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G = C42.533C23  order 128 = 27

394th non-split extension by C42 of C23 acting via C23/C2=C22

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

Aliases: C42.533C23, C4.1542+ (1+4), (C4×D8)⋊36C2, C4⋊D8⋊44C2, C8⋊3D4⋊23C2, C4⋊C4.190D4, C8⋊4Q8⋊25C2, C2.73(D4○D8), C8.24(C4○D4), (C2×Q8).145D4, D4.2D4⋊51C2, C8.12D4⋊26C2, C4⋊C4.454C23, C4⋊C8.156C22, (C2×C4).595C24, (C4×C8).213C22, (C2×C8).124C23, Q8.D4⋊51C2, SD16⋊C4⋊52C2, C8⋊C4.82C22, C2.49(Q8⋊6D4), (C2×D4).289C23, (C4×D4).228C22, (C2×D8).170C22, (C2×Q16).42C22, (C2×Q8).274C23, (C4×Q8).218C22, C2.D8.231C22, C4⋊1D4.112C22, Q8⋊C4.98C22, (C2×SD16).79C22, C4.4D4.95C22, C22.855(C22×D4), D4⋊C4.103C22, C2.110(D8⋊C22), C22.53C24⋊10C2, C4.173(C2×C4○D4), (C2×C4).659(C2×D4), SmallGroup(128,2135)

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

Derived series
C1 — C2×C4 — C42.533C23
Chief series
C1 — C2 — C4 — C2×C4 — C42 — C4×D4 — C22.53C24 — C42.533C23
Lower central
C1 — C2 — C2×C4 — C42.533C23
Upper central
C1 — C22 — C4×Q8 — C42.533C23
Jennings
C1 — C2 — C2 — C2×C4 — C42.533C23

Subgroups: 408 in 190 conjugacy classes, 86 normal (26 characteristic)
C1, C2 [×3], C2 [×4], C4 [×2], C4 [×10], C22, C22 [×12], C8 [×2], C8 [×3], C2×C4 [×3], C2×C4 [×4], C2×C4 [×8], D4 [×14], Q8 [×5], C23 [×4], C42, C42 [×2], C42 [×2], C22⋊C4 [×12], C4⋊C4, C4⋊C4 [×4], C4⋊C4 [×2], C2×C8 [×2], C2×C8 [×2], D8 [×5], SD16 [×6], Q16, C22×C4 [×4], C2×D4 [×4], C2×D4 [×4], C2×Q8, C2×Q8 [×2], C4×C8, C8⋊C4 [×2], D4⋊C4 [×4], Q8⋊C4 [×2], C4⋊C8, C4⋊C8 [×2], C2.D8, C4×D4 [×4], C4×D4 [×2], C4×Q8, C4×Q8 [×2], C22.D4 [×4], C4.4D4 [×4], C4.4D4 [×2], C4⋊1D4 [×2], C2×D8 [×2], C2×D8 [×2], C2×SD16 [×4], C2×Q16, C4×D8, SD16⋊C4 [×2], C8⋊4Q8, C4⋊D8 [×2], D4.2D4 [×2], Q8.D4 [×2], C8.12D4, C8⋊3D4 [×2], C22.53C24 [×2], C42.533C23

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⋊6D4, D8⋊C22, D4○D8, C42.533C23

Generators and relations
 G = < a,b,c,d,e | a4=b4=c2=1, d2=a2b2, e2=a2, ab=ba, ac=ca, dad-1=a-1b2, ae=ea, cbc=ebe-1=b-1, bd=db, dcd-1=a2b2c, ece-1=bc, ede-1=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 28 20 23)(2 25 17 24)(3 26 18 21)(4 27 19 22)(5 12 15 62)(6 9 16 63)(7 10 13 64)(8 11 14 61)(29 40 33 41)(30 37 34 42)(31 38 35 43)(32 39 36 44)(45 56 49 57)(46 53 50 58)(47 54 51 59)(48 55 52 60)

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

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

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

Matrix representation ►G ⊆ GL6(𝔽17)

120000
16160000
0001300
004000
0000013
000040
,
100000
010000
000100
0016000
0000016
000010
,
1390000
440000
00001414
0000314
0014300
00141400
,
1300000
440000
0000160
000001
001000
0001600
,
1300000
0130000
000010
000001
001000
000100

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

Character table of C42.533C23

 class 12A2B2C2D2E2F2G4A4B4C4D4E4F4G4H4I4J4K4L4M4N4O8A8B8C8D8E8F
 size 11118888222244444444488444488
ρ111111111111111111111111111111    trivial
ρ2111111-1-11111111-11111-11-1-1-1-1-1-1-1    linear of order 2
ρ31111111-111-1-1-1-11-1-1-1-11-1-1111-1-11-1    linear of order 2
ρ4111111-1111-1-1-1-111-1-1-111-1-1-1-111-11    linear of order 2
ρ511111-1-1111111-1-1-111-1-1-1-11-1-1-1-111    linear of order 2
ρ611111-11-111111-1-1111-1-11-1-11111-1-1    linear of order 2
ρ711111-1-1-111-1-1-11-11-1-11-1111-1-1111-1    linear of order 2
ρ811111-11111-1-1-11-1-1-1-11-1-11-111-1-1-11    linear of order 2
ρ91111-1-11-111-1-11-11-1-11-11-111-1-111-11    linear of order 2
ρ101111-1-1-1111-1-11-111-11-1111-111-1-11-1    linear of order 2
ρ111111-1-1111111-11111-1111-11-1-1-1-1-1-1    linear of order 2
ρ121111-1-1-1-11111-111-11-111-1-1-1111111    linear of order 2
ρ131111-11-1-111-1-111-11-111-11-1111-1-1-11    linear of order 2
ρ141111-111111-1-111-1-1-111-1-1-1-1-1-1111-1    linear of order 2
ρ151111-11-111111-1-1-1-11-1-1-1-1111111-1-1    linear of order 2
ρ161111-111-11111-1-1-111-1-1-111-1-1-1-1-111    linear of order 2
ρ1722220000-2-2-2-202-2020-22000000000    orthogonal lifted from D4
ρ1822220000-2-2220220-20-2-2000000000    orthogonal lifted from D4
ρ1922220000-2-2220-2-20-2022000000000    orthogonal lifted from D4
ρ2022220000-2-2-2-20-220202-2000000000    orthogonal lifted from D4
ρ212-22-200002-2002i002i02i002i002-20000    complex lifted from C4○D4
ρ222-22-200002-2002i002i02i002i00-220000    complex lifted from C4○D4
ρ232-22-200002-2002i002i02i002i002-20000    complex lifted from C4○D4
ρ242-22-200002-2002i002i02i002i00-220000    complex lifted from C4○D4
ρ254-44-40000-440000000000000000000    orthogonal lifted from 2+ (1+4)
ρ2644-4-40000000000000000000002√22√200    orthogonal lifted from D4○D8
ρ2744-4-40000000000000000000002√22√200    orthogonal lifted from D4○D8
ρ284-4-440000004i4i00000000000000000    complex lifted from D8⋊C22
ρ294-4-440000004i4i00000000000000000    complex lifted from D8⋊C22

In GAP, Magma, Sage, TeX 

C_4^2._{533}C_2^3
 
% in TeX
 
G:=Group("C4^2.533C2^3");
 
// GroupNames label
 
G:=SmallGroup(128,2135);
 
// by ID
 
G=gap.SmallGroup(128,2135);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,120,758,723,436,346,304,4037,1027,124]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^4=b^4=c^2=1,d^2=a^2*b^2,e^2=a^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^-1=b^-1,b*d=d*b,d*c*d^-1=a^2*b^2*c,e*c*e^-1=b*c,e*d*e^-1=b^2*d>;
 
// generators/relations
 

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