Copied to
clipboard

?

G = C42.514C23  order 128 = 27

375th non-split extension by C42 of C23 acting via C23/C2=C22

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

Aliases: C42.514C23, C4.352- (1+4), (D4×Q8)⋊12C2, C4⋊C4.179D4, C8⋊4Q8⋊10C2, D8⋊C4⋊29C2, D4⋊2Q8⋊27C2, Q8⋊Q8⋊26C2, (C4×SD16)⋊63C2, C4.4D8⋊37C2, C4⋊D8.13C2, (C2×Q8).245D4, D4.39(C4○D4), D4.2D4⋊47C2, C4⋊C8.138C22, C4⋊C4.439C23, C4.79(C8⋊C22), (C4×C8).300C22, (C2×C8).119C23, (C2×C4).565C24, (C2×D8).93C22, C4⋊Q8.194C22, C8⋊C4.64C22, C2.73(Q8⋊5D4), (C4×D4).204C22, (C2×D4).275C23, (C2×Q8).259C23, (C4×Q8).196C22, C4.Q8.182C22, C2.104(D4○SD16), D4⋊C4.89C22, C4⋊1D4.102C22, C4.4D4.82C22, C22.825(C22×D4), C22.53C24⋊5C2, Q8⋊C4.210C22, (C2×SD16).174C22, C42.28C22⋊23C2, C4.266(C2×C4○D4), (C2×C4).641(C2×D4), C2.89(C2×C8⋊C22), SmallGroup(128,2105)

Series: Derived ►Chief ►Lower central ►Upper central ►Jennings

Derived series
C1 — C2×C4 — C42.514C23
Chief series
C1 — C2 — C4 — C2×C4 — C42 — C4×D4 — D4×Q8 — C42.514C23
Lower central
C1 — C2 — C2×C4 — C42.514C23
Upper central
C1 — C22 — C4×Q8 — C42.514C23
Jennings
C1 — C2 — C2 — C2×C4 — C42.514C23

Subgroups: 392 in 192 conjugacy classes, 88 normal (38 characteristic)
C1, C2 [×3], C2 [×4], C4 [×2], C4 [×2], C4 [×10], C22, C22 [×10], C8 [×4], C2×C4 [×3], C2×C4 [×4], C2×C4 [×13], D4 [×2], D4 [×8], Q8 [×10], C23 [×3], C42, C42 [×2], C42, C22⋊C4 [×9], C4⋊C4 [×3], C4⋊C4 [×4], C4⋊C4 [×4], C2×C8 [×2], C2×C8 [×2], D8 [×4], SD16 [×2], C22×C4 [×5], C2×D4, C2×D4 [×2], C2×D4 [×2], C2×Q8 [×2], C2×Q8 [×7], C4×C8, C8⋊C4 [×2], D4⋊C4, D4⋊C4 [×6], Q8⋊C4, Q8⋊C4 [×2], C4⋊C8, C4⋊C8 [×2], C4.Q8, C4.Q8 [×2], C4×D4, C4×D4 [×4], C4×D4, C4×Q8 [×2], C22⋊Q8 [×3], C22.D4 [×2], C4.4D4 [×2], C4.4D4, C4⋊1D4, C4⋊Q8, C4⋊Q8 [×2], C2×D8 [×2], C2×SD16, C22×Q8, C4×SD16, D8⋊C4 [×2], C8⋊4Q8, C4⋊D8, D4.2D4 [×2], Q8⋊Q8, D4⋊2Q8 [×2], C4.4D8, C42.28C22 [×2], D4×Q8, C22.53C24, C42.514C23

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), Q8⋊5D4, C2×C8⋊C22, D4○SD16, C42.514C23

Generators and relations
 G = < a,b,c,d,e | a4=b4=e2=1, c2=b2, d2=a2b2, ab=ba, ac=ca, dad-1=a-1, ae=ea, cbc-1=ebe=b-1, bd=db, dcd-1=a2c, ece=bc, ede=b2d >

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

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

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

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

Matrix representation ►G ⊆ GL8(𝔽17)

11000000
1516000000
0161150000
101160000
000016000
000001600
000000160
000000016
,
160000000
016000000
001600000
000160000
00000100
000016000
00001112
00000161616
,
1010100000
1460150000
1610730000
10117110000
00004911
0000813016
00001313510
000004812
,
1010100000
081520000
1710140000
10160000
00001381616
0000813016
000098120
0000413913
,
00100000
0160150000
10000000
00010000
00000010
000016161615
00001000
00000001

G:=sub<GL(8,GF(17))| [1,15,0,1,0,0,0,0,1,16,16,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,15,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16],[16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,0,16,1,0,0,0,0,0,1,0,1,16,0,0,0,0,0,0,1,16,0,0,0,0,0,0,2,16],[10,14,16,10,0,0,0,0,10,6,10,11,0,0,0,0,1,0,7,7,0,0,0,0,0,15,3,11,0,0,0,0,0,0,0,0,4,8,13,0,0,0,0,0,9,13,13,4,0,0,0,0,1,0,5,8,0,0,0,0,1,16,10,12],[10,0,1,1,0,0,0,0,10,8,7,0,0,0,0,0,1,15,10,1,0,0,0,0,0,2,14,6,0,0,0,0,0,0,0,0,13,8,9,4,0,0,0,0,8,13,8,13,0,0,0,0,16,0,12,9,0,0,0,0,16,16,0,13],[0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,15,0,1,0,0,0,0,0,0,0,0,0,16,1,0,0,0,0,0,0,16,0,0,0,0,0,0,1,16,0,0,0,0,0,0,0,15,0,1] >;
 

Character table of C42.514C23

 class 12A2B2C2D2E2F2G4A4B4C4D4E4F4G4H4I4J4K4L4M4N4O8A8B8C8D8E8F
 size 11114488222244444448888444488
ρ111111111111111111111111111111    trivial
ρ21111-1-1-11-111-1-11-1111-1-1-1111-11-1-11    linear of order 2
ρ3111111111111-1-1-1-1-1-11-1-1-11-1-1-1-111    linear of order 2
ρ41111-1-1-11-111-11-11-1-1-1-111-11-11-11-11    linear of order 2
ρ51111-1-1-1-111111-111-111-1-1-1-1111111    linear of order 2
ρ61111111-1-111-1-1-1-11-11-111-1-11-11-1-11    linear of order 2
ρ71111-1-1-1-11111-11-1-11-11111-1-1-1-1-111    linear of order 2
ρ81111111-1-111-1111-11-1-1-1-11-1-11-11-11    linear of order 2
ρ91111-1-1111111-1-1-1-1-1-111-11-11111-1-1    linear of order 2
ρ10111111-11-111-11-11-1-1-1-1-111-11-11-11-1    linear of order 2
ρ111111-1-11111111111111-11-1-1-1-1-1-1-1-1    linear of order 2
ρ12111111-11-111-1-11-1111-11-1-1-1-11-111-1    linear of order 2
ρ13111111-1-11111-11-1-11-11-11-111111-1-1    linear of order 2
ρ141111-1-11-1-111-1111-11-1-11-1-111-11-11-1    linear of order 2
ρ15111111-1-111111-111-1111-111-1-1-1-1-1-1    linear of order 2
ρ161111-1-11-1-111-1-1-1-11-11-1-1111-11-111-1    linear of order 2
ρ1722220000-2-2-2-2-202-20220000000000    orthogonal lifted from D4
ρ18222200002-2-2220-2-202-20000000000    orthogonal lifted from D4
ρ19222200002-2-22-20220-2-20000000000    orthogonal lifted from D4
ρ2022220000-2-2-2-220-220-220000000000    orthogonal lifted from D4
ρ212-22-22-2000-22002i002i00000002i02i00    complex lifted from C4○D4
ρ222-22-2-22000-22002i002i00000002i02i00    complex lifted from C4○D4
ρ232-22-2-22000-22002i002i00000002i02i00    complex lifted from C4○D4
ρ242-22-22-2000-22002i002i00000002i02i00    complex lifted from C4○D4
ρ254-4-440000400-400000000000000000    orthogonal lifted from C8⋊C22
ρ264-4-440000-400400000000000000000    orthogonal lifted from C8⋊C22
ρ274-44-4000004-4000000000000000000    symplectic lifted from 2- (1+4), Schur index 2
ρ2844-4-400000000000000000002√-202√-2000    complex lifted from D4○SD16
ρ2944-4-400000000000000000002√-202√-2000    complex lifted from D4○SD16

In GAP, Magma, Sage, TeX 

C_4^2._{514}C_2^3
 
% in TeX
 
G:=Group("C4^2.514C2^3");
 
// GroupNames label
 
G:=SmallGroup(128,2105);
 
// by ID
 
G=gap.SmallGroup(128,2105);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,-2,448,253,568,758,723,346,80,4037,1027,124]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^4=b^4=e^2=1,c^2=b^2,d^2=a^2*b^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
 

׿
×
⋊
ℤ
𝔽
○
≀