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G = C23.322C24  order 128 = 27

39th central stem extension by C23 of C24

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

Aliases: C23.322C24, C24.256C23, C22.972- 1+4, C22.1352+ 1+4, C2.11D42, C22⋊C4⋊21D4, (C2×D4).283D4, C23.37(C2×D4), C42⋊9C4⋊15C2, C2.15(D4⋊6D4), C23.11D4⋊8C2, C4⋊1(C22.D4), (C23×C4).74C22, C23.8Q8⋊32C2, C23.7Q8⋊41C2, C23.10D4⋊14C2, (C2×C42).470C22, (C22×C4).505C23, C22.202(C22×D4), C24.3C22⋊31C2, (C22×D4).123C22, C23.65C23⋊40C2, C2.C42.85C22, C2.12(C22.26C24), C2.15(C22.47C24), C2.12(C22.36C24), (C2×C4×D4)⋊25C2, (C2×C4).47(C2×D4), (C4×C22⋊C4)⋊51C2, (C2×C4).96(C4○D4), (C2×C4⋊D4).23C2, (C2×C4⋊C4).210C22, C22.201(C2×C4○D4), (C2×C22.D4)⋊7C2, C2.19(C2×C22.D4), (C2×C22⋊C4).114C22, SmallGroup(128,1154)

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

Derived series
C1 — C23 — C23.322C24
Chief series
C1 — C2 — C22 — C23 — C22×C4 — C23×C4 — C4×C22⋊C4 — C23.322C24
Lower central
C1 — C23 — C23.322C24
Upper central
C1 — C23 — C23.322C24
Jennings
C1 — C23 — C23.322C24

Generators and relations for C23.322C24
 G = < a,b,c,d,e,f,g | a2=b2=c2=d2=f2=1, e2=cb=bc, g2=b, eae-1=gag-1=ab=ba, faf=ac=ca, ad=da, bd=db, be=eb, bf=fb, bg=gb, cd=dc, ce=ec, cf=fc, cg=gc, fef=de=ed, df=fd, dg=gd, eg=ge, fg=gf >

Subgroups: 644 in 326 conjugacy classes, 112 normal (42 characteristic)
C1, C2, C2, C4, C4, C22, C22, C2×C4, C2×C4, D4, C23, C23, C23, C42, C22⋊C4, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C22×C4, C2×D4, C2×D4, 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.D4, C23×C4, C23×C4, C22×D4, C22×D4, C4×C22⋊C4, C23.7Q8, C42⋊9C4, C23.8Q8, C23.65C23, C24.3C22, C23.10D4, C23.11D4, C2×C4×D4, C2×C4⋊D4, C2×C22.D4, C23.322C24
Quotients: C1, C2, C22, D4, C23, C2×D4, C4○D4, C24, C22.D4, C22×D4, C2×C4○D4, 2+ 1+4, 2- 1+4, C2×C22.D4, C22.26C24, C22.36C24, D42, D4⋊6D4, C22.47C24, C23.322C24

Smallest permutation representation of C23.322C24
►On 64 points
Generators in S64
(1 47)(2 16)(3 45)(4 14)(5 39)(6 34)(7 37)(8 36)(9 51)(10 20)(11 49)(12 18)(13 43)(15 41)(17 21)(19 23)(22 50)(24 52)(25 55)(26 58)(27 53)(28 60)(29 59)(30 54)(31 57)(32 56)(33 62)(35 64)(38 61)(40 63)(42 48)(44 46)

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

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

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

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

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

38 conjugacy classes

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

38 irreducible representations

dim11111111111122244
type+++++++++++++++-
imageC1C2C2C2C2C2C2C2C2C2C2C2D4D4C4○D42+ 1+42- 1+4
kernelC23.322C24C4×C22⋊C4C23.7Q8C42⋊9C4C23.8Q8C23.65C23C24.3C22C23.10D4C23.11D4C2×C4×D4C2×C4⋊D4C2×C22.D4C22⋊C4C2×D4C2×C4C22C22
# reps111121122112441211

Matrix representation of C23.322C24 ►in GL6(𝔽5)

100000
010000
001200
000400
000040
000031
,
100000
010000
004000
000400
000010
000001
,
100000
010000
001000
000100
000040
000004
,
400000
040000
001000
000100
000040
000004
,
100000
340000
001200
004400
000030
000012
,
110000
040000
004000
000400
000032
000012
,
100000
010000
001200
004400
000040
000004

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

C23.322C24 in GAP, Magma, Sage, TeX 

C_2^3._{322}C_2^4
 
% in TeX
 
G:=Group("C2^3.322C2^4");
 
// GroupNames label
 
G:=SmallGroup(128,1154);
 
// by ID
 
G=gap.SmallGroup(128,1154);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,2,253,344,758,723,675,80]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=d^2=f^2=1,e^2=c*b=b*c,g^2=b,e*a*e^-1=g*a*g^-1=a*b=b*a,f*a*f=a*c=c*a,a*d=d*a,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=d*e=e*d,d*f=f*d,d*g=g*d,e*g=g*e,f*g=g*f>;
 
// generators/relations
 

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