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G = C23.668C24order 128 = 27

385th central stem extension by C23 of C24

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

Aliases: C23.668C24, C24.444C23, C22.4412+ 1+4, C22.3342- 1+4, C22⋊C4.2Q8, C428C464C2, C23.41(C2×Q8), C2.58(D43Q8), (C22×C4).209C23, (C23×C4).490C22, (C2×C42).700C22, C23.7Q8.73C2, C23.4Q8.25C2, C23.Q8.36C2, C23.8Q8.61C2, C22.156(C22×Q8), C2.93(C22.32C24), C24.C22.68C2, C23.65C23144C2, C23.63C23173C2, C23.81C23118C2, C2.C42.372C22, C2.39(C23.41C23), C2.58(C22.34C24), C2.36(C22.56C24), C2.106(C22.46C24), (C2×C4).80(C2×Q8), (C2×C4).462(C4○D4), (C2×C4⋊C4).478C22, C22.529(C2×C4○D4), (C2×C22⋊C4).312C22, SmallGroup(128,1500)

Series: Derived Chief Lower central Upper central Jennings

C1C23 — C23.668C24
C1C2C22C23C22×C4C2×C42C23.65C23 — C23.668C24
C1C23 — C23.668C24
C1C23 — C23.668C24
C1C23 — C23.668C24

Generators and relations for C23.668C24
 G = < a,b,c,d,e,f,g | a2=b2=c2=1, d2=ca=ac, e2=ba=ab, f2=b, g2=a, ede-1=ad=da, geg-1=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-1=abd, fg=gf >

Subgroups: 388 in 204 conjugacy classes, 96 normal (82 characteristic)
C1, C2 [×7], C2 [×2], C4 [×18], C22 [×7], C22 [×10], C2×C4 [×8], C2×C4 [×42], C23, C23 [×2], C23 [×6], C42 [×3], C22⋊C4 [×4], C22⋊C4 [×6], C4⋊C4 [×17], C22×C4 [×14], C22×C4 [×4], C24, C2.C42 [×12], C2×C42 [×2], C2×C22⋊C4 [×6], C2×C4⋊C4 [×14], C23×C4, C23.7Q8, C428C4, C23.8Q8 [×2], C23.63C23, C24.C22 [×2], C23.65C23 [×2], C23.Q8, C23.81C23 [×4], C23.4Q8, C23.668C24
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, C22.34C24, C23.41C23, C22.46C24, D43Q8 [×2], C22.56C24, C23.668C24

Smallest permutation representation of C23.668C24
On 64 points
Generators in S64
(1 25)(2 26)(3 27)(4 28)(5 60)(6 57)(7 58)(8 59)(9 47)(10 48)(11 45)(12 46)(13 51)(14 52)(15 49)(16 50)(17 44)(18 41)(19 42)(20 43)(21 37)(22 38)(23 39)(24 40)(29 34)(30 35)(31 36)(32 33)(53 63)(54 64)(55 61)(56 62)
(1 58)(2 59)(3 60)(4 57)(5 27)(6 28)(7 25)(8 26)(9 18)(10 19)(11 20)(12 17)(13 24)(14 21)(15 22)(16 23)(29 54)(30 55)(31 56)(32 53)(33 63)(34 64)(35 61)(36 62)(37 52)(38 49)(39 50)(40 51)(41 47)(42 48)(43 45)(44 46)
(1 27)(2 28)(3 25)(4 26)(5 58)(6 59)(7 60)(8 57)(9 45)(10 46)(11 47)(12 48)(13 49)(14 50)(15 51)(16 52)(17 42)(18 43)(19 44)(20 41)(21 39)(22 40)(23 37)(24 38)(29 36)(30 33)(31 34)(32 35)(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 46 7 17)(2 9 8 41)(3 48 5 19)(4 11 6 43)(10 60 42 27)(12 58 44 25)(13 30 40 61)(14 36 37 56)(15 32 38 63)(16 34 39 54)(18 26 47 59)(20 28 45 57)(21 62 52 31)(22 53 49 33)(23 64 50 29)(24 55 51 35)
(1 50 58 39)(2 40 59 51)(3 52 60 37)(4 38 57 49)(5 21 27 14)(6 15 28 22)(7 23 25 16)(8 13 26 24)(9 53 18 32)(10 29 19 54)(11 55 20 30)(12 31 17 56)(33 47 63 41)(34 42 64 48)(35 45 61 43)(36 44 62 46)
(1 63 25 53)(2 29 26 34)(3 61 27 55)(4 31 28 36)(5 30 60 35)(6 62 57 56)(7 32 58 33)(8 64 59 54)(9 39 47 23)(10 13 48 51)(11 37 45 21)(12 15 46 49)(14 20 52 43)(16 18 50 41)(17 22 44 38)(19 24 42 40)

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

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

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

32 conjugacy classes

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

32 irreducible representations

dim11111111112244
type++++++++++-+-
imageC1C2C2C2C2C2C2C2C2C2Q8C4○D42+ 1+42- 1+4
kernelC23.668C24C23.7Q8C428C4C23.8Q8C23.63C23C24.C22C23.65C23C23.Q8C23.81C23C23.4Q8C22⋊C4C2×C4C22C22
# reps11121221414831

Matrix representation of C23.668C24 in GL6(𝔽5)

100000
010000
001000
000100
000040
000004
,
400000
040000
001000
000100
000010
000001
,
100000
010000
004000
000400
000010
000001
,
100000
040000
002000
000200
000002
000020
,
300000
030000
000200
003000
000020
000003
,
020000
200000
000100
001000
000010
000001
,
010000
100000
001000
000100
000001
000040

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

C23.668C24 in GAP, Magma, Sage, TeX

C_2^3._{668}C_2^4
% in TeX

G:=Group("C2^3.668C2^4");
// GroupNames label

G:=SmallGroup(128,1500);
// by ID

G=gap.SmallGroup(128,1500);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,560,253,344,758,723,184,1571,346,80]);
// Polycyclic

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

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