Copied to
clipboard

?

G = C42.61C23  order 128 = 27

61st non-split extension by C42 of C23 acting faithfully

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

Aliases: C42.61C23, C4.822- (1+4), C8⋊Q8⋊27C2, C8⋊2D4⋊32C2, C8⋊7D4⋊32C2, C8⋊9D4⋊27C2, C4⋊C4.379D4, D4.Q8⋊42C2, D8⋊C4⋊26C2, D4⋊Q8⋊39C2, (C2×D4).179D4, C8.34(C4○D4), C2.58(D4○D8), C22⋊C4.63D4, C4⋊C4.252C23, C4⋊C8.120C22, (C2×C8).106C23, (C2×C4).539C24, (C2×D8).90C22, C23.344(C2×D4), C4⋊Q8.171C22, C2.92(D4⋊6D4), C8⋊C4.53C22, (C2×D4).257C23, (C4×D4).179C22, C22.D8⋊34C2, C22⋊C8.98C22, M4(2)⋊C4⋊34C2, C2.D8.132C22, C4.Q8.138C22, D4⋊C4.81C22, C4⋊D4.106C22, C23.25D4⋊30C2, C23.19D4⋊44C2, (C22×C8).290C22, C22.799(C22×D4), C42.C2.52C22, C2.94(D8⋊C22), C22.49C24⋊9C2, (C22×C4).1167C23, C42⋊C2.210C22, C22.47C24⋊10C2, (C2×M4(2)).132C22, C4.121(C2×C4○D4), (C2×C4).623(C2×D4), (C2×C4⋊C4).688C22, SmallGroup(128,2079)

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

Derived series
C1 — C2×C4 — C42.61C23
Chief series
C1 — C2 — C4 — C2×C4 — C22×C4 — C42⋊C2 — C22.49C24 — C42.61C23
Lower central
C1 — C2 — C2×C4 — C42.61C23
Upper central
C1 — C22 — C4×D4 — C42.61C23
Jennings
C1 — C2 — C2 — C2×C4 — C42.61C23

Subgroups: 360 in 180 conjugacy classes, 86 normal (84 characteristic)
C1, C2 [×3], C2 [×4], C4 [×2], C4 [×10], C22, C22 [×12], C8 [×2], C8 [×3], C2×C4 [×5], C2×C4 [×14], D4 [×10], Q8, C23 [×2], C23 [×2], C42, C42 [×3], C22⋊C4 [×2], C22⋊C4 [×9], C4⋊C4 [×7], C4⋊C4 [×4], C2×C8 [×4], C2×C8 [×2], M4(2) [×2], D8 [×2], C22×C4 [×2], C22×C4 [×3], C2×D4 [×3], C2×D4 [×4], C2×Q8, C8⋊C4, C22⋊C8 [×2], D4⋊C4 [×6], C4⋊C8, C4.Q8 [×4], C2.D8 [×5], C2×C4⋊C4, C42⋊C2 [×3], C42⋊C2, C4×D4 [×3], C4×D4, C4⋊D4 [×4], C4⋊D4 [×2], C22.D4, C4.4D4 [×2], C42.C2, C42⋊2C2, C4⋊Q8, C22×C8, C2×M4(2), C2×D8, C23.25D4, M4(2)⋊C4, C8⋊9D4, D8⋊C4, C8⋊7D4, C8⋊2D4, D4⋊Q8, D4.Q8, C22.D8, C23.19D4 [×3], C8⋊Q8, C22.47C24, C22.49C24, C42.61C23

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), D4⋊6D4, D8⋊C22, D4○D8, C42.61C23

Generators and relations
 G = < a,b,c,d,e | a4=b4=e2=1, c2=d2=a2b2, ab=ba, cac-1=eae=a-1b2, dad-1=ab2, cbc-1=dbd-1=b-1, be=eb, dcd-1=bc, ece=a2b2c, 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 52 28 31)(2 49 25 32)(3 50 26 29)(4 51 27 30)(5 60 61 54)(6 57 62 55)(7 58 63 56)(8 59 64 53)(9 33 14 40)(10 34 15 37)(11 35 16 38)(12 36 13 39)(17 45 24 41)(18 46 21 42)(19 47 22 43)(20 48 23 44)

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

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

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

Matrix representation ►G ⊆ GL8(𝔽17)

001600000
00010000
10000000
016000000
000001300
00004000
00000004
000000130
,
160000000
016000000
001600000
000160000
00000100
000016000
00000001
000000160
,
50500000
05050000
501200000
050120000
00005500
000051200
00000055
000000512
,
00100000
00010000
160000000
016000000
000001300
000013000
000000013
000000130
,
00010000
001600000
016000000
10000000
000000013
00000040
000001300
00004000

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

Character table of C42.61C23

 class 12A2B2C2D2E2F2G4A4B4C4D4E4F4G4H4I4J4K4L4M4N4O8A8B8C8D8E8F
 size 11114488222244444448888444488
ρ111111111111111111111111111111    trivial
ρ21111-111111-1-1-1-1-1-11-1-1-1-111-111-1-11    linear of order 2
ρ31111111-11111-111-1111-111-1-1-1-1-1-1-1    linear of order 2
ρ41111-111-111-1-11-1-111-1-11-11-11-1-111-1    linear of order 2
ρ51111-1-1-1-111-1-1-11-1-1-11-11111-111-11-1    linear of order 2
ρ611111-1-1-111111-111-1-11-1-1111111-1-1    linear of order 2
ρ71111-1-1-1111-1-111-11-11-1-111-11-1-11-11    linear of order 2
ρ811111-1-111111-1-11-1-1-111-11-1-1-1-1-111    linear of order 2
ρ91111-11-1-111-1-11-1111-1111-1-1-111-1-11    linear of order 2
ρ10111111-1-11111-11-1-111-1-1-1-1-1111111    linear of order 2
ρ111111-11-1111-1-1-1-11-11-11-11-111-1-111-1    linear of order 2
ρ12111111-11111111-1111-11-1-11-1-1-1-1-1-1    linear of order 2
ρ1311111-1111111-1-1-1-1-1-1-111-1-11111-1-1    linear of order 2
ρ141111-1-11111-1-11111-111-1-1-1-1-111-11-1    linear of order 2
ρ1511111-11-111111-1-11-1-1-1-11-11-1-1-1-111    linear of order 2
ρ161111-1-11-111-1-1-111-1-1111-1-111-1-11-11    linear of order 2
ρ1722222200-2-2-2-20-200-2200000000000    orthogonal lifted from D4
ρ182222-2-200-2-2220-2002200000000000    orthogonal lifted from D4
ρ1922222-200-2-2-2-202002-200000000000    orthogonal lifted from D4
ρ202222-2200-2-2220200-2-200000000000    orthogonal lifted from D4
ρ212-22-200002-2002i02i2i002i00000-22000    complex lifted from C4○D4
ρ222-22-200002-2002i02i2i002i000002-2000    complex lifted from C4○D4
ρ232-22-200002-2002i02i2i002i00000-22000    complex lifted from C4○D4
ρ242-22-200002-2002i02i2i002i000002-2000    complex lifted from C4○D4
ρ2544-4-400000000000000000002√2002√200    orthogonal lifted from D4○D8
ρ2644-4-400000000000000000002√2002√200    orthogonal lifted from D4○D8
ρ274-44-40000-440000000000000000000    symplectic lifted from 2- (1+4), Schur index 2
ρ284-4-440000004i4i00000000000000000    complex lifted from D8⋊C22
ρ294-4-440000004i4i00000000000000000    complex lifted from D8⋊C22

In GAP, Magma, Sage, TeX 

C_4^2._{61}C_2^3
 
% in TeX
 
G:=Group("C4^2.61C2^3");
 
// GroupNames label
 
G:=SmallGroup(128,2079);
 
// by ID
 
G=gap.SmallGroup(128,2079);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,-2,112,253,758,723,100,346,248,4037,1027,124]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^4=b^4=e^2=1,c^2=d^2=a^2*b^2,a*b=b*a,c*a*c^-1=e*a*e=a^-1*b^2,d*a*d^-1=a*b^2,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*b^2*c,e*d*e=b^2*d>;
 
// generators/relations
 

׿
×
⋊
ℤ
𝔽
○
≀