear of order 2
ρ10111111111111-11-111-11-1-111-1-1-1-1-1-1    linear of order 2
ρ11111111-111-1111-111-11-1-1-1-111111-1-1    linear of order 2
ρ121111-1-1-111-1-1-1-11-1-1111-11-1111-1-1-11    linear of order 2
ρ1311111111111-1-11-1-11-1-1-1-1-1-1111111    linear of order 2
ρ141111-1-11111-111-111-1-11-11-1-111-1-11-1    linear of order 2
ρ151111-1-1-111-1-11-11-1111-1-111-1-1-1111-1    linear of order 2
ρ16111111-111-11-11-11-1-111-1-11-1-1-1-1-111    linear of order 2
ρ17222222-2-2-22-200-2002000000000000    orthogonal lifted from D4
ρ182222-2-22-2-2-2200-2002000000000000    orthogonal lifted from D4
ρ192222-2-2-2-2-22200200-2000000000000    orthogonal lifted from D4
ρ202222222-2-2-2-200200-2000000000000    orthogonal lifted from D4
ρ212-22-2000-2200-22i0-2i200000002i-2i0000    complex lifted from C4○D4
ρ222-22-2000-2200-2-2i02i20000000-2i2i0000    complex lifted from C4○D4
ρ232-22-2000-220022i0-2i-20000000-2i2i0000    complex lifted from C4○D4
ρ242-22-2000-22002-2i02i-200000002i-2i0000    complex lifted from C4○D4
ρ254-44-40004-400000000000000000000    orthogonal lifted from 2+ 1+4
ρ264-4-444-400000000000000000000000    symplectic lifted from C8.C22, Schur index 2
ρ274-4-44-4400000000000000000000000    symplectic lifted from C8.C22, Schur index 2
ρ2844-4-400000000000000000000022-2200    symplectic lifted from Q8○D8, Schur index 2
ρ2944-4-4000000000000000000000-222200    symplectic lifted from Q8○D8, Schur index 2

Smallest permutation representation of C42.47C23
On 64 points
Generators in S64
(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)(33 34 35 36)(37 38 39 40)(41 42 43 44)(45 46 47 48)(49 50 51 52)(53 54 55 56)(57 58 59 60)(61 62 63 64)
(1 19 24 25)(2 20 21 26)(3 17 22 27)(4 18 23 28)(5 11 61 14)(6 12 62 15)(7 9 63 16)(8 10 64 13)(29 34 42 39)(30 35 43 40)(31 36 44 37)(32 33 41 38)(45 51 58 54)(46 52 59 55)(47 49 60 56)(48 50 57 53)
(1 55 22 50)(2 51 23 56)(3 53 24 52)(4 49 21 54)(5 31 63 42)(6 43 64 32)(7 29 61 44)(8 41 62 30)(9 39 14 36)(10 33 15 40)(11 37 16 34)(12 35 13 38)(17 57 25 46)(18 47 26 58)(19 59 27 48)(20 45 28 60)
(1 38 24 33)(2 34 21 39)(3 40 22 35)(4 36 23 37)(5 54 61 51)(6 52 62 55)(7 56 63 49)(8 50 64 53)(9 60 16 47)(10 48 13 57)(11 58 14 45)(12 46 15 59)(17 43 27 30)(18 31 28 44)(19 41 25 32)(20 29 26 42)
(1 25 24 19)(2 18 21 28)(3 27 22 17)(4 20 23 26)(5 11 61 14)(6 13 62 10)(7 9 63 16)(8 15 64 12)(29 36 42 37)(30 40 43 35)(31 34 44 39)(32 38 41 33)(45 51 58 54)(46 53 59 50)(47 49 60 56)(48 55 57 52)

G:=sub<Sym(64)| (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)(33,34,35,36)(37,38,39,40)(41,42,43,44)(45,46,47,48)(49,50,51,52)(53,54,55,56)(57,58,59,60)(61,62,63,64), (1,19,24,25)(2,20,21,26)(3,17,22,27)(4,18,23,28)(5,11,61,14)(6,12,62,15)(7,9,63,16)(8,10,64,13)(29,34,42,39)(30,35,43,40)(31,36,44,37)(32,33,41,38)(45,51,58,54)(46,52,59,55)(47,49,60,56)(48,50,57,53), (1,55,22,50)(2,51,23,56)(3,53,24,52)(4,49,21,54)(5,31,63,42)(6,43,64,32)(7,29,61,44)(8,41,62,30)(9,39,14,36)(10,33,15,40)(11,37,16,34)(12,35,13,38)(17,57,25,46)(18,47,26,58)(19,59,27,48)(20,45,28,60), (1,38,24,33)(2,34,21,39)(3,40,22,35)(4,36,23,37)(5,54,61,51)(6,52,62,55)(7,56,63,49)(8,50,64,53)(9,60,16,47)(10,48,13,57)(11,58,14,45)(12,46,15,59)(17,43,27,30)(18,31,28,44)(19,41,25,32)(20,29,26,42), (1,25,24,19)(2,18,21,28)(3,27,22,17)(4,20,23,26)(5,11,61,14)(6,13,62,10)(7,9,63,16)(8,15,64,12)(29,36,42,37)(30,40,43,35)(31,34,44,39)(32,38,41,33)(45,51,58,54)(46,53,59,50)(47,49,60,56)(48,55,57,52)>;

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)(33,34,35,36)(37,38,39,40)(41,42,43,44)(45,46,47,48)(49,50,51,52)(53,54,55,56)(57,58,59,60)(61,62,63,64), (1,19,24,25)(2,20,21,26)(3,17,22,27)(4,18,23,28)(5,11,61,14)(6,12,62,15)(7,9,63,16)(8,10,64,13)(29,34,42,39)(30,35,43,40)(31,36,44,37)(32,33,41,38)(45,51,58,54)(46,52,59,55)(47,49,60,56)(48,50,57,53), (1,55,22,50)(2,51,23,56)(3,53,24,52)(4,49,21,54)(5,31,63,42)(6,43,64,32)(7,29,61,44)(8,41,62,30)(9,39,14,36)(10,33,15,40)(11,37,16,34)(12,35,13,38)(17,57,25,46)(18,47,26,58)(19,59,27,48)(20,45,28,60), (1,38,24,33)(2,34,21,39)(3,40,22,35)(4,36,23,37)(5,54,61,51)(6,52,62,55)(7,56,63,49)(8,50,64,53)(9,60,16,47)(10,48,13,57)(11,58,14,45)(12,46,15,59)(17,43,27,30)(18,31,28,44)(19,41,25,32)(20,29,26,42), (1,25,24,19)(2,18,21,28)(3,27,22,17)(4,20,23,26)(5,11,61,14)(6,13,62,10)(7,9,63,16)(8,15,64,12)(29,36,42,37)(30,40,43,35)(31,34,44,39)(32,38,41,33)(45,51,58,54)(46,53,59,50)(47,49,60,56)(48,55,57,52) );

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),(33,34,35,36),(37,38,39,40),(41,42,43,44),(45,46,47,48),(49,50,51,52),(53,54,55,56),(57,58,59,60),(61,62,63,64)], [(1,19,24,25),(2,20,21,26),(3,17,22,27),(4,18,23,28),(5,11,61,14),(6,12,62,15),(7,9,63,16),(8,10,64,13),(29,34,42,39),(30,35,43,40),(31,36,44,37),(32,33,41,38),(45,51,58,54),(46,52,59,55),(47,49,60,56),(48,50,57,53)], [(1,55,22,50),(2,51,23,56),(3,53,24,52),(4,49,21,54),(5,31,63,42),(6,43,64,32),(7,29,61,44),(8,41,62,30),(9,39,14,36),(10,33,15,40),(11,37,16,34),(12,35,13,38),(17,57,25,46),(18,47,26,58),(19,59,27,48),(20,45,28,60)], [(1,38,24,33),(2,34,21,39),(3,40,22,35),(4,36,23,37),(5,54,61,51),(6,52,62,55),(7,56,63,49),(8,50,64,53),(9,60,16,47),(10,48,13,57),(11,58,14,45),(12,46,15,59),(17,43,27,30),(18,31,28,44),(19,41,25,32),(20,29,26,42)], [(1,25,24,19),(2,18,21,28),(3,27,22,17),(4,20,23,26),(5,11,61,14),(6,13,62,10),(7,9,63,16),(8,15,64,12),(29,36,42,37),(30,40,43,35),(31,34,44,39),(32,38,41,33),(45,51,58,54),(46,53,59,50),(47,49,60,56),(48,55,57,52)])

Matrix representation of C42.47C23 in GL6(𝔽17)

0160000
100000
0001600
001000
00107016
007710
,
100000
010000
000100
0016000
0078016
009710
,
040000
400000
0013150
00015015
00071614
0001002
,
1600000
0160000
004000
0001300
0027013
00210130
,
100000
0160000
0001600
001000
00107016
007710

G:=sub<GL(6,GF(17))| [0,1,0,0,0,0,16,0,0,0,0,0,0,0,0,1,10,7,0,0,16,0,7,7,0,0,0,0,0,1,0,0,0,0,16,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,16,7,9,0,0,1,0,8,7,0,0,0,0,0,1,0,0,0,0,16,0],[0,4,0,0,0,0,4,0,0,0,0,0,0,0,1,0,0,0,0,0,3,15,7,10,0,0,15,0,16,0,0,0,0,15,14,2],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,4,0,2,2,0,0,0,13,7,10,0,0,0,0,0,13,0,0,0,0,13,0],[1,0,0,0,0,0,0,16,0,0,0,0,0,0,0,1,10,7,0,0,16,0,7,7,0,0,0,0,0,1,0,0,0,0,16,0] >;

C42.47C23 in GAP, Magma, Sage, TeX

C_4^2._{47}C_2^3
% in TeX

G:=Group("C4^2.47C2^3");
// GroupNames label

G:=SmallGroup(128,2044);
// by ID

G=gap.SmallGroup(128,2044);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,112,253,456,758,723,352,346,4037,1027,124]);
// Polycyclic

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

Export

Character table of C42.47C23 in TeX

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