Copied to
clipboard

G = C23.688C24order 128 = 27

405th central stem extension by C23 of C24

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

Aliases: C23.688C24, C24.453C23, C22.3512- 1+4, C22.4612+ 1+4, C22⋊C48Q8, C23.44(C2×Q8), C2.62(D43Q8), C23⋊Q8.29C2, (C2×C42).715C22, (C23×C4).497C22, (C22×C4).215C23, C23.8Q8.65C2, C23.Q8.41C2, C23.7Q8.76C2, C22.160(C22×Q8), (C22×Q8).220C22, C23.78C2361C2, C24.C22.77C2, C23.63C23187C2, C23.67C23102C2, C23.83C23122C2, C23.65C23155C2, C23.81C23128C2, C2.105(C22.32C24), C2.C42.392C22, C2.42(C22.56C24), C2.71(C22.50C24), C2.41(C23.41C23), (C2×C4).84(C2×Q8), (C2×C4).472(C4○D4), (C2×C4⋊C4).498C22, C22.549(C2×C4○D4), (C2×C22⋊C4).324C22, SmallGroup(128,1520)

Series: Derived Chief Lower central Upper central Jennings

C1C23 — C23.688C24
C1C2C22C23C22×C4C23×C4C23.8Q8 — C23.688C24
C1C23 — C23.688C24
C1C23 — C23.688C24
C1C23 — C23.688C24

Generators and relations for C23.688C24
 G = < a,b,c,d,e,f,g | a2=b2=c2=d2=1, e2=bcd, f2=g2=cb=bc, ab=ba, eae-1=ac=ca, faf-1=ad=da, ag=ga, bd=db, fef-1=be=eb, gfg-1=bf=fb, bg=gb, cd=dc, ce=ec, cf=fc, cg=gc, de=ed, df=fd, dg=gd, geg-1=bce >

Subgroups: 404 in 208 conjugacy classes, 96 normal (82 characteristic)
C1, C2 [×7], C2 [×2], C4 [×18], C22 [×7], C22 [×10], C2×C4 [×8], C2×C4 [×42], Q8 [×4], C23, C23 [×2], C23 [×6], C42 [×2], C22⋊C4 [×4], C22⋊C4 [×6], C4⋊C4 [×14], C22×C4 [×14], C22×C4 [×4], C2×Q8 [×4], C24, C2.C42 [×14], C2×C42 [×2], C2×C22⋊C4 [×6], C2×C4⋊C4 [×11], C23×C4, C22×Q8, C23.7Q8, C23.8Q8 [×2], C23.63C23, C24.C22 [×2], C23.65C23 [×2], C23.67C23, C23⋊Q8, C23.78C23, C23.Q8, C23.81C23, C23.83C23 [×2], C23.688C24
Quotients: C1, C2 [×15], C22 [×35], Q8 [×4], C23 [×15], C2×Q8 [×6], C4○D4 [×4], C24, C22×Q8, C2×C4○D4 [×2], 2+ 1+4 [×3], 2- 1+4, C22.32C24 [×2], C23.41C23, D43Q8 [×2], C22.50C24, C22.56C24, C23.688C24

Smallest permutation representation of C23.688C24
On 64 points
Generators in S64
(2 41)(4 43)(5 18)(6 64)(7 20)(8 62)(9 54)(11 56)(14 32)(16 30)(17 51)(19 49)(21 34)(22 57)(23 36)(24 59)(25 46)(27 48)(33 38)(35 40)(37 58)(39 60)(50 61)(52 63)
(1 55)(2 56)(3 53)(4 54)(5 61)(6 62)(7 63)(8 64)(9 43)(10 44)(11 41)(12 42)(13 45)(14 46)(15 47)(16 48)(17 49)(18 50)(19 51)(20 52)(21 58)(22 59)(23 60)(24 57)(25 32)(26 29)(27 30)(28 31)(33 40)(34 37)(35 38)(36 39)
(1 44)(2 41)(3 42)(4 43)(5 52)(6 49)(7 50)(8 51)(9 54)(10 55)(11 56)(12 53)(13 31)(14 32)(15 29)(16 30)(17 62)(18 63)(19 64)(20 61)(21 39)(22 40)(23 37)(24 38)(25 46)(26 47)(27 48)(28 45)(33 59)(34 60)(35 57)(36 58)
(1 12)(2 9)(3 10)(4 11)(5 18)(6 19)(7 20)(8 17)(13 26)(14 27)(15 28)(16 25)(21 34)(22 35)(23 36)(24 33)(29 45)(30 46)(31 47)(32 48)(37 58)(38 59)(39 60)(40 57)(41 54)(42 55)(43 56)(44 53)(49 64)(50 61)(51 62)(52 63)
(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 18 10 7)(2 51 11 64)(3 20 12 5)(4 49 9 62)(6 54 17 43)(8 56 19 41)(13 36 28 21)(14 40 25 59)(15 34 26 23)(16 38 27 57)(22 46 33 32)(24 48 35 30)(29 60 47 37)(31 58 45 39)(42 61 53 52)(44 63 55 50)
(1 45 10 31)(2 32 11 46)(3 47 12 29)(4 30 9 48)(5 23 20 34)(6 35 17 24)(7 21 18 36)(8 33 19 22)(13 44 28 55)(14 56 25 41)(15 42 26 53)(16 54 27 43)(37 61 60 52)(38 49 57 62)(39 63 58 50)(40 51 59 64)

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

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

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

32 conjugacy classes

class 1 2A···2G2H2I4A···4P4Q···4V
order12···2224···44···4
size11···1444···48···8

32 irreducible representations

dim1111111111112244
type++++++++++++-+-
imageC1C2C2C2C2C2C2C2C2C2C2C2Q8C4○D42+ 1+42- 1+4
kernelC23.688C24C23.7Q8C23.8Q8C23.63C23C24.C22C23.65C23C23.67C23C23⋊Q8C23.78C23C23.Q8C23.81C23C23.83C23C22⋊C4C2×C4C22C22
# reps1121221111124831

Matrix representation of C23.688C24 in GL6(𝔽5)

100000
040000
001000
000400
000010
000001
,
100000
010000
001000
000100
000040
000004
,
400000
040000
001000
000100
000010
000001
,
100000
010000
004000
000400
000010
000001
,
010000
400000
003000
000300
000024
000003
,
300000
030000
000100
001000
000043
000011
,
200000
030000
001000
000100
000020
000033

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

C23.688C24 in GAP, Magma, Sage, TeX

C_2^3._{688}C_2^4
% in TeX

G:=Group("C2^3.688C2^4");
// GroupNames label

G:=SmallGroup(128,1520);
// by ID

G=gap.SmallGroup(128,1520);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,784,253,344,758,723,1571,346,192]);
// Polycyclic

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

׿
×
𝔽