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G = C24.587C23order 128 = 27

68th non-split extension by C24 of C23 acting via C23/C22=C2

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

Aliases: C24.587C23, C23.518C24, C22.2962+ 1+4, C22.2162- 1+4, (C22×C4)⋊34D4, C23.624(C2×D4), C23.Q836C2, C23.7Q877C2, C23.239(C4○D4), C23.23D467C2, C23.10D455C2, C22.30(C4⋊D4), C2.23(C233D4), (C22×C4).128C23, (C23×C4).421C22, C22.343(C22×D4), (C22×D4).190C22, C23.81C2356C2, C2.C42.245C22, C2.33(C23.38C23), C2.36(C22.33C24), C2.23(C22.31C24), (C22×C4⋊C4)⋊29C2, (C2×C4).378(C2×D4), C2.42(C2×C4⋊D4), (C2×C4⋊D4).38C2, (C2×C4⋊C4).885C22, C22.390(C2×C4○D4), (C2×C22.D4)⋊25C2, (C2×C22⋊C4).210C22, SmallGroup(128,1350)

Series: Derived Chief Lower central Upper central Jennings

C1C23 — C24.587C23
C1C2C22C23C24C22×D4C2×C22.D4 — C24.587C23
C1C23 — C24.587C23
C1C23 — C24.587C23
C1C23 — C24.587C23

Generators and relations for C24.587C23
 G = < a,b,c,d,e,f,g | a2=b2=c2=d2=g2=1, e2=c, f2=b, gag=ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, fef-1=be=eb, bf=fb, bg=gb, cd=dc, geg=ce=ec, cf=fc, cg=gc, de=ed, gfg=df=fd, dg=gd >

Subgroups: 644 in 316 conjugacy classes, 108 normal (20 characteristic)
C1, C2 [×3], C2 [×4], C2 [×6], C4 [×16], C22 [×3], C22 [×8], C22 [×26], C2×C4 [×8], C2×C4 [×48], D4 [×12], C23, C23 [×6], C23 [×18], C22⋊C4 [×20], C4⋊C4 [×20], C22×C4 [×2], C22×C4 [×18], C22×C4 [×10], C2×D4 [×16], C24, C24 [×2], C2.C42 [×6], C2×C22⋊C4 [×12], C2×C4⋊C4, C2×C4⋊C4 [×10], C2×C4⋊C4 [×4], C4⋊D4 [×4], C22.D4 [×8], C23×C4 [×3], C22×D4, C22×D4 [×2], C23.7Q8, C23.23D4 [×2], C23.10D4 [×2], C23.Q8 [×4], C23.81C23 [×2], C22×C4⋊C4, C2×C4⋊D4, C2×C22.D4 [×2], C24.587C23
Quotients: C1, C2 [×15], C22 [×35], D4 [×8], C23 [×15], C2×D4 [×12], C4○D4 [×2], C24, C4⋊D4 [×4], C22×D4 [×2], C2×C4○D4, 2+ 1+4 [×2], 2- 1+4 [×2], C2×C4⋊D4, C233D4, C23.38C23, C22.31C24 [×2], C22.33C24 [×2], C24.587C23

Smallest permutation representation of C24.587C23
On 64 points
Generators in S64
(1 11)(2 12)(3 9)(4 10)(5 45)(6 46)(7 47)(8 48)(13 17)(14 18)(15 19)(16 20)(21 25)(22 26)(23 27)(24 28)(29 36)(30 33)(31 34)(32 35)(37 41)(38 42)(39 43)(40 44)(49 53)(50 54)(51 55)(52 56)(57 61)(58 62)(59 63)(60 64)
(1 39)(2 40)(3 37)(4 38)(5 17)(6 18)(7 19)(8 20)(9 41)(10 42)(11 43)(12 44)(13 45)(14 46)(15 47)(16 48)(21 49)(22 50)(23 51)(24 52)(25 53)(26 54)(27 55)(28 56)(29 57)(30 58)(31 59)(32 60)(33 62)(34 63)(35 64)(36 61)
(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 51)(2 52)(3 49)(4 50)(5 36)(6 33)(7 34)(8 35)(9 53)(10 54)(11 55)(12 56)(13 57)(14 58)(15 59)(16 60)(17 61)(18 62)(19 63)(20 64)(21 37)(22 38)(23 39)(24 40)(25 41)(26 42)(27 43)(28 44)(29 45)(30 46)(31 47)(32 48)
(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 47 39 15)(2 16 40 48)(3 45 37 13)(4 14 38 46)(5 41 17 9)(6 10 18 42)(7 43 19 11)(8 12 20 44)(21 57 49 29)(22 30 50 58)(23 59 51 31)(24 32 52 60)(25 61 53 36)(26 33 54 62)(27 63 55 34)(28 35 56 64)
(2 4)(5 61)(6 64)(7 63)(8 62)(9 41)(10 44)(11 43)(12 42)(13 57)(14 60)(15 59)(16 58)(17 36)(18 35)(19 34)(20 33)(22 24)(25 53)(26 56)(27 55)(28 54)(29 45)(30 48)(31 47)(32 46)(38 40)(50 52)

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

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

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

32 conjugacy classes

class 1 2A···2G2H2I2J2K2L2M4A···4L4M···4R
order12···22222224···44···4
size11···12222884···48···8

32 irreducible representations

dim1111111112244
type+++++++++++-
imageC1C2C2C2C2C2C2C2C2D4C4○D42+ 1+42- 1+4
kernelC24.587C23C23.7Q8C23.23D4C23.10D4C23.Q8C23.81C23C22×C4⋊C4C2×C4⋊D4C2×C22.D4C22×C4C23C22C22
# reps1122421128422

Matrix representation of C24.587C23 in GL8(𝔽5)

40000000
04000000
00100000
00010000
00004300
00000100
00000401
00000110
,
10000000
01000000
00100000
00010000
00004000
00000400
00000040
00000004
,
40000000
04000000
00100000
00010000
00004000
00000400
00000040
00000004
,
10000000
01000000
00400000
00040000
00004000
00000400
00000040
00000004
,
34000000
02000000
00400000
00040000
00002010
00000022
00000030
00000220
,
10000000
01000000
00240000
00330000
00001030
00000011
00001040
00004410
,
10000000
14000000
00100000
00440000
00001000
00004400
00001040
00000001

G:=sub<GL(8,GF(5))| [4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,3,1,4,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4],[4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4],[3,0,0,0,0,0,0,0,4,2,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,1,2,3,2,0,0,0,0,0,2,0,0],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,2,3,0,0,0,0,0,0,4,3,0,0,0,0,0,0,0,0,1,0,1,4,0,0,0,0,0,0,0,4,0,0,0,0,3,1,4,1,0,0,0,0,0,1,0,0],[1,1,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1,4,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1,4,1,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1] >;

C24.587C23 in GAP, Magma, Sage, TeX

C_2^4._{587}C_2^3
% in TeX

G:=Group("C2^4.587C2^3");
// GroupNames label

G:=SmallGroup(128,1350);
// by ID

G=gap.SmallGroup(128,1350);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,224,253,120,758,723,185]);
// Polycyclic

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

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