Copied to
clipboard

G = C23.543C24order 128 = 27

260th central stem extension by C23 of C24

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

Aliases: C24.44C23, C23.543C24, C22.3182+ 1+4, C22.2352- 1+4, C23.Q841C2, C23.8Q889C2, C23.245(C4○D4), C23.11D465C2, (C23×C4).141C22, (C22×C4).153C23, C23.84C237C2, C22.3(C422C2), C23.23D4.46C2, (C22×D4).200C22, C23.83C2364C2, C2.44(C22.32C24), C2.C42.264C22, C2.44(C22.33C24), (C2×C4⋊C4).369C22, C2.22(C2×C422C2), C22.415(C2×C4○D4), (C2×C2.C42)⋊11C2, (C2×C22⋊C4).230C22, SmallGroup(128,1375)

Series: Derived Chief Lower central Upper central Jennings

C1C23 — C23.543C24
C1C2C22C23C24C22×D4C23.23D4 — C23.543C24
C1C23 — C23.543C24
C1C23 — C23.543C24
C1C23 — C23.543C24

Generators and relations for C23.543C24
 G = < a,b,c,d,e,f,g | a2=b2=c2=f2=g2=1, d2=cb=bc, e2=b, ab=ba, ac=ca, ede-1=ad=da, ae=ea, gfg=af=fa, ag=ga, fdf=bd=db, be=eb, bf=fb, bg=gb, cd=dc, fef=ce=ec, cf=fc, cg=gc, dg=gd, eg=ge >

Subgroups: 484 in 231 conjugacy classes, 92 normal (13 characteristic)
C1, C2, C2 [×6], C2 [×5], C4 [×13], C22, C22 [×10], C22 [×19], C2×C4 [×51], D4 [×4], C23, C23 [×6], C23 [×11], C22⋊C4 [×12], C4⋊C4 [×6], C22×C4, C22×C4 [×12], C22×C4 [×12], C2×D4 [×6], C24 [×2], C2.C42, C2.C42 [×15], C2×C22⋊C4 [×9], C2×C4⋊C4 [×6], C23×C4 [×3], C22×D4, C2×C2.C42, C23.8Q8 [×3], C23.23D4 [×3], C23.Q8, C23.11D4 [×3], C23.83C23 [×3], C23.84C23, C23.543C24
Quotients: C1, C2 [×15], C22 [×35], C23 [×15], C4○D4 [×6], C24, C422C2 [×4], C2×C4○D4 [×3], 2+ 1+4 [×3], 2- 1+4, C2×C422C2, C22.32C24 [×3], C22.33C24 [×3], C23.543C24

Smallest permutation representation of C23.543C24
On 64 points
Generators in S64
(1 23)(2 24)(3 21)(4 22)(5 37)(6 38)(7 39)(8 40)(9 41)(10 42)(11 43)(12 44)(13 45)(14 46)(15 47)(16 48)(17 49)(18 50)(19 51)(20 52)(25 53)(26 54)(27 55)(28 56)(29 57)(30 58)(31 59)(32 60)(33 61)(34 62)(35 63)(36 64)
(1 5)(2 6)(3 7)(4 8)(9 55)(10 56)(11 53)(12 54)(13 59)(14 60)(15 57)(16 58)(17 63)(18 64)(19 61)(20 62)(21 39)(22 40)(23 37)(24 38)(25 43)(26 44)(27 41)(28 42)(29 47)(30 48)(31 45)(32 46)(33 51)(34 52)(35 49)(36 50)
(1 7)(2 8)(3 5)(4 6)(9 53)(10 54)(11 55)(12 56)(13 57)(14 58)(15 59)(16 60)(17 61)(18 62)(19 63)(20 64)(21 37)(22 38)(23 39)(24 40)(25 41)(26 42)(27 43)(28 44)(29 45)(30 46)(31 47)(32 48)(33 49)(34 50)(35 51)(36 52)
(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 13 5 59)(2 46 6 32)(3 15 7 57)(4 48 8 30)(9 63 55 17)(10 36 56 50)(11 61 53 19)(12 34 54 52)(14 38 60 24)(16 40 58 22)(18 42 64 28)(20 44 62 26)(21 47 39 29)(23 45 37 31)(25 51 43 33)(27 49 41 35)
(2 6)(4 8)(9 41)(10 28)(11 43)(12 26)(13 57)(14 16)(15 59)(17 33)(18 52)(19 35)(20 50)(22 40)(24 38)(25 53)(27 55)(29 45)(30 32)(31 47)(34 64)(36 62)(42 56)(44 54)(46 48)(49 61)(51 63)(58 60)
(1 55)(2 56)(3 53)(4 54)(5 9)(6 10)(7 11)(8 12)(13 17)(14 18)(15 19)(16 20)(21 25)(22 26)(23 27)(24 28)(29 33)(30 34)(31 35)(32 36)(37 41)(38 42)(39 43)(40 44)(45 49)(46 50)(47 51)(48 52)(57 61)(58 62)(59 63)(60 64)

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

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

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

32 conjugacy classes

class 1 2A···2G2H2I2J2K2L4A···4L4M···4S
order12···2222224···44···4
size11···1222284···48···8

32 irreducible representations

dim11111111244
type+++++++++-
imageC1C2C2C2C2C2C2C2C4○D42+ 1+42- 1+4
kernelC23.543C24C2×C2.C42C23.8Q8C23.23D4C23.Q8C23.11D4C23.83C23C23.84C23C23C22C22
# reps113313311231

Matrix representation of C23.543C24 in GL8(𝔽5)

10000000
01000000
00100000
00010000
00004000
00000400
00000040
00000004
,
40000000
04000000
00400000
00040000
00004000
00000400
00000040
00000004
,
10000000
01000000
00400000
00040000
00004000
00000400
00000040
00000004
,
42000000
41000000
00020000
00300000
00000100
00001000
00000004
00000040
,
30000000
03000000
00010000
00400000
00000010
00000001
00004000
00000400
,
10000000
14000000
00100000
00040000
00001000
00000400
00000040
00000001
,
40000000
04000000
00100000
00010000
00000100
00001000
00000001
00000010

G:=sub<GL(8,GF(5))| [1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4],[4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4],[4,4,0,0,0,0,0,0,2,1,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0],[3,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0],[1,1,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1],[4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0] >;

C23.543C24 in GAP, Magma, Sage, TeX

C_2^3._{543}C_2^4
% in TeX

G:=Group("C2^3.543C2^4");
// GroupNames label

G:=SmallGroup(128,1375);
// by ID

G=gap.SmallGroup(128,1375);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,672,253,232,758,723,185]);
// Polycyclic

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

׿
×
𝔽