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G = C42.511C23  order 128 = 27

372nd non-split extension by C42 of C23 acting via C23/C2=C22

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

Aliases: C42.511C23, C4.322- (1+4), (C4×D8)⋊46C2, C8⋊4Q8⋊7C2, C4⋊C4.176D4, D4.Q8⋊48C2, Q8.Q8⋊46C2, D8⋊C4⋊28C2, D4⋊Q8⋊44C2, D4⋊3Q8⋊13C2, C2.64(D4○D8), C4⋊D8.12C2, (C2×Q8).136D4, D4.37(C4○D4), D4.2D4⋊46C2, C4⋊C8.135C22, C4⋊C4.263C23, (C2×C4).562C24, (C4×C8).232C22, (C2×C8).208C23, (C2×D8).92C22, C4⋊Q8.191C22, SD16⋊C4⋊45C2, C8⋊C4.61C22, C4.Q8.74C22, C2.70(Q8⋊5D4), (C2×D4).273C23, (C4×D4).201C22, (C2×Q8).257C23, (C4×Q8).193C22, C2.D8.206C22, C4⋊1D4.100C22, Q8⋊C4.88C22, (C2×SD16).69C22, C4.4D4.80C22, C22.822(C22×D4), C42.C2.65C22, D4⋊C4.211C22, C22.53C24⋊4C2, C2.101(D8⋊C22), C42.78C22⋊13C2, C42.28C22⋊21C2, C42.29C22⋊14C2, C4.263(C2×C4○D4), (C2×C4).638(C2×D4), SmallGroup(128,2102)

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

Derived series
C1 — C2×C4 — C42.511C23
Chief series
C1 — C2 — C4 — C2×C4 — C42 — C4×D4 — D4⋊3Q8 — C42.511C23
Lower central
C1 — C2 — C2×C4 — C42.511C23
Upper central
C1 — C22 — C4×Q8 — C42.511C23
Jennings
C1 — C2 — C2 — C2×C4 — C42.511C23

Subgroups: 360 in 179 conjugacy classes, 86 normal (84 characteristic)
C1, C2 [×3], C2 [×4], C4 [×2], C4 [×11], C22, C22 [×10], C8 [×4], C2×C4 [×7], C2×C4 [×11], D4 [×2], D4 [×8], Q8 [×4], C23 [×3], C42 [×3], C42, C22⋊C4 [×9], C4⋊C4 [×7], C4⋊C4 [×6], C2×C8 [×4], D8 [×4], SD16 [×2], C22×C4 [×5], C2×D4 [×3], C2×D4 [×2], C2×Q8 [×2], C2×Q8, C4×C8, C8⋊C4 [×2], D4⋊C4 [×7], Q8⋊C4 [×3], C4⋊C8 [×3], C4.Q8, C2.D8 [×2], C2×C4⋊C4, C4×D4 [×5], C4×D4, C4×Q8 [×2], C22⋊Q8 [×3], C22.D4 [×2], C4.4D4 [×2], C4.4D4, C42.C2 [×2], C4⋊1D4, C4⋊Q8, C2×D8 [×2], C2×SD16, C4×D8, SD16⋊C4, D8⋊C4, C8⋊4Q8, C4⋊D8, D4.2D4 [×2], D4⋊Q8, D4.Q8, Q8.Q8, C42.78C22, C42.28C22, C42.29C22, D4⋊3Q8, C22.53C24, C42.511C23

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), Q8⋊5D4, D8⋊C22, D4○D8, C42.511C23

Generators and relations
 G = < a,b,c,d,e | a4=b4=c2=e2=1, d2=a2b2, ab=ba, ac=ca, dad-1=a-1b2, ae=ea, cbc=ebe=b-1, bd=db, dcd-1=a2b2c, 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 20 26 23)(2 17 27 24)(3 18 28 21)(4 19 25 22)(5 12 63 15)(6 9 64 16)(7 10 61 13)(8 11 62 14)(29 37 41 36)(30 38 42 33)(31 39 43 34)(32 40 44 35)(45 56 60 51)(46 53 57 52)(47 54 58 49)(48 55 59 50)

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

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

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

Matrix representation ►G ⊆ GL6(𝔽17)

010000
1600000
004000
000400
000040
000004
,
100000
010000
0011500
0011600
00116016
0001610
,
0130000
400000
00110611
00146011
0014338
00031414
,
0130000
1300000
00110611
00141160
001114914
00141433
,
1600000
0160000
0010150
0000161
0000160
0001160

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

Character table of C42.511C23

 class 12A2B2C2D2E2F2G4A4B4C4D4E4F4G4H4I4J4K4L4M4N4O8A8B8C8D8E8F
 size 11114488222244444448888444488
ρ111111111111111111111111111111    trivial
ρ21111111-11111-1-1-1-1-1-11-11-11-1-1-1-111    linear of order 2
ρ3111111-1-111111111-1-111-111-1-1-1-1-1-1    linear of order 2
ρ4111111-111111-1-1-1-1111-1-1-111111-1-1    linear of order 2
ρ511111111-111-11-11-1-1-1-11-1-1-11-11-11-1    linear of order 2
ρ61111111-1-111-1-11-1111-1-1-11-1-11-111-1    linear of order 2
ρ7111111-1-1-111-11-11-111-111-1-1-11-11-11    linear of order 2
ρ8111111-11-111-1-11-11-1-1-1-111-11-11-1-11    linear of order 2
ρ91111-1-11111111111111-11-1-1-1-1-1-1-1-1    linear of order 2
ρ101111-1-11-11111-1-1-1-1-1-11111-11111-1-1    linear of order 2
ρ111111-1-1-1-111111111-1-11-1-1-1-1111111    linear of order 2
ρ121111-1-1-111111-1-1-1-11111-11-1-1-1-1-111    linear of order 2
ρ131111-1-111-111-11-11-1-1-1-1-1-111-11-11-11    linear of order 2
ρ141111-1-11-1-111-1-11-1111-11-1-111-11-1-11    linear of order 2
ρ151111-1-1-1-1-111-11-11-111-1-11111-11-11-1    linear of order 2
ρ161111-1-1-11-111-1-11-11-1-1-111-11-11-111-1    linear of order 2
ρ1722220000-2-2-2-22-2-220020000000000    orthogonal lifted from D4
ρ18222200002-2-22-2-22200-20000000000    orthogonal lifted from D4
ρ1922220000-2-2-2-2-222-20020000000000    orthogonal lifted from D4
ρ20222200002-2-2222-2-200-20000000000    orthogonal lifted from D4
ρ212-22-22-2000-22000002i2i0000002i02i00    complex lifted from C4○D4
ρ222-22-22-2000-22000002i2i0000002i02i00    complex lifted from C4○D4
ρ232-22-2-22000-22000002i2i0000002i02i00    complex lifted from C4○D4
ρ242-22-2-22000-22000002i2i0000002i02i00    complex lifted from C4○D4
ρ2544-4-400000000000000000002√202√2000    orthogonal lifted from D4○D8
ρ2644-4-400000000000000000002√202√2000    orthogonal lifted from D4○D8
ρ274-44-4000004-4000000000000000000    symplectic lifted from 2- (1+4), Schur index 2
ρ284-4-4400004i004i00000000000000000    complex lifted from D8⋊C22
ρ294-4-4400004i004i00000000000000000    complex lifted from D8⋊C22

In GAP, Magma, Sage, TeX 

C_4^2._{511}C_2^3
 
% in TeX
 
G:=Group("C4^2.511C2^3");
 
// GroupNames label
 
G:=SmallGroup(128,2102);
 
// by ID
 
G=gap.SmallGroup(128,2102);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,120,758,723,346,304,4037,1027,124]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^4=b^4=c^2=e^2=1,d^2=a^2*b^2,a*b=b*a,a*c=c*a,d*a*d^-1=a^-1*b^2,a*e=e*a,c*b*c=e*b*e=b^-1,b*d=d*b,d*c*d^-1=a^2*b^2*c,e*c*e=b*c,e*d*e=b^2*d>;
 
// generators/relations
 

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