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G = C23.544C24  order 128 = 27

261st central stem extension by C23 of C24

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

Aliases: C24.45C23, C23.544C24, C22.2362- 1+4, C22.3192+ 1+4, C42⋊4C4⋊32C2, C4.9(C42⋊2C2), C23.Q8⋊42C2, C23.11D4⋊66C2, (C2×C42).619C22, (C22×C4).154C23, (C22×D4).201C22, C24.3C22.57C2, C23.65C23⋊106C2, C2.C42.497C22, C2.53(C22.36C24), C2.30(C22.34C24), (C2×C4).665(C4○D4), (C2×C4⋊C4).370C22, C2.23(C2×C42⋊2C2), C22.416(C2×C4○D4), (C2×C22⋊C4).231C22, SmallGroup(128,1376)

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

Derived series
C1 — C23 — C23.544C24
Chief series
C1 — C2 — C22 — C23 — C22×C4 — C22×D4 — C24.3C22 — C23.544C24
Lower central
C1 — C23 — C23.544C24
Upper central
C1 — C23 — C23.544C24
Jennings
C1 — C23 — C23.544C24

Generators and relations for C23.544C24
 G = < a,b,c,d,e,f,g | a2=b2=c2=f2=1, d2=cb=bc, e2=b, g2=a, ab=ba, ac=ca, ede-1=ad=da, ae=ea, gfg-1=af=fa, ag=ga, fdf=bd=db, be=eb, bf=fb, bg=gb, cd=dc, fef=ce=ec, cf=fc, cg=gc, dg=gd, eg=ge >

Subgroups: 436 in 210 conjugacy classes, 92 normal (9 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C2×C4, C2×C4, D4, C23, C23, C42, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C24, C2.C42, C2.C42, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C22×D4, C42⋊4C4, C23.65C23, C24.3C22, C23.Q8, C23.11D4, C23.544C24
Quotients: C1, C2, C22, C23, C4○D4, C24, C42⋊2C2, C2×C4○D4, 2+ 1+4, 2- 1+4, C2×C42⋊2C2, C22.34C24, C22.36C24, C23.544C24

Smallest permutation representation of C23.544C24
►On 64 points
Generators in S64
(1 23)(2 24)(3 21)(4 22)(5 37)(6 38)(7 39)(8 40)(9 41)(10 42)(11 43)(12 44)(13 45)(14 46)(15 47)(16 48)(17 49)(18 50)(19 51)(20 52)(25 53)(26 54)(27 55)(28 56)(29 57)(30 58)(31 59)(32 60)(33 61)(34 62)(35 63)(36 64)

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

(1 7)(2 8)(3 5)(4 6)(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)(33 49)(34 50)(35 51)(36 52)

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

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

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

32 conjugacy classes

class 1 2A···2G2H2I4A4B4C4D4E···4P4Q···4V
order12···22244444···44···4
size11···18822224···48···8

32 irreducible representations

dim111111244
type+++++++-
imageC1C2C2C2C2C2C4○D42+ 1+42- 1+4
kernelC23.544C24C42⋊4C4C23.65C23C24.3C22C23.Q8C23.11D4C2×C4C22C22
# reps1133261231

Matrix representation of C23.544C24 ►in GL8(𝔽5)

10000000
01000000
00100000
00010000
00004000
00000400
00000040
00000004
,
40000000
04000000
00100000
00010000
00004000
00000400
00000040
00000004
,
10000000
01000000
00400000
00040000
00004000
00000400
00000040
00000004
,
13000000
14000000
00300000
00030000
00003400
00003200
00000021
00000023
,
20000000
02000000
00010000
00100000
00000010
00000001
00004000
00000400
,
10000000
14000000
00100000
00040000
00001000
00001400
00000040
00000041
,
40000000
04000000
00100000
00010000
00001300
00001400
00000013
00000014

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

C23.544C24 in GAP, Magma, Sage, TeX 

C_2^3._{544}C_2^4
 
% in TeX
 
G:=Group("C2^3.544C2^4");
 
// GroupNames label
 
G:=SmallGroup(128,1376);
 
// by ID
 
G=gap.SmallGroup(128,1376);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,2,672,253,232,758,723,185,80]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=f^2=1,d^2=c*b=b*c,e^2=b,g^2=a,a*b=b*a,a*c=c*a,e*d*e^-1=a*d=d*a,a*e=e*a,g*f*g^-1=a*f=f*a,a*g=g*a,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,d*g=g*d,e*g=g*e>;
 
// generators/relations
 

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