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G = C23.327C24  order 128 = 27

44th central stem extension by C23 of C24

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

Aliases: C23.327C24, C24.260C23, C22.1402+ 1+4, C22.1012- 1+4, C4⋊C4⋊37D4, (C2×D4).209D4, C23.39(C2×D4), C4.31(C4⋊D4), C2.27(D4⋊5D4), C2.17(D4⋊6D4), C2.14(Q8⋊5D4), C23.25(C4○D4), C23.7Q8⋊43C2, C23.10D4⋊18C2, C23.11D4⋊10C2, C23.23D4⋊33C2, (C22×C4).508C23, (C23×C4).340C22, (C2×C42).474C22, C22.207(C22×D4), (C22×Q8).99C22, (C22×D4).125C22, C23.65C23⋊44C2, C23.67C23⋊33C2, C2.C42.88C22, C2.23(C23.36C23), C2.15(C22.36C24), (C2×C4×D4)⋊28C2, (C2×C22⋊Q8)⋊7C2, (C4×C22⋊C4)⋊53C2, (C2×C4.4D4)⋊5C2, (C2×C4).314(C2×D4), C2.25(C2×C4⋊D4), (C2×C4⋊D4).26C2, (C2×C4).654(C4○D4), (C2×C4⋊C4).214C22, C22.206(C2×C4○D4), (C2×C22⋊C4).117C22, SmallGroup(128,1159)

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

Derived series
C1 — C23 — C23.327C24
Chief series
C1 — C2 — C22 — C23 — C22×C4 — C23×C4 — C4×C22⋊C4 — C23.327C24
Lower central
C1 — C23 — C23.327C24
Upper central
C1 — C23 — C23.327C24
Jennings
C1 — C23 — C23.327C24

Generators and relations for C23.327C24
 G = < a,b,c,d,e,f,g | a2=b2=c2=f2=1, d2=ba=ab, e2=b, g2=a, ac=ca, ede-1=gdg-1=ad=da, ae=ea, af=fa, ag=ga, bc=cb, fdf=bd=db, be=eb, bf=fb, bg=gb, cd=dc, fef=ce=ec, cf=fc, cg=gc, eg=ge, fg=gf >

Subgroups: 660 in 330 conjugacy classes, 112 normal (42 characteristic)
C1, C2, C2, C4, C4, 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, C24, C2.C42, C2.C42, C2×C42, C2×C22⋊C4, C2×C22⋊C4, C2×C4⋊C4, C2×C4⋊C4, C4×D4, C4⋊D4, C22⋊Q8, C4.4D4, C23×C4, C23×C4, C22×D4, C22×D4, C22×Q8, C4×C22⋊C4, C23.7Q8, C23.23D4, C23.65C23, C23.67C23, C23.10D4, C23.11D4, C2×C4×D4, C2×C4⋊D4, C2×C22⋊Q8, C2×C4.4D4, C23.327C24
Quotients: C1, C2, C22, D4, C23, C2×D4, C4○D4, C24, C4⋊D4, C22×D4, C2×C4○D4, 2+ 1+4, 2- 1+4, C2×C4⋊D4, C23.36C23, C22.36C24, D4⋊5D4, D4⋊6D4, Q8⋊5D4, C23.327C24

Smallest permutation representation of C23.327C24
►On 64 points
Generators in S64
(1 15)(2 16)(3 13)(4 14)(5 43)(6 44)(7 41)(8 42)(9 36)(10 33)(11 34)(12 35)(17 60)(18 57)(19 58)(20 59)(21 56)(22 53)(23 54)(24 55)(25 52)(26 49)(27 50)(28 51)(29 40)(30 37)(31 38)(32 39)(45 62)(46 63)(47 64)(48 61)

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

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

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

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

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

38 conjugacy classes

class 1 2A···2G2H···2M4A···4H4I···4T4U4V4W4X
order12···22···24···44···44444
size11···14···42···24···48888

38 irreducible representations

dim111111111111222244
type+++++++++++++++-
imageC1C2C2C2C2C2C2C2C2C2C2C2D4D4C4○D4C4○D42+ 1+42- 1+4
kernelC23.327C24C4×C22⋊C4C23.7Q8C23.23D4C23.65C23C23.67C23C23.10D4C23.11D4C2×C4×D4C2×C4⋊D4C2×C22⋊Q8C2×C4.4D4C4⋊C4C2×D4C2×C4C23C22C22
# reps112211221111448411

Matrix representation of C23.327C24 ►in GL6(𝔽5)

400000
040000
001000
000100
000010
000001
,
100000
010000
004000
000400
000010
000001
,
100000
010000
001000
000100
000040
000004
,
300000
020000
002000
004300
000010
000001
,
030000
200000
003000
000300
000010
000004
,
400000
040000
003200
001200
000001
000010
,
010000
400000
001000
000100
000010
000001

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

C23.327C24 in GAP, Magma, Sage, TeX 

C_2^3._{327}C_2^4
 
% in TeX
 
G:=Group("C2^3.327C2^4");
 
// GroupNames label
 
G:=SmallGroup(128,1159);
 
// by ID
 
G=gap.SmallGroup(128,1159);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,2,336,253,232,758,723,675,80]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=f^2=1,d^2=b*a=a*b,e^2=b,g^2=a,a*c=c*a,e*d*e^-1=g*d*g^-1=a*d=d*a,a*e=e*a,a*f=f*a,a*g=g*a,b*c=c*b,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,e*g=g*e,f*g=g*f>;
 
// generators/relations
 

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