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G = C23.259C24  order 128 = 27

112nd central extension by C23 of C24

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

Aliases: C23.259C24, C24.226C23, C22.652- 1+4, C22.902+ 1+4, C4⋊D4⋊22C4, C23.28(C22×C4), C23.8Q8⋊20C2, C23.7Q8⋊29C2, C23.34D4⋊16C2, C23.23D4⋊18C2, (C2×C42).447C22, C22.150(C23×C4), (C23×C4).314C22, (C22×C4).487C23, C24.C22⋊27C2, C24.3C22⋊24C2, (C22×D4).114C22, C23.63C23⋊25C2, C2.3(C22.54C24), C2.39(C22.11C24), C2.C42.67C22, C2.2(C22.56C24), C2.39(C23.33C23), C4⋊C4⋊20(C2×C4), (C2×D4)⋊22(C2×C4), C22⋊C4⋊21(C2×C4), (C22×C4)⋊39(C2×C4), (C2×C4⋊D4).21C2, (C2×C4).56(C22×C4), (C2×C4⋊C4).195C22, (C2×C22⋊C4).41C22, SmallGroup(128,1109)

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

Derived series
C1 — C22 — C23.259C24
Chief series
C1 — C2 — C22 — C23 — C22×C4 — C23×C4 — C2×C4⋊D4 — C23.259C24
Lower central
C1 — C22 — C23.259C24
Upper central
C1 — C23 — C23.259C24
Jennings
C1 — C23 — C23.259C24

Generators and relations for C23.259C24
 G = < a,b,c,d,e,f,g | a2=b2=c2=d2=f2=1, e2=d, g2=b, eae-1=gag-1=ab=ba, ac=ca, ad=da, faf=abc, bc=cb, bd=db, be=eb, bf=fb, bg=gb, cd=dc, fef=ce=ec, cf=fc, cg=gc, de=ed, df=fd, dg=gd, geg-1=bce, fg=gf >

Subgroups: 604 in 288 conjugacy classes, 132 normal (22 characteristic)
C1, C2, C2, C2, C4, C22, C22, C22, C2×C4, C2×C4, D4, C23, C23, C23, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C22×C4, C2×D4, C2×D4, C24, C24, C2.C42, C2.C42, C2×C42, C2×C22⋊C4, C2×C22⋊C4, C2×C4⋊C4, C2×C4⋊C4, C4⋊D4, C23×C4, C23×C4, C22×D4, C22×D4, C23.7Q8, C23.34D4, C23.8Q8, C23.23D4, C23.63C23, C24.C22, C24.3C22, C2×C4⋊D4, C23.259C24
Quotients: C1, C2, C4, C22, C2×C4, C23, C22×C4, C24, C23×C4, 2+ 1+4, 2- 1+4, C22.11C24, C23.33C23, C22.54C24, C22.56C24, C23.259C24

Smallest permutation representation of C23.259C24
►On 64 points
Generators in S64
(1 43)(2 16)(3 41)(4 14)(5 30)(6 59)(7 32)(8 57)(9 13)(10 42)(11 15)(12 44)(17 61)(18 36)(19 63)(20 34)(21 25)(22 54)(23 27)(24 56)(26 50)(28 52)(29 37)(31 39)(33 47)(35 45)(38 58)(40 60)(46 62)(48 64)(49 53)(51 55)

(1 11)(2 12)(3 9)(4 10)(5 38)(6 39)(7 40)(8 37)(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 53)(26 54)(27 55)(28 56)(29 57)(30 58)(31 59)(32 60)(33 63)(34 64)(35 61)(36 62)

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

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

38 conjugacy classes

class 1 2A···2G2H···2M4A···4X
order12···22···24···4
size11···14···44···4

38 irreducible representations

dim111111111144
type++++++++++-
imageC1C2C2C2C2C2C2C2C2C42+ 1+42- 1+4
kernelC23.259C24C23.7Q8C23.34D4C23.8Q8C23.23D4C23.63C23C24.C22C24.3C22C2×C4⋊D4C4⋊D4C22C22
# reps1112422211651

Matrix representation of C23.259C24 ►in GL9(𝔽5)

400000000
010000000
014000000
000100000
040140000
000000100
000001000
000000242
000004101
,
100000000
010000000
001000000
000100000
000010000
000004000
000000400
000000040
000000004
,
100000000
040000000
004000000
000400000
000040000
000001000
000000100
000000010
000000001
,
400000000
010000000
001000000
000100000
000010000
000004000
000000400
000000040
000000004
,
200000000
040400000
000410000
000100000
001100000
000000321
000003020
000000021
000000003
,
100000000
013000000
004000000
032130000
032040000
000004010
000000413
000000010
000000001
,
100000000
010000000
001000000
030400000
030040000
000000100
000004000
000000013
000000014

G:=sub<GL(9,GF(5))| [4,0,0,0,0,0,0,0,0,0,1,1,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,1,0,4,0,0,0,0,0,1,0,2,1,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,2,1],[1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,4],[1,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1],[4,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,4],[2,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,4,4,1,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,2,2,2,0,0,0,0,0,0,1,0,1,3],[1,0,0,0,0,0,0,0,0,0,1,0,3,3,0,0,0,0,0,3,4,2,2,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,3,4,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,1,1,1,0,0,0,0,0,0,0,3,0,1],[1,0,0,0,0,0,0,0,0,0,1,0,3,3,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,3,4] >;
 

C23.259C24 in GAP, Magma, Sage, TeX 

C_2^3._{259}C_2^4
 
% in TeX
 
G:=Group("C2^3.259C2^4");
 
// GroupNames label
 
G:=SmallGroup(128,1109);
 
// by ID
 
G=gap.SmallGroup(128,1109);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,2,448,253,758,555,1571,346,80]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=d^2=f^2=1,e^2=d,g^2=b,e*a*e^-1=g*a*g^-1=a*b=b*a,a*c=c*a,a*d=d*a,f*a*f=a*b*c,b*c=c*b,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,d*e=e*d,d*f=f*d,d*g=g*d,g*e*g^-1=b*c*e,f*g=g*f>;
 
// generators/relations
 

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