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G = C23.591C24  order 128 = 27

308th central stem extension by C23 of C24

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

Aliases: C23.591C24, C24.398C23, C22.3652+ (1+4), C22.2712- (1+4), C22⋊C4.14D4, C23.66(C2×D4), C2.96(D4⋊5D4), C23.78(C4○D4), C23.7Q8⋊85C2, C23.Q8⋊56C2, C23.11D4⋊81C2, C23.23D4⋊86C2, C23.10D4⋊83C2, (C23×C4).148C22, (C22×C4).182C23, (C2×C42).645C22, C23.8Q8⋊103C2, C22.400(C22×D4), C24.3C22⋊78C2, (C22×D4).228C22, C23.81C23⋊82C2, C23.63C23⋊131C2, C2.13(C22.54C24), C2.C42.298C22, C2.83(C22.47C24), C2.62(C22.33C24), C2.43(C22.31C24), C2.71(C22.36C24), (C2×C4).419(C2×D4), (C2×C4⋊D4).45C2, (C2×C42⋊2C2)⋊18C2, (C2×C4).422(C4○D4), (C2×C4⋊C4).405C22, C22.453(C2×C4○D4), (C2×C22⋊C4).258C22, SmallGroup(128,1423)

Series: Derived ►Chief ►Lower central ►Upper central ►Jennings

Derived series
C1 — C23 — C23.591C24
Chief series
C1 — C2 — C22 — C23 — C22×C4 — C23×C4 — C23.7Q8 — C23.591C24
Lower central
C1 — C23 — C23.591C24
Upper central
C1 — C23 — C23.591C24
Jennings
C1 — C23 — C23.591C24

Subgroups: 596 in 277 conjugacy classes, 96 normal (82 characteristic)
C1, C2 [×7], C2 [×5], C4 [×15], C22 [×7], C22 [×27], C2×C4 [×6], C2×C4 [×41], D4 [×12], C23, C23 [×4], C23 [×19], C42 [×2], C22⋊C4 [×4], C22⋊C4 [×16], C4⋊C4 [×13], C22×C4 [×12], C22×C4 [×8], C2×D4 [×15], C24 [×3], C2.C42 [×8], C2×C42, C2×C22⋊C4 [×13], C2×C4⋊C4 [×8], C4⋊D4 [×4], C42⋊2C2 [×4], C23×C4 [×2], C22×D4 [×3], C23.7Q8, C23.8Q8 [×2], C23.23D4 [×2], C23.63C23, C24.3C22, C23.10D4 [×3], C23.Q8, C23.11D4, C23.81C23, C2×C4⋊D4, C2×C42⋊2C2, C23.591C24

Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], C2×D4 [×6], C4○D4 [×4], C24, C22×D4, C2×C4○D4 [×2], 2+ (1+4) [×3], 2- (1+4), C22.31C24, C22.33C24, C22.36C24, D4⋊5D4 [×2], C22.47C24, C22.54C24, C23.591C24

Generators and relations
 G = < a,b,c,d,e,f,g | a2=b2=c2=d2=f2=1, e2=ba=ab, g2=a, ac=ca, ede-1=ad=da, geg-1=ae=ea, af=fa, ag=ga, bc=cb, fdf=bd=db, be=eb, bf=fb, bg=gb, cd=dc, fef=ce=ec, cf=fc, cg=gc, gdg-1=abd, fg=gf >

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

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

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

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

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

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

Matrix representation ►G ⊆ GL6(𝔽5)

400000
040000
001000
000100
000010
000001
,
100000
010000
001000
000100
000040
000004
,
100000
010000
004000
000400
000010
000001
,
120000
040000
004000
000400
000040
000001
,
200000
330000
000100
001000
000030
000003
,
100000
010000
001000
000400
000004
000040
,
120000
440000
004000
000400
000001
000010

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

32 conjugacy classes

class 1 2A···2G2H2I2J2K2L4A···4N4O···4S
order12···2222224···44···4
size11···1444484···48···8

32 irreducible representations

dim11111111111122244
type++++++++++++++-
imageC1C2C2C2C2C2C2C2C2C2C2C2D4C4○D4C4○D42+ (1+4)2- (1+4)
kernelC23.591C24C23.7Q8C23.8Q8C23.23D4C23.63C23C24.3C22C23.10D4C23.Q8C23.11D4C23.81C23C2×C4⋊D4C2×C42⋊2C2C22⋊C4C2×C4C23C22C22
# reps11221131111144431

In GAP, Magma, Sage, TeX 

C_2^3._{591}C_2^4
 
% in TeX
 
G:=Group("C2^3.591C2^4");
 
// GroupNames label
 
G:=SmallGroup(128,1423);
 
// by ID
 
G=gap.SmallGroup(128,1423);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,2,253,344,758,723,1571,346,80]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=d^2=f^2=1,e^2=b*a=a*b,g^2=a,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=b*d=d*b,b*e=e*b,b*f=f*b,b*g=g*b,c*d=d*c,f*e*f=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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