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G = C23.583C24  order 128 = 27

300th central stem extension by C23 of C24

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

Aliases: C23.583C24, C24.390C23, C22.3572+ 1+4, C22.2662- 1+4, C2.47D42, C4⋊C4⋊12D4, (C2×D4)⋊4Q8, C2.27(D4×Q8), C23.32(C2×Q8), C23.Q8⋊54C2, (C2×C42).640C22, (C22×C4).869C23, (C23×C4).451C22, C2.13(C23⋊2Q8), C22.392(C22×D4), C22.143(C22×Q8), C23.23D4.49C2, (C22×D4).222C22, (C22×Q8).178C22, C23.78C23⋊40C2, C24.3C22.59C2, C23.65C23⋊117C2, C2.C42.292C22, C2.42(C22.31C24), C2.10(C22.57C24), (C2×C4).91(C2×D4), (C2×C4).67(C2×Q8), (C2×C22⋊Q8)⋊37C2, (C2×C4⋊C4).399C22, (C2×C22⋊C4).252C22, SmallGroup(128,1415)

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

Derived series
C1 — C23 — C23.583C24
Chief series
C1 — C2 — C22 — C23 — C22×C4 — C23×C4 — C2×C22⋊Q8 — C23.583C24
Lower central
C1 — C23 — C23.583C24
Upper central
C1 — C23 — C23.583C24
Jennings
C1 — C23 — C23.583C24

Generators and relations for C23.583C24
 G = < a,b,c,d,e,f,g | a2=b2=c2=d2=f2=1, e2=b, g2=cb=bc, ab=ba, gag-1=ac=ca, ad=da, ae=ea, faf=acd, bd=db, geg-1=be=eb, bf=fb, bg=gb, cd=dc, ce=ec, cf=fc, cg=gc, fef=de=ed, df=fd, dg=gd, gfg-1=cdf >

Subgroups: 580 in 296 conjugacy classes, 112 normal (14 characteristic)
C1, C2, C2, C2, C4, C22, C22, C22, C2×C4, C2×C4, D4, Q8, C23, C23, C23, C42, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C24, C2.C42, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C2×C4⋊C4, C22⋊Q8, C23×C4, C22×D4, C22×Q8, C23.23D4, C23.65C23, C24.3C22, C23.78C23, C23.Q8, C2×C22⋊Q8, C23.583C24
Quotients: C1, C2, C22, D4, Q8, C23, C2×D4, C2×Q8, C24, C22×D4, C22×Q8, 2+ 1+4, 2- 1+4, C22.31C24, C23⋊2Q8, D42, D4×Q8, C22.57C24, C23.583C24

Smallest permutation representation of C23.583C24
►On 64 points
Generators in S64
(5 62)(6 63)(7 64)(8 61)(9 26)(10 27)(11 28)(12 25)(13 30)(14 31)(15 32)(16 29)(17 37)(18 38)(19 39)(20 40)(21 33)(22 34)(23 35)(24 36)(45 51)(46 52)(47 49)(48 50)

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

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

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

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

32 conjugacy classes

class 1 2A···2G2H2I2J2K4A···4N4O···4T
order12···222224···44···4
size11···144444···48···8

32 irreducible representations

dim11111112244
type++++++++-+-
imageC1C2C2C2C2C2C2D4Q82+ 1+42- 1+4
kernelC23.583C24C23.23D4C23.65C23C24.3C22C23.78C23C23.Q8C2×C22⋊Q8C4⋊C4C2×D4C22C22
# reps12212448422

Matrix representation of C23.583C24 ►in GL6(𝔽5)

100000
010000
001000
000400
000010
000004
,
400000
040000
001000
000100
000010
000001
,
100000
010000
001000
000100
000040
000004
,
100000
010000
004000
000400
000010
000001
,
010000
400000
004000
000100
000040
000004
,
100000
010000
000200
003000
000004
000040
,
300000
020000
004000
000100
000004
000010

G:=sub<GL(6,GF(5))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,4],[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],[0,4,0,0,0,0,1,0,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,3,0,0,0,0,2,0,0,0,0,0,0,0,0,4,0,0,0,0,4,0],[3,0,0,0,0,0,0,2,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,4,0] >;
 

C23.583C24 in GAP, Magma, Sage, TeX 

C_2^3._{583}C_2^4
 
% in TeX
 
G:=Group("C2^3.583C2^4");
 
// GroupNames label
 
G:=SmallGroup(128,1415);
 
// by ID
 
G=gap.SmallGroup(128,1415);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,2,336,253,120,758,723,1571,346,80]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=d^2=f^2=1,e^2=b,g^2=c*b=b*c,a*b=b*a,g*a*g^-1=a*c=c*a,a*d=d*a,a*e=e*a,f*a*f=a*c*d,b*d=d*b,g*e*g^-1=b*e=e*b,b*f=f*b,b*g=g*b,c*d=d*c,c*e=e*c,c*f=f*c,c*g=g*c,f*e*f=d*e=e*d,d*f=f*d,d*g=g*d,g*f*g^-1=c*d*f>;
 
// generators/relations
 

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