Copied to
clipboard

G = C23.641C24order 128 = 27

358th central stem extension by C23 of C24

p-group, metabelian, nilpotent (class 2), monomial

Aliases: C24.78C23, C23.641C24, C22.3132- 1+4, C22.4142+ 1+4, C23.87(C4○D4), (C2×C42).90C22, C23.4Q856C2, (C23×C4).483C22, (C22×C4).566C23, C23.7Q8101C2, C23.11D4106C2, C23.23D4.62C2, C23.10D4.54C2, (C22×D4).262C22, C24.C22154C2, C23.83C2393C2, C2.79(C22.32C24), C23.81C23107C2, C23.63C23157C2, C2.93(C22.45C24), C2.C42.345C22, C2.92(C22.47C24), C2.91(C22.46C24), C2.91(C22.36C24), C2.50(C22.34C24), C2.27(C22.56C24), (C2×C4).442(C4○D4), (C2×C4⋊C4).452C22, C22.502(C2×C4○D4), (C2×C22⋊C4).301C22, SmallGroup(128,1473)

Series: Derived Chief Lower central Upper central Jennings

C1C23 — C23.641C24
C1C2C22C23C22×C4C23×C4C23.7Q8 — C23.641C24
C1C23 — C23.641C24
C1C23 — C23.641C24
C1C23 — C23.641C24

Generators and relations for C23.641C24
 G = < a,b,c,d,e,f,g | a2=b2=c2=f2=1, d2=ca=ac, e2=b, g2=a, ab=ba, ede-1=ad=da, geg-1=ae=ea, af=fa, ag=ga, bc=cb, fdf=bd=db, be=eb, bf=fb, bg=gb, cd=dc, fef=ce=ec, cf=fc, cg=gc, gdg-1=abd, fg=gf >

Subgroups: 452 in 217 conjugacy classes, 88 normal (82 characteristic)
C1, C2 [×7], C2 [×3], C4 [×15], C22 [×7], C22 [×17], C2×C4 [×4], C2×C4 [×41], D4 [×4], C23, C23 [×2], C23 [×13], C42 [×2], C22⋊C4 [×15], C4⋊C4 [×10], C22×C4 [×13], C22×C4 [×5], C2×D4 [×4], C24 [×2], C2.C42 [×12], C2×C42 [×2], C2×C22⋊C4 [×11], C2×C4⋊C4 [×8], C23×C4, C22×D4, C23.7Q8 [×2], C23.23D4, C23.63C23 [×2], C24.C22 [×4], C23.10D4 [×2], C23.11D4, C23.81C23, C23.4Q8, C23.83C23, C23.641C24
Quotients: C1, C2 [×15], C22 [×35], C23 [×15], C4○D4 [×6], C24, C2×C4○D4 [×3], 2+ 1+4 [×3], 2- 1+4, C22.32C24, C22.34C24, C22.36C24, C22.45C24, C22.46C24, C22.47C24, C22.56C24, C23.641C24

Smallest permutation representation of C23.641C24
On 64 points
Generators in S64
(1 21)(2 22)(3 23)(4 24)(5 52)(6 49)(7 50)(8 51)(9 59)(10 60)(11 57)(12 58)(13 47)(14 48)(15 45)(16 46)(17 34)(18 35)(19 36)(20 33)(25 44)(26 41)(27 42)(28 43)(29 38)(30 39)(31 40)(32 37)(53 63)(54 64)(55 61)(56 62)
(1 55)(2 56)(3 53)(4 54)(5 35)(6 36)(7 33)(8 34)(9 42)(10 43)(11 44)(12 41)(13 38)(14 39)(15 40)(16 37)(17 51)(18 52)(19 49)(20 50)(21 61)(22 62)(23 63)(24 64)(25 57)(26 58)(27 59)(28 60)(29 47)(30 48)(31 45)(32 46)
(1 23)(2 24)(3 21)(4 22)(5 50)(6 51)(7 52)(8 49)(9 57)(10 58)(11 59)(12 60)(13 45)(14 46)(15 47)(16 48)(17 36)(18 33)(19 34)(20 35)(25 42)(26 43)(27 44)(28 41)(29 40)(30 37)(31 38)(32 39)(53 61)(54 62)(55 63)(56 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 16 55 37)(2 47 56 29)(3 14 53 39)(4 45 54 31)(5 59 35 27)(6 10 36 43)(7 57 33 25)(8 12 34 41)(9 18 42 52)(11 20 44 50)(13 62 38 22)(15 64 40 24)(17 26 51 58)(19 28 49 60)(21 46 61 32)(23 48 63 30)
(1 47)(2 30)(3 45)(4 32)(5 43)(6 11)(7 41)(8 9)(10 35)(12 33)(13 21)(14 62)(15 23)(16 64)(17 27)(18 60)(19 25)(20 58)(22 39)(24 37)(26 50)(28 52)(29 55)(31 53)(34 42)(36 44)(38 61)(40 63)(46 54)(48 56)(49 57)(51 59)
(1 7 21 50)(2 17 22 34)(3 5 23 52)(4 19 24 36)(6 54 49 64)(8 56 51 62)(9 48 59 14)(10 40 60 31)(11 46 57 16)(12 38 58 29)(13 26 47 41)(15 28 45 43)(18 53 35 63)(20 55 33 61)(25 37 44 32)(27 39 42 30)

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

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

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

32 conjugacy classes

class 1 2A···2G2H2I2J4A···4P4Q···4U
order12···22224···44···4
size11···14484···48···8

32 irreducible representations

dim11111111112244
type+++++++++++-
imageC1C2C2C2C2C2C2C2C2C2C4○D4C4○D42+ 1+42- 1+4
kernelC23.641C24C23.7Q8C23.23D4C23.63C23C24.C22C23.10D4C23.11D4C23.81C23C23.4Q8C23.83C23C2×C4C23C22C22
# reps12124211118431

Matrix representation of C23.641C24 in GL6(𝔽5)

100000
010000
004000
000400
000010
000001
,
100000
010000
001000
000100
000040
000004
,
400000
040000
001000
000100
000010
000001
,
200000
020000
000100
004000
000034
000032
,
020000
300000
000100
001000
000030
000003
,
010000
100000
004000
000400
000013
000004
,
400000
040000
002000
000300
000042
000001

G:=sub<GL(6,GF(5))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[2,0,0,0,0,0,0,2,0,0,0,0,0,0,0,4,0,0,0,0,1,0,0,0,0,0,0,0,3,3,0,0,0,0,4,2],[0,3,0,0,0,0,2,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,3,0,0,0,0,0,0,3],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,3,4],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,2,0,0,0,0,0,0,3,0,0,0,0,0,0,4,0,0,0,0,0,2,1] >;

C23.641C24 in GAP, Magma, Sage, TeX

C_2^3._{641}C_2^4
% in TeX

G:=Group("C2^3.641C2^4");
// GroupNames label

G:=SmallGroup(128,1473);
// by ID

G=gap.SmallGroup(128,1473);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,560,253,232,758,723,1571,346,80]);
// Polycyclic

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

׿
×
𝔽