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G = C42.513C23order 128 = 27

374th non-split extension by C42 of C23 acting via C23/C2=C22

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

Aliases: C42.513C23, C4.342- 1+4, C84Q89C2, C4⋊C4.178D4, Q83Q87C2, Q8.Q848C2, C4.Q1643C2, D43Q8.8C2, (C2×Q8).244D4, D4.38(C4○D4), C4⋊C4.438C23, C4⋊C8.137C22, (C2×C8).118C23, (C4×C8).299C22, (C2×C4).564C24, C4.SD1637C2, (C4×SD16).17C2, D4.D4.1C2, C4⋊Q8.193C22, C8⋊C4.63C22, C2.72(Q85D4), SD16⋊C4.1C2, (C2×D4).431C23, (C4×D4).203C22, C4.52(C8.C22), (C2×Q8).258C23, (C4×Q8).195C22, C4.Q8.181C22, C2.D8.136C22, C2.103(D4○SD16), Q8⋊C4.90C22, (C2×SD16).71C22, C22.824(C22×D4), C42.C2.67C22, D4⋊C4.213C22, C42.30C2212C2, C4.265(C2×C4○D4), (C2×C4).640(C2×D4), C2.87(C2×C8.C22), SmallGroup(128,2104)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C42.513C23
C1C2C4C2×C4C42C4×D4D43Q8 — C42.513C23
C1C2C2×C4 — C42.513C23
C1C22C4×Q8 — C42.513C23
C1C2C2C2×C4 — C42.513C23

Generators and relations for C42.513C23
 G = < a,b,c,d,e | a4=b4=e2=1, c2=b2, d2=a2, ab=ba, ac=ca, dad-1=a-1, ae=ea, cbc-1=ebe=b-1, bd=db, dcd-1=a2c, ece=bc, ede=b2d >

Subgroups: 296 in 170 conjugacy classes, 88 normal (38 characteristic)
C1, C2 [×3], C2 [×2], C4 [×2], C4 [×2], C4 [×12], C22, C22 [×4], C8 [×4], C2×C4 [×3], C2×C4 [×4], C2×C4 [×11], D4 [×2], D4, Q8 [×9], C23, C42, C42 [×2], C42 [×3], C22⋊C4 [×3], C4⋊C4 [×3], C4⋊C4 [×4], C4⋊C4 [×14], C2×C8 [×2], C2×C8 [×2], SD16 [×6], C22×C4 [×3], C2×D4, C2×Q8 [×2], C2×Q8 [×2], C2×Q8, C4×C8, C8⋊C4 [×2], D4⋊C4, Q8⋊C4, Q8⋊C4 [×8], C4⋊C8, C4⋊C8 [×2], C4.Q8, C2.D8 [×2], C2×C4⋊C4, C4×D4, C4×D4 [×2], C4×Q8 [×2], C4×Q8 [×2], C4×Q8, C22⋊Q8 [×3], C42.C2 [×2], C42.C2 [×3], C4⋊Q8 [×2], C4⋊Q8 [×2], C2×SD16, C2×SD16 [×2], C4×SD16, SD16⋊C4 [×2], C84Q8, D4.D4, D4.D4 [×2], C4.Q16, Q8.Q8 [×2], C4.SD16, C42.30C22 [×2], D43Q8, Q83Q8, C42.513C23
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, Q85D4, C2×C8.C22, D4○SD16, C42.513C23

Character table of C42.513C23

 class 12A2B2C2D2E4A4B4C4D4E4F4G4H4I4J4K4L4M4N4O4P4Q8A8B8C8D8E8F
 size 11114422224444444888888444488
ρ111111111111111111111111111111    trivial
ρ21111111111-11-1-11-111-1-1-1-111111-1-1    linear of order 2
ρ311111111111-111-111-11-1-111-1-1-1-1-1-1    linear of order 2
ρ41111111111-1-1-1-1-1-11-1-111-11-1-1-1-111    linear of order 2
ρ51111-1-111111-111-111-1-1-1-1-1-1111111    linear of order 2
ρ61111-1-11111-1-1-1-1-1-11-11111-11111-1-1    linear of order 2
ρ71111-1-1111111111111-111-1-1-1-1-1-1-1-1    linear of order 2
ρ81111-1-11111-11-1-11-1111-1-11-1-1-1-1-111    linear of order 2
ρ91111-1-11-11-1111-11-1-1-1-1-1111-1-111-11    linear of order 2
ρ101111-1-11-11-1-11-1111-1-111-1-11-1-1111-1    linear of order 2
ρ111111-1-11-11-11-11-1-1-1-11-11-11111-1-11-1    linear of order 2
ρ121111-1-11-11-1-1-1-11-11-111-11-1111-1-1-11    linear of order 2
ρ131111111-11-11-11-1-1-1-1111-1-1-1-1-111-11    linear of order 2
ρ141111111-11-1-1-1-11-11-11-1-111-1-1-1111-1    linear of order 2
ρ151111111-11-1111-11-1-1-11-11-1-111-1-11-1    linear of order 2
ρ161111111-11-1-11-1111-1-1-11-11-111-1-1-11    linear of order 2
ρ17222200-22-22-20220-2-2000000000000    orthogonal lifted from D4
ρ18222200-22-2220-2-202-2000000000000    orthogonal lifted from D4
ρ19222200-2-2-2-2-202-2022000000000000    orthogonal lifted from D4
ρ20222200-2-2-2-220-220-22000000000000    orthogonal lifted from D4
ρ212-22-22-2-202002i00-2i000000002i-2i0000    complex lifted from C4○D4
ρ222-22-2-22-20200-2i002i000000002i-2i0000    complex lifted from C4○D4
ρ232-22-2-22-202002i00-2i00000000-2i2i0000    complex lifted from C4○D4
ρ242-22-22-2-20200-2i002i00000000-2i2i0000    complex lifted from C4○D4
ρ254-4-4400040-40000000000000000000    symplectic lifted from C8.C22, Schur index 2
ρ264-4-44000-4040000000000000000000    symplectic lifted from C8.C22, Schur index 2
ρ274-44-40040-400000000000000000000    symplectic lifted from 2- 1+4, Schur index 2
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 C42.513C23
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 27 23)(2 20 28 24)(3 17 25 21)(4 18 26 22)(5 15 9 64)(6 16 10 61)(7 13 11 62)(8 14 12 63)(29 33 39 41)(30 34 40 42)(31 35 37 43)(32 36 38 44)(45 51 53 57)(46 52 54 58)(47 49 55 59)(48 50 56 60)
(1 45 27 53)(2 46 28 54)(3 47 25 55)(4 48 26 56)(5 44 9 36)(6 41 10 33)(7 42 11 34)(8 43 12 35)(13 40 62 30)(14 37 63 31)(15 38 64 32)(16 39 61 29)(17 59 21 49)(18 60 22 50)(19 57 23 51)(20 58 24 52)
(1 33 3 35)(2 36 4 34)(5 46 7 48)(6 45 8 47)(9 54 11 56)(10 53 12 55)(13 50 15 52)(14 49 16 51)(17 37 19 39)(18 40 20 38)(21 31 23 29)(22 30 24 32)(25 43 27 41)(26 42 28 44)(57 63 59 61)(58 62 60 64)
(5 15)(6 16)(7 13)(8 14)(9 64)(10 61)(11 62)(12 63)(17 21)(18 22)(19 23)(20 24)(33 41)(34 42)(35 43)(36 44)(45 57)(46 58)(47 59)(48 60)(49 55)(50 56)(51 53)(52 54)

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

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

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

Matrix representation of C42.513C23 in GL6(𝔽17)

010000
1600000
0016000
0001600
0000160
0000016
,
100000
010000
00161500
001100
00001615
000011
,
0130000
400000
006609
001411130
000966
001301411
,
7160000
16100000
000010
000001
001000
000100
,
1600000
0160000
001000
00161600
0000160
000011

G:=sub<GL(6,GF(17))| [0,16,0,0,0,0,1,0,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,1,0,0,0,0,15,1,0,0,0,0,0,0,16,1,0,0,0,0,15,1],[0,4,0,0,0,0,13,0,0,0,0,0,0,0,6,14,0,13,0,0,6,11,9,0,0,0,0,13,6,14,0,0,9,0,6,11],[7,16,0,0,0,0,16,10,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,1,0,0],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,16,0,0,0,0,0,16,0,0,0,0,0,0,16,1,0,0,0,0,0,1] >;

C42.513C23 in GAP, Magma, Sage, TeX

C_4^2._{513}C_2^3
% in TeX

G:=Group("C4^2.513C2^3");
// GroupNames label

G:=SmallGroup(128,2104);
// by ID

G=gap.SmallGroup(128,2104);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,448,253,120,758,723,346,80,4037,1027,124]);
// Polycyclic

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

Export

Character table of C42.513C23 in TeX

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