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G = C23.663C24  order 128 = 27

380th central stem extension by C23 of C24

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

Aliases: C23.663C24, C24.441C23, C22.4362+ 1+4, C22.3292- 1+4, C22⋊C4⋊7Q8, C42⋊8C4⋊62C2, C23.40(C2×Q8), C2.57(D4⋊3Q8), C23⋊Q8.26C2, (C22×C4).208C23, (C2×C42).695C22, (C23×C4).168C22, C23.8Q8.58C2, C22.155(C22×Q8), (C22×Q8).214C22, C23.78C23⋊57C2, C23.67C23⋊99C2, C2.10(C24⋊C22), C24.C22.65C2, C23.83C23⋊105C2, C2.C42.367C22, C2.115(C22.45C24), C2.38(C23.41C23), C2.104(C22.36C24), (C2×C4).79(C2×Q8), (C2×C4).457(C4○D4), (C2×C4⋊C4).473C22, C22.524(C2×C4○D4), (C2×C22⋊C4).310C22, SmallGroup(128,1495)

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

Derived series
C1 — C23 — C23.663C24
Chief series
C1 — C2 — C22 — C23 — C22×C4 — C2×C42 — C23.67C23 — C23.663C24
Lower central
C1 — C23 — C23.663C24
Upper central
C1 — C23 — C23.663C24
Jennings
C1 — C23 — C23.663C24

Generators and relations for C23.663C24
 G = < a,b,c,d,e,f,g | a2=b2=c2=d2=1, e2=b, f2=c, g2=cb=bc, ab=ba, gag-1=ac=ca, eae-1=ad=da, faf-1=acd, bd=db, be=eb, bf=fb, bg=gb, cd=dc, ce=ec, cf=fc, cg=gc, fef-1=de=ed, df=fd, dg=gd, geg-1=bde, gfg-1=cdf >

Subgroups: 420 in 212 conjugacy classes, 96 normal (22 characteristic)
C1, C2, C2, C2, C4, C22, C22, C22, C2×C4, C2×C4, Q8, C23, C23, C23, C42, C22⋊C4, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C22×C4, C2×Q8, C24, C2.C42, C2.C42, C2×C42, C2×C22⋊C4, C2×C22⋊C4, C2×C4⋊C4, C23×C4, C22×Q8, C42⋊8C4, C23.8Q8, C23.8Q8, C24.C22, C23.67C23, C23⋊Q8, C23.78C23, C23.83C23, C23.663C24
Quotients: C1, C2, C22, Q8, C23, C2×Q8, C4○D4, C24, C22×Q8, C2×C4○D4, 2+ 1+4, 2- 1+4, C22.36C24, C23.41C23, C22.45C24, D4⋊3Q8, C24⋊C22, C23.663C24

Smallest permutation representation of C23.663C24
►On 64 points
Generators in S64
(2 9)(4 11)(5 21)(6 63)(7 23)(8 61)(13 43)(15 41)(17 62)(18 22)(19 64)(20 24)(25 31)(26 47)(27 29)(28 45)(30 49)(32 51)(33 55)(35 53)(37 57)(39 59)(46 50)(48 52)

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

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

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

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

32 conjugacy classes

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

32 irreducible representations

dim111111112244
type++++++++-+-
imageC1C2C2C2C2C2C2C2Q8C4○D42+ 1+42- 1+4
kernelC23.663C24C42⋊8C4C23.8Q8C24.C22C23.67C23C23⋊Q8C23.78C23C23.83C23C22⋊C4C2×C4C22C22
# reps123222224831

Matrix representation of C23.663C24 ►in GL6(𝔽5)

100000
040000
001000
000100
000010
000004
,
100000
010000
004000
000400
000010
000001
,
400000
040000
001000
000100
000010
000001
,
100000
010000
001000
000100
000040
000004
,
400000
040000
000300
003000
000003
000020
,
030000
300000
004000
000400
000001
000010
,
010000
400000
000100
004000
000010
000004

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

C23.663C24 in GAP, Magma, Sage, TeX 

C_2^3._{663}C_2^4
 
% in TeX
 
G:=Group("C2^3.663C2^4");
 
// GroupNames label
 
G:=SmallGroup(128,1495);
 
// by ID
 
G=gap.SmallGroup(128,1495);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,2,784,253,120,758,723,184,1571,346,80]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=d^2=1,e^2=b,f^2=c,g^2=c*b=b*c,a*b=b*a,g*a*g^-1=a*c=c*a,e*a*e^-1=a*d=d*a,f*a*f^-1=a*c*d,b*d=d*b,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*e*g^-1=b*d*e,g*f*g^-1=c*d*f>;
 
// generators/relations
 

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