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G = C23.643C24order 128 = 27

360th central stem extension by C23 of C24

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

Aliases: C23.643C24, C24.429C23, C22.4162+ 1+4, C22.3152- 1+4, (C2×C42).91C22, C23.Q877C2, C23.4Q857C2, C23.185(C4○D4), C23.34D453C2, (C22×C4).204C23, (C23×C4).158C22, C23.8Q8123C2, C23.7Q8102C2, C23.23D4.63C2, C23.10D4.55C2, (C22×D4).263C22, C24.C22155C2, C23.83C2395C2, C2.80(C22.32C24), C23.63C23159C2, C2.95(C22.45C24), C2.C42.347C22, C2.51(C22.34C24), C2.92(C22.46C24), C2.28(C22.56C24), C2.85(C22.33C24), (C2×C4).444(C4○D4), (C2×C4⋊C4).454C22, C22.504(C2×C4○D4), (C2×C22⋊C4).61C22, SmallGroup(128,1475)

Series: Derived Chief Lower central Upper central Jennings

C1C23 — C23.643C24
C1C2C22C23C24C23×C4C23.34D4 — C23.643C24
C1C23 — C23.643C24
C1C23 — C23.643C24
C1C23 — C23.643C24

Generators and relations for C23.643C24
 G = < a,b,c,d,e,f,g | a2=b2=c2=g2=1, d2=cb=bc, e2=b, f2=a, ab=ba, ac=ca, ede-1=ad=da, geg=ae=ea, af=fa, ag=ga, 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: 468 in 224 conjugacy classes, 88 normal (82 characteristic)
C1, C2 [×7], C2 [×4], C4 [×14], C22 [×7], C22 [×20], C2×C4 [×2], C2×C4 [×46], D4 [×4], C23, C23 [×4], C23 [×12], C42, C22⋊C4 [×13], C4⋊C4 [×8], C22×C4 [×13], C22×C4 [×9], C2×D4 [×5], C24 [×2], C2.C42 [×14], C2×C42, C2×C22⋊C4 [×10], C2×C4⋊C4 [×7], C23×C4 [×2], C22×D4, C23.7Q8, C23.34D4 [×2], C23.8Q8, C23.23D4 [×2], C23.63C23, C24.C22 [×2], C23.10D4, C23.Q8, C23.4Q8, C23.83C23 [×3], C23.643C24
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, C22.33C24, C22.34C24, C22.45C24 [×2], C22.46C24, C22.56C24, C23.643C24

Smallest permutation representation of C23.643C24
On 64 points
Generators in S64
(1 58)(2 59)(3 60)(4 57)(5 56)(6 53)(7 54)(8 55)(9 46)(10 47)(11 48)(12 45)(13 50)(14 51)(15 52)(16 49)(17 41)(18 42)(19 43)(20 44)(21 39)(22 40)(23 37)(24 38)(25 35)(26 36)(27 33)(28 34)(29 63)(30 64)(31 61)(32 62)
(1 31)(2 32)(3 29)(4 30)(5 27)(6 28)(7 25)(8 26)(9 41)(10 42)(11 43)(12 44)(13 37)(14 38)(15 39)(16 40)(17 46)(18 47)(19 48)(20 45)(21 52)(22 49)(23 50)(24 51)(33 56)(34 53)(35 54)(36 55)(57 64)(58 61)(59 62)(60 63)
(1 29)(2 30)(3 31)(4 32)(5 25)(6 26)(7 27)(8 28)(9 43)(10 44)(11 41)(12 42)(13 39)(14 40)(15 37)(16 38)(17 48)(18 45)(19 46)(20 47)(21 50)(22 51)(23 52)(24 49)(33 54)(34 55)(35 56)(36 53)(57 62)(58 63)(59 64)(60 61)
(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 17 31 46)(2 42 32 10)(3 19 29 48)(4 44 30 12)(5 49 27 22)(6 13 28 37)(7 51 25 24)(8 15 26 39)(9 58 41 61)(11 60 43 63)(14 35 38 54)(16 33 40 56)(18 62 47 59)(20 64 45 57)(21 55 52 36)(23 53 50 34)
(1 51 58 14)(2 21 59 39)(3 49 60 16)(4 23 57 37)(5 41 56 17)(6 10 53 47)(7 43 54 19)(8 12 55 45)(9 33 46 27)(11 35 48 25)(13 30 50 64)(15 32 52 62)(18 28 42 34)(20 26 44 36)(22 63 40 29)(24 61 38 31)
(1 34)(2 7)(3 36)(4 5)(6 61)(8 63)(9 50)(10 38)(11 52)(12 40)(13 46)(14 42)(15 48)(16 44)(17 37)(18 51)(19 39)(20 49)(21 43)(22 45)(23 41)(24 47)(25 32)(26 60)(27 30)(28 58)(29 55)(31 53)(33 64)(35 62)(54 59)(56 57)

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

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

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

32 conjugacy classes

class 1 2A···2G2H2I2J2K4A···4N4O···4T
order12···222224···44···4
size11···144444···48···8

32 irreducible representations

dim111111111112244
type++++++++++++-
imageC1C2C2C2C2C2C2C2C2C2C2C4○D4C4○D42+ 1+42- 1+4
kernelC23.643C24C23.7Q8C23.34D4C23.8Q8C23.23D4C23.63C23C24.C22C23.10D4C23.Q8C23.4Q8C23.83C23C2×C4C23C22C22
# reps112121211134831

Matrix representation of C23.643C24 in GL6(𝔽5)

400000
040000
001000
000100
000010
000001
,
100000
010000
004000
000400
000010
000001
,
100000
010000
001000
000100
000040
000004
,
030000
200000
002000
003300
000020
000002
,
010000
100000
003000
000300
000031
000022
,
200000
020000
004300
000100
000043
000001
,
400000
010000
004300
000100
000010
000001

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

C23.643C24 in GAP, Magma, Sage, TeX

C_2^3._{643}C_2^4
% in TeX

G:=Group("C2^3.643C2^4");
// GroupNames label

G:=SmallGroup(128,1475);
// by ID

G=gap.SmallGroup(128,1475);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,672,253,232,758,723,100,1571,346]);
// Polycyclic

G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=g^2=1,d^2=c*b=b*c,e^2=b,f^2=a,a*b=b*a,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,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

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