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G = C23.620C24  order 128 = 27

337th central stem extension by C23 of C24

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

Aliases: C23.620C24, C24.416C23, C22.2952- 1+4, C22.3932+ 1+4, C22⋊C4⋊4Q8, C2.31(D4×Q8), C4⋊C4.125D4, C42⋊9C4⋊34C2, C23.35(C2×Q8), C2.71(D4⋊6D4), C2.51(D4⋊3Q8), (C2×C42).671C22, (C23×C4).472C22, (C22×C4).884C23, C22.429(C22×D4), C23.4Q8.20C2, C23.8Q8.49C2, C23.Q8.29C2, C22.147(C22×Q8), (C22×Q8).195C22, C23.78C23⋊50C2, C23.81C23⋊96C2, C23.83C23⋊87C2, C2.65(C22.29C24), C24.C22.53C2, C23.65C23⋊130C2, C23.63C23⋊145C2, C2.C42.326C22, C2.22(C22.57C24), C2.73(C22.33C24), C2.32(C23.41C23), (C2×C4).71(C2×Q8), (C2×C4).116(C2×D4), (C2×C42.C2)⋊25C2, (C2×C22⋊Q8).46C2, (C2×C4).436(C4○D4), (C2×C4⋊C4).433C22, C22.482(C2×C4○D4), (C2×C22⋊C4).284C22, SmallGroup(128,1452)

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

Derived series
C1 — C23 — C23.620C24
Chief series
C1 — C2 — C22 — C23 — C22×C4 — C2×C42 — C23.65C23 — C23.620C24
Lower central
C1 — C23 — C23.620C24
Upper central
C1 — C23 — C23.620C24
Jennings
C1 — C23 — C23.620C24

Generators and relations for C23.620C24
 G = < a,b,c,d,e,f,g | a2=b2=c2=d2=1, e2=g2=cb=bc, f2=c, ab=ba, gag-1=ac=ca, ad=da, ae=ea, faf-1=acd, bd=db, geg-1=be=eb, bf=fb, bg=gb, cd=dc, ce=ec, cf=fc, cg=gc, fef-1=de=ed, df=fd, dg=gd, gfg-1=cdf >

Subgroups: 436 in 236 conjugacy classes, 104 normal (82 characteristic)
C1, C2, C2, C4, C22, C22, C2×C4, C2×C4, Q8, C23, C23, C23, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C2×Q8, C24, C2.C42, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C22⋊Q8, C42.C2, C23×C4, C22×Q8, C42⋊9C4, C23.8Q8, C23.63C23, C24.C22, C23.65C23, C23.78C23, C23.Q8, C23.81C23, C23.4Q8, C23.83C23, C2×C22⋊Q8, C2×C42.C2, C23.620C24
Quotients: C1, C2, C22, D4, Q8, C23, C2×D4, C2×Q8, C4○D4, C24, C22×D4, C22×Q8, C2×C4○D4, 2+ 1+4, 2- 1+4, C22.29C24, C22.33C24, C23.41C23, D4⋊6D4, D4×Q8, D4⋊3Q8, C22.57C24, C23.620C24

Smallest permutation representation of C23.620C24
►On 64 points
Generators in S64
(5 22)(6 23)(7 24)(8 21)(13 47)(14 48)(15 45)(16 46)(17 34)(18 35)(19 36)(20 33)(29 57)(30 58)(31 59)(32 60)(37 44)(38 41)(39 42)(40 43)(49 61)(50 62)(51 63)(52 64)

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

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

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

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

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

32 conjugacy classes

class 1 2A···2G2H2I4A···4P4Q···4V
order12···2224···44···4
size11···1444···48···8

32 irreducible representations

dim111111111111122244
type+++++++++++++-++-
imageC1C2C2C2C2C2C2C2C2C2C2C2C2Q8D4C4○D42+ 1+42- 1+4
kernelC23.620C24C42⋊9C4C23.8Q8C23.63C23C24.C22C23.65C23C23.78C23C23.Q8C23.81C23C23.4Q8C23.83C23C2×C22⋊Q8C2×C42.C2C22⋊C4C4⋊C4C2×C4C22C22
# reps112121211111144422

Matrix representation of C23.620C24 ►in GL6(𝔽5)

100000
010000
001000
001400
000010
000034
,
400000
040000
001000
000100
000010
000001
,
100000
010000
004000
000400
000010
000001
,
100000
010000
001000
000100
000040
000004
,
200000
230000
003000
000300
000010
000034
,
100000
010000
003400
000200
000022
000013
,
420000
410000
004200
004100
000040
000021

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

C23.620C24 in GAP, Magma, Sage, TeX 

C_2^3._{620}C_2^4
 
% in TeX
 
G:=Group("C2^3.620C2^4");
 
// GroupNames label
 
G:=SmallGroup(128,1452);
 
// by ID
 
G=gap.SmallGroup(128,1452);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,2,112,253,344,758,723,184,1571,346,80]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=d^2=1,e^2=g^2=c*b=b*c,f^2=c,a*b=b*a,g*a*g^-1=a*c=c*a,a*d=d*a,a*e=e*a,f*a*f^-1=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^-1=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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