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G = C42.475C23order 128 = 27

336th non-split extension by C42 of C23 acting via C23/C2=C22

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

Aliases: C42.475C23, C4.712+ 1+4, C8⋊D445C2, C82D429C2, C86D414C2, C4⋊C4.163D4, Q8.Q838C2, D4⋊D448C2, Q85D411C2, Q8⋊D423C2, (C4×SD16)⋊57C2, (C2×D4).177D4, D4.2D445C2, C4⋊C4.418C23, C4⋊C8.108C22, (C2×C8).104C23, (C2×C4).518C24, (C4×C8).292C22, Q8.28(C4○D4), C22⋊C4.173D4, (C2×D8).87C22, C23.335(C2×D4), C4.Q8.60C22, C2.82(D4○SD16), (C2×D4).242C23, (C4×D4).167C22, C4⋊D4.91C22, C22⋊C8.86C22, (C2×Q8).399C23, (C4×Q8).163C22, C2.154(D45D4), C2.D8.123C22, C22⋊Q8.89C22, C23.19D440C2, C23.38D416C2, C23.36D423C2, C23.47D418C2, (C22×C4).331C23, C4.4D4.72C22, C22.778(C22×D4), C42.C2.43C22, D4⋊C4.122C22, C2.92(D8⋊C22), C22.47C247C2, Q8⋊C4.115C22, (C2×SD16).160C22, (C22×Q8).347C22, C42⋊C2.196C22, C42.78C2222C2, (C2×M4(2)).120C22, C4.243(C2×C4○D4), (C2×C4).929(C2×D4), (C2×C4⋊C4).672C22, (C2×C4○D4).218C22, SmallGroup(128,2058)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C42.475C23
C1C2C4C2×C4C22×C4C22×Q8Q85D4 — C42.475C23
C1C2C2×C4 — C42.475C23
C1C22C4×D4 — C42.475C23
C1C2C2C2×C4 — C42.475C23

Generators and relations for C42.475C23
 G = < a,b,c,d,e | a4=b4=e2=1, c2=a2, d2=b2, ab=ba, cac-1=a-1b2, dad-1=ab2, eae=a-1, cbc-1=dbd-1=b-1, be=eb, dcd-1=bc, ece=a2c, ede=b2d >

Subgroups: 400 in 196 conjugacy classes, 86 normal (84 characteristic)
C1, C2 [×3], C2 [×4], C4 [×2], C4 [×11], C22, C22 [×12], C8 [×4], C2×C4 [×5], C2×C4 [×16], D4 [×12], Q8 [×2], Q8 [×5], C23 [×2], C23 [×2], C42, C42 [×2], C22⋊C4 [×2], C22⋊C4 [×8], C4⋊C4 [×5], C4⋊C4 [×5], C2×C8 [×4], M4(2) [×2], D8, SD16 [×3], C22×C4 [×2], C22×C4 [×4], C2×D4 [×3], C2×D4 [×4], C2×Q8 [×2], C2×Q8 [×4], C4○D4 [×3], C4×C8, C22⋊C8 [×2], D4⋊C4 [×4], Q8⋊C4 [×6], C4⋊C8, C4.Q8 [×2], C2.D8, C2×C4⋊C4, C42⋊C2, C4×D4 [×2], C4×D4 [×2], C4×Q8, C4⋊D4 [×3], C4⋊D4 [×2], C22⋊Q8, C22⋊Q8, C22.D4, C4.4D4, C4.4D4, C42.C2, C422C2, C2×M4(2) [×2], C2×D8, C2×SD16 [×2], C22×Q8, C2×C4○D4, C23.36D4, C23.38D4, C86D4, C4×SD16, Q8⋊D4, D4⋊D4, D4.2D4, C8⋊D4, C82D4, Q8.Q8, C23.19D4, C23.47D4, C42.78C22, Q85D4, C22.47C24, C42.475C23
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, C42.475C23

Character table of C42.475C23

 class 12A2B2C2D2E2F2G4A4B4C4D4E4F4G4H4I4J4K4L4M4N4O8A8B8C8D8E8F
 size 11114488222244444448888444488
ρ111111111111111111111111111111    trivial
ρ21111111-11111-111-1111-111-1-1-1-1-1-1-1    linear of order 2
ρ31111-111-1-111-1-1-1-1-11-1-11-111-11-11-11    linear of order 2
ρ41111-1111-111-11-1-111-1-1-1-11-11-11-11-1    linear of order 2
ρ511111-11-1-111-11-111-1-1-1-11-111-11-1-11    linear of order 2
ρ611111-111-111-1-1-11-1-1-1-111-1-1-11-111-1    linear of order 2
ρ71111-1-1111111-11-1-1-111-1-1-11-1-1-1-111    linear of order 2
ρ81111-1-11-1111111-11-1111-1-1-11111-1-1    linear of order 2
ρ91111-1-1-1-1111111-11-1-1-1111-1-1-1-1-111    linear of order 2
ρ101111-1-1-111111-11-1-1-1-1-1-11111111-1-1    linear of order 2
ρ1111111-1-11-111-1-1-11-1-1111-11-11-11-1-11    linear of order 2
ρ1211111-1-1-1-111-11-111-111-1-111-11-111-1    linear of order 2
ρ131111-11-11-111-11-1-11111-11-1-1-11-11-11    linear of order 2
ρ141111-11-1-1-111-1-1-1-1-111111-111-11-11-1    linear of order 2
ρ15111111-1-11111-111-11-1-1-1-1-1-1111111    linear of order 2
ρ16111111-11111111111-1-11-1-11-1-1-1-1-1-1    linear of order 2
ρ1722222-2002-2-220-2-202000000000000    orthogonal lifted from D4
ρ182222-22002-2-220-220-2000000000000    orthogonal lifted from D4
ρ192222-2-200-2-2-2-202202000000000000    orthogonal lifted from D4
ρ2022222200-2-2-2-202-20-2000000000000    orthogonal lifted from D4
ρ212-22-2000002-20200-20-2i2i00000-2i02i00    complex lifted from C4○D4
ρ222-22-2000002-20-20020-2i2i000002i0-2i00    complex lifted from C4○D4
ρ232-22-2000002-20200-202i-2i000002i0-2i00    complex lifted from C4○D4
ρ242-22-2000002-20-200202i-2i00000-2i02i00    complex lifted from C4○D4
ρ254-44-400000-44000000000000000000    orthogonal lifted from 2+ 1+4
ρ264-4-4400004i00-4i00000000000000000    complex lifted from D8⋊C22
ρ274-4-440000-4i004i00000000000000000    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 C42.475C23
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 9 51 41)(2 10 52 42)(3 11 49 43)(4 12 50 44)(5 36 56 27)(6 33 53 28)(7 34 54 25)(8 35 55 26)(13 21 47 17)(14 22 48 18)(15 23 45 19)(16 24 46 20)(29 62 58 38)(30 63 59 39)(31 64 60 40)(32 61 57 37)
(1 8 3 6)(2 54 4 56)(5 52 7 50)(9 26 11 28)(10 34 12 36)(13 32 15 30)(14 60 16 58)(17 61 19 63)(18 40 20 38)(21 37 23 39)(22 64 24 62)(25 44 27 42)(29 48 31 46)(33 41 35 43)(45 59 47 57)(49 53 51 55)
(1 24 51 20)(2 17 52 21)(3 22 49 18)(4 19 50 23)(5 59 56 30)(6 31 53 60)(7 57 54 32)(8 29 55 58)(9 16 41 46)(10 47 42 13)(11 14 43 48)(12 45 44 15)(25 37 34 61)(26 62 35 38)(27 39 36 63)(28 64 33 40)
(1 20)(2 19)(3 18)(4 17)(5 39)(6 38)(7 37)(8 40)(9 16)(10 15)(11 14)(12 13)(21 50)(22 49)(23 52)(24 51)(25 57)(26 60)(27 59)(28 58)(29 33)(30 36)(31 35)(32 34)(41 46)(42 45)(43 48)(44 47)(53 62)(54 61)(55 64)(56 63)

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

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

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

Matrix representation of C42.475C23 in GL8(𝔽17)

01000000
160000000
000160000
00100000
00000400
000013000
00000004
000000130
,
160000000
016000000
001600000
000160000
00000100
000016000
00000001
000000160
,
00100000
00010000
160000000
016000000
0000512134
0000121244
0000413125
0000131355
,
160000000
016000000
00100000
00010000
00000004
00000040
00000400
00004000
,
160000000
01000000
00100000
000160000
00000004
000000130
00000400
000013000

G:=sub<GL(8,GF(17))| [0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,0,0,13,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,13,0,0,0,0,0,0,4,0],[16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0],[0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,5,12,4,13,0,0,0,0,12,12,13,13,0,0,0,0,13,4,12,5,0,0,0,0,4,4,5,5],[16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0],[16,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,0,0,0,13,0,0,0,0,0,0,4,0,0,0,0,0,0,13,0,0,0,0,0,0,4,0,0,0] >;

C42.475C23 in GAP, Magma, Sage, TeX

C_4^2._{475}C_2^3
% in TeX

G:=Group("C4^2.475C2^3");
// GroupNames label

G:=SmallGroup(128,2058);
// by ID

G=gap.SmallGroup(128,2058);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,112,253,758,723,352,2019,346,4037,1027,124]);
// Polycyclic

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

Export

Character table of C42.475C23 in TeX

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