Copied to
clipboard

G = C23.618C24order 128 = 27

335th central stem extension by C23 of C24

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

Aliases: C24.70C23, C23.618C24, C22.2932- 1+4, C22.3922+ 1+4, C22⋊C4.19D4, C429C433C2, C23.220(C2×D4), C2.69(D46D4), C23.Q870C2, C23.4Q849C2, (C2×C42).669C22, (C23×C4).156C22, (C22×C4).883C23, C23.8Q8118C2, C22.427(C22×D4), C23.10D4.47C2, (C22×D4).251C22, C23.83C2386C2, C24.C22143C2, C23.81C2395C2, C2.64(C22.29C24), C23.65C23129C2, C2.C42.324C22, C2.89(C22.47C24), C2.72(C22.33C24), C2.21(C22.57C24), C2.83(C22.36C24), (C2×C4).114(C2×D4), (C2×C422C2)⋊24C2, (C2×C4).203(C4○D4), (C2×C4⋊C4).431C22, C22.480(C2×C4○D4), (C2×C22⋊C4).283C22, (C2×C22.D4).29C2, SmallGroup(128,1450)

Series: Derived Chief Lower central Upper central Jennings

C1C23 — C23.618C24
C1C2C22C23C24C23×C4C23.8Q8 — C23.618C24
C1C23 — C23.618C24
C1C23 — C23.618C24
C1C23 — C23.618C24

Generators and relations for C23.618C24
 G = < a,b,c,d,e,f,g | a2=b2=c2=g2=1, d2=a, e2=f2=ba=ab, ac=ca, ede-1=ad=da, geg=ae=ea, af=fa, ag=ga, bc=cb, fdf-1=bd=db, be=eb, bf=fb, bg=gb, cd=dc, fef-1=ce=ec, cf=fc, cg=gc, gdg=abd, fg=gf >

Subgroups: 484 in 245 conjugacy classes, 96 normal (82 characteristic)
C1, C2 [×7], C2 [×3], C4 [×17], C22 [×7], C22 [×17], C2×C4 [×8], C2×C4 [×39], D4 [×4], C23, C23 [×2], C23 [×13], C42 [×4], C22⋊C4 [×4], C22⋊C4 [×18], C4⋊C4 [×18], C22×C4 [×13], C22×C4 [×4], C2×D4 [×4], C24 [×2], C2.C42 [×8], C2×C42 [×2], C2×C22⋊C4 [×11], C2×C4⋊C4 [×12], C22.D4 [×4], C422C2 [×4], C23×C4, C22×D4, C429C4, C23.8Q8 [×2], C24.C22 [×3], C23.65C23, C23.10D4 [×2], C23.Q8, C23.81C23, C23.4Q8, C23.83C23, C2×C22.D4, C2×C422C2, C23.618C24
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], C2×D4 [×6], C4○D4 [×4], C24, C22×D4, C2×C4○D4 [×2], 2+ 1+4 [×2], 2- 1+4 [×2], C22.29C24, C22.33C24, C22.36C24, D46D4 [×2], C22.47C24, C22.57C24, C23.618C24

Smallest permutation representation of C23.618C24
On 64 points
Generators in S64
(1 3)(2 4)(5 7)(6 8)(9 11)(10 12)(13 15)(14 16)(17 19)(18 20)(21 23)(22 24)(25 27)(26 28)(29 31)(30 32)(33 35)(34 36)(37 39)(38 40)(41 43)(42 44)(45 47)(46 48)(49 51)(50 52)(53 55)(54 56)(57 59)(58 60)(61 63)(62 64)
(1 26)(2 27)(3 28)(4 25)(5 54)(6 55)(7 56)(8 53)(9 35)(10 36)(11 33)(12 34)(13 51)(14 52)(15 49)(16 50)(17 43)(18 44)(19 41)(20 42)(21 47)(22 48)(23 45)(24 46)(29 37)(30 38)(31 39)(32 40)(57 61)(58 62)(59 63)(60 64)
(1 15)(2 16)(3 13)(4 14)(5 58)(6 59)(7 60)(8 57)(9 37)(10 38)(11 39)(12 40)(17 45)(18 46)(19 47)(20 48)(21 41)(22 42)(23 43)(24 44)(25 52)(26 49)(27 50)(28 51)(29 35)(30 36)(31 33)(32 34)(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 59 28 61)(2 58 25 64)(3 57 26 63)(4 60 27 62)(5 52 56 16)(6 51 53 15)(7 50 54 14)(8 49 55 13)(9 24 33 48)(10 23 34 47)(11 22 35 46)(12 21 36 45)(17 40 41 30)(18 39 42 29)(19 38 43 32)(20 37 44 31)
(1 25 28 2)(3 27 26 4)(5 59 56 61)(6 64 53 58)(7 57 54 63)(8 62 55 60)(9 10 33 34)(11 12 35 36)(13 50 49 14)(15 52 51 16)(17 22 41 46)(18 45 42 21)(19 24 43 48)(20 47 44 23)(29 30 39 40)(31 32 37 38)
(1 11)(2 36)(3 9)(4 34)(5 19)(6 44)(7 17)(8 42)(10 27)(12 25)(13 37)(14 32)(15 39)(16 30)(18 55)(20 53)(21 62)(22 57)(23 64)(24 59)(26 33)(28 35)(29 51)(31 49)(38 50)(40 52)(41 54)(43 56)(45 60)(46 63)(47 58)(48 61)

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

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

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

32 conjugacy classes

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

32 irreducible representations

dim1111111111112244
type++++++++++++++-
imageC1C2C2C2C2C2C2C2C2C2C2C2D4C4○D42+ 1+42- 1+4
kernelC23.618C24C429C4C23.8Q8C24.C22C23.65C23C23.10D4C23.Q8C23.81C23C23.4Q8C23.83C23C2×C22.D4C2×C422C2C22⋊C4C2×C4C22C22
# reps1123121111114822

Matrix representation of C23.618C24 in GL6(𝔽5)

100000
010000
004000
000400
000010
000001
,
100000
010000
001000
000100
000040
000004
,
400000
040000
001000
000100
000010
000001
,
100000
010000
002000
001300
000031
000022
,
010000
100000
004400
002100
000020
000002
,
100000
040000
003000
000300
000031
000002
,
100000
010000
004400
000100
000043
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],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,2,1,0,0,0,0,0,3,0,0,0,0,0,0,3,2,0,0,0,0,1,2],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,4,2,0,0,0,0,4,1,0,0,0,0,0,0,2,0,0,0,0,0,0,2],[1,0,0,0,0,0,0,4,0,0,0,0,0,0,3,0,0,0,0,0,0,3,0,0,0,0,0,0,3,0,0,0,0,0,1,2],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,4,1,0,0,0,0,0,0,4,0,0,0,0,0,3,1] >;

C23.618C24 in GAP, Magma, Sage, TeX

C_2^3._{618}C_2^4
% in TeX

G:=Group("C2^3.618C2^4");
// GroupNames label

G:=SmallGroup(128,1450);
// by ID

G=gap.SmallGroup(128,1450);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,112,253,344,758,723,268,1571,346]);
// Polycyclic

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

׿
×
𝔽