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G = C24.589C23  order 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, C22.2192- (1+4), C22.3002+ (1+4), (C22×C4).401D4, C23.625(C2×D4), C23.4Q8⋊26C2, C23.8Q8⋊83C2, C23.241(C4○D4), C23.11D4⋊57C2, C23.34D4⋊42C2, (C22×C4).133C23, (C23×C4).425C22, C22.348(C22×D4), C23.23D4.45C2, (C22×D4).194C22, C23.81C23⋊58C2, C23.83C23⋊57C2, 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

Derived series
C1 — C23 — C24.589C23
Chief series
C1 — C2 — C22 — C23 — C24 — C22×D4 — C23.23D4 — C24.589C23
Lower central
C1 — C23 — C24.589C23
Upper central
C1 — C23 — C24.589C23
Jennings
C1 — C23 — C24.589C23

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

Generators and relations
 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 >

Smallest permutation representation
►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)])
 

Matrix representation ►G ⊆ 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] >;
 

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+4)2- (1+4)
kernelC24.589C23C23.34D4C23.8Q8C23.23D4C23.11D4C23.81C23C23.4Q8C23.83C23C22×C4⋊C4C2×C22.D4C22×C4C23C22C22
# reps11222222114822

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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