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G = C23.685C24order 128 = 27

402nd central stem extension by C23 of C24

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

Aliases: C24.91C23, C23.685C24, C22.4582+ 1+4, C22.3482- 1+4, C428C471C2, C23⋊Q857C2, (C2×C42).712C22, (C22×C4).598C23, C23.10D4.64C2, (C22×D4).281C22, (C22×Q8).219C22, C24.C22169C2, C23.78C2360C2, C2.12(C24⋊C22), C2.C42.389C22, C2.46(C22.49C24), C2.104(C22.33C24), (C2×C4).471(C4○D4), (C2×C4⋊C4).495C22, C22.546(C2×C4○D4), (C2×C22⋊C4).321C22, SmallGroup(128,1517)

Series: Derived Chief Lower central Upper central Jennings

C1C23 — C23.685C24
C1C2C22C23C22×C4C2×C42C24.C22 — C23.685C24
C1C23 — C23.685C24
C1C23 — C23.685C24
C1C23 — C23.685C24

Generators and relations for C23.685C24
 G = < a,b,c,d,e,f,g | a2=b2=c2=1, d2=c, e2=b, f2=a, g2=ba=ab, ac=ca, 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: 452 in 214 conjugacy classes, 88 normal (9 characteristic)
C1, C2, C2 [×6], C2 [×2], C4 [×16], C22, C22 [×6], C22 [×14], C2×C4 [×6], C2×C4 [×36], D4 [×4], Q8 [×4], C23, C23 [×14], C42 [×6], C22⋊C4 [×18], C4⋊C4 [×6], C22×C4, C22×C4 [×12], C2×D4 [×3], C2×Q8 [×3], C24 [×2], C2.C42 [×12], C2×C42 [×3], C2×C22⋊C4 [×12], C2×C4⋊C4 [×6], C22×D4, C22×Q8, C428C4 [×3], C24.C22 [×6], C23⋊Q8 [×2], C23.10D4 [×3], C23.78C23, C23.685C24
Quotients: C1, C2 [×15], C22 [×35], C23 [×15], C4○D4 [×6], C24, C2×C4○D4 [×3], 2+ 1+4 [×3], 2- 1+4, C22.33C24 [×3], C22.49C24 [×3], C24⋊C22, C23.685C24

Smallest permutation representation of C23.685C24
On 64 points
Generators in S64
(1 21)(2 22)(3 23)(4 24)(5 46)(6 47)(7 48)(8 45)(9 56)(10 53)(11 54)(12 55)(13 60)(14 57)(15 58)(16 59)(17 35)(18 36)(19 33)(20 34)(25 41)(26 42)(27 43)(28 44)(29 40)(30 37)(31 38)(32 39)(49 64)(50 61)(51 62)(52 63)
(1 57)(2 58)(3 59)(4 60)(5 42)(6 43)(7 44)(8 41)(9 37)(10 38)(11 39)(12 40)(13 24)(14 21)(15 22)(16 23)(17 51)(18 52)(19 49)(20 50)(25 45)(26 46)(27 47)(28 48)(29 55)(30 56)(31 53)(32 54)(33 64)(34 61)(35 62)(36 63)
(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 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 57 26)(2 6 58 43)(3 48 59 28)(4 8 60 41)(5 14 42 21)(7 16 44 23)(9 18 37 52)(10 33 38 64)(11 20 39 50)(12 35 40 62)(13 25 24 45)(15 27 22 47)(17 29 51 55)(19 31 49 53)(30 63 56 36)(32 61 54 34)
(1 47 21 6)(2 28 22 44)(3 45 23 8)(4 26 24 42)(5 60 46 13)(7 58 48 15)(9 62 56 51)(10 36 53 18)(11 64 54 49)(12 34 55 20)(14 43 57 27)(16 41 59 25)(17 37 35 30)(19 39 33 32)(29 50 40 61)(31 52 38 63)
(1 20 14 61)(2 62 15 17)(3 18 16 63)(4 64 13 19)(5 39 26 54)(6 55 27 40)(7 37 28 56)(8 53 25 38)(9 48 30 44)(10 41 31 45)(11 46 32 42)(12 43 29 47)(21 34 57 50)(22 51 58 35)(23 36 59 52)(24 49 60 33)

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

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

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

32 conjugacy classes

class 1 2A···2G2H2I4A···4R4S4T4U4V
order12···2224···44444
size11···1884···48888

32 irreducible representations

dim111111244
type+++++++-
imageC1C2C2C2C2C2C4○D42+ 1+42- 1+4
kernelC23.685C24C428C4C24.C22C23⋊Q8C23.10D4C23.78C23C2×C4C22C22
# reps1362311231

Matrix representation of C23.685C24 in GL6(𝔽5)

400000
040000
001000
000100
000010
000001
,
100000
010000
001000
000100
000040
000004
,
100000
010000
004000
000400
000010
000001
,
210000
230000
003000
000300
000001
000010
,
130000
040000
002400
003300
000030
000003
,
200000
020000
004300
000100
000002
000030
,
200000
230000
004000
000400
000001
000040

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],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[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],[2,2,0,0,0,0,1,3,0,0,0,0,0,0,3,0,0,0,0,0,0,3,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[1,0,0,0,0,0,3,4,0,0,0,0,0,0,2,3,0,0,0,0,4,3,0,0,0,0,0,0,3,0,0,0,0,0,0,3],[2,0,0,0,0,0,0,2,0,0,0,0,0,0,4,0,0,0,0,0,3,1,0,0,0,0,0,0,0,3,0,0,0,0,2,0],[2,2,0,0,0,0,0,3,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,0,4,0,0,0,0,1,0] >;

C23.685C24 in GAP, Magma, Sage, TeX

C_2^3._{685}C_2^4
% in TeX

G:=Group("C2^3.685C2^4");
// GroupNames label

G:=SmallGroup(128,1517);
// by ID

G=gap.SmallGroup(128,1517);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,448,253,232,758,723,100,1571,346,192]);
// Polycyclic

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