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G = C24.589C23order 128 = 27

70th non-split extension by C24 of C23 acting via C23/C22=C2

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

Aliases: C24.589C23, C23.523C24, C22.3002+ 1+4, C22.2192- 1+4, (C22×C4).401D4, C23.625(C2×D4), C23.8Q883C2, C23.4Q826C2, C23.241(C4○D4), C23.11D457C2, C23.34D442C2, (C23×C4).425C22, (C22×C4).133C23, C22.348(C22×D4), C23.23D4.45C2, (C22×D4).194C22, C23.81C2358C2, C23.83C2357C2, C2.36(C22.29C24), C2.C42.249C22, C22.38(C22.D4), C2.37(C22.33C24), C2.36(C23.38C23), (C22×C4⋊C4)⋊30C2, (C2×C4).382(C2×D4), (C2×C4⋊C4).889C22, C22.395(C2×C4○D4), C2.41(C2×C22.D4), (C2×C22⋊C4).214C22, (C2×C22.D4).20C2, SmallGroup(128,1355)

Series: Derived Chief Lower central Upper central Jennings

C1C23 — C24.589C23
C1C2C22C23C24C22×D4C23.23D4 — C24.589C23
C1C23 — C24.589C23
C1C23 — C24.589C23
C1C23 — C24.589C23

Generators and relations for C24.589C23
 G = < a,b,c,d,e,f,g | a2=b2=c2=d2=g2=1, e2=b, f2=c, 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: 516 in 263 conjugacy classes, 100 normal (24 characteristic)
C1, C2 [×3], C2 [×4], C2 [×5], C4 [×15], C22, C22 [×10], C22 [×19], C2×C4 [×4], C2×C4 [×53], D4 [×4], C23, C23 [×6], C23 [×11], C22⋊C4 [×14], C4⋊C4 [×16], C22×C4 [×5], C22×C4 [×12], C22×C4 [×12], C2×D4 [×6], C24 [×2], C2.C42 [×12], C2×C22⋊C4 [×3], C2×C22⋊C4 [×6], C2×C4⋊C4 [×2], C2×C4⋊C4 [×8], C2×C4⋊C4 [×4], C22.D4 [×4], C23×C4, C23×C4 [×2], C22×D4, C23.34D4, C23.8Q8 [×2], C23.23D4 [×2], C23.11D4 [×2], C23.81C23 [×2], C23.4Q8 [×2], C23.83C23 [×2], C22×C4⋊C4, C2×C22.D4, C24.589C23
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], C2×D4 [×6], C4○D4 [×4], C24, C22.D4 [×4], C22×D4, C2×C4○D4 [×2], 2+ 1+4 [×2], 2- 1+4 [×2], C2×C22.D4, C22.29C24, C23.38C23, C22.33C24 [×4], C24.589C23

Smallest permutation representation of C24.589C23
On 64 points
Generators in S64
(1 41)(2 42)(3 43)(4 44)(5 64)(6 61)(7 62)(8 63)(9 13)(10 14)(11 15)(12 16)(17 21)(18 22)(19 23)(20 24)(25 29)(26 30)(27 31)(28 32)(33 37)(34 38)(35 39)(36 40)(45 49)(46 50)(47 51)(48 52)(53 57)(54 58)(55 59)(56 60)
(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 9)(2 10)(3 11)(4 12)(5 39)(6 40)(7 37)(8 38)(13 41)(14 42)(15 43)(16 44)(17 45)(18 46)(19 47)(20 48)(21 49)(22 50)(23 51)(24 52)(25 55)(26 56)(27 53)(28 54)(29 59)(30 60)(31 57)(32 58)(33 62)(34 63)(35 64)(36 61)
(1 53)(2 54)(3 55)(4 56)(5 52)(6 49)(7 50)(8 51)(9 27)(10 28)(11 25)(12 26)(13 31)(14 32)(15 29)(16 30)(17 36)(18 33)(19 34)(20 35)(21 40)(22 37)(23 38)(24 39)(41 57)(42 58)(43 59)(44 60)(45 61)(46 62)(47 63)(48 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 17 9 45)(2 20 10 48)(3 19 11 47)(4 18 12 46)(5 58 39 32)(6 57 40 31)(7 60 37 30)(8 59 38 29)(13 49 41 21)(14 52 42 24)(15 51 43 23)(16 50 44 22)(25 63 55 34)(26 62 56 33)(27 61 53 36)(28 64 54 35)
(2 10)(4 12)(5 22)(6 51)(7 24)(8 49)(13 15)(14 44)(16 42)(17 36)(18 62)(19 34)(20 64)(21 38)(23 40)(26 56)(28 54)(29 31)(30 58)(32 60)(33 46)(35 48)(37 52)(39 50)(41 43)(45 61)(47 63)(57 59)

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

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

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

32 conjugacy classes

class 1 2A···2G2H2I2J2K2L4A···4L4M···4S
order12···2222224···44···4
size11···1222284···48···8

32 irreducible representations

dim11111111112244
type++++++++++++-
imageC1C2C2C2C2C2C2C2C2C2D4C4○D42+ 1+42- 1+4
kernelC24.589C23C23.34D4C23.8Q8C23.23D4C23.11D4C23.81C23C23.4Q8C23.83C23C22×C4⋊C4C2×C22.D4C22×C4C23C22C22
# reps11222222114822

Matrix representation of C24.589C23 in GL8(𝔽5)

40000000
04000000
00400000
00040000
00004300
00000100
00000301
00000210
,
10000000
01000000
00100000
00010000
00004000
00000400
00000040
00000004
,
40000000
04000000
00100000
00010000
00004000
00000400
00000040
00000004
,
40000000
04000000
00400000
00040000
00004000
00000400
00000040
00000004
,
21000000
23000000
00100000
00010000
00001040
00000033
00002040
00003310
,
13000000
14000000
00010000
00100000
00002030
00000011
00000030
00000420
,
10000000
14000000
00100000
00040000
00001000
00004400
00002040
00000001

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

C24.589C23 in GAP, Magma, Sage, TeX

C_2^4._{589}C_2^3
% in TeX

G:=Group("C2^4.589C2^3");
// GroupNames label

G:=SmallGroup(128,1355);
// by ID

G=gap.SmallGroup(128,1355);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,112,253,232,758,723,185]);
// Polycyclic

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