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G = C23.288C24  order 128 = 27

5th central stem extension by C23 of C24

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

Aliases: C23.288C24, C24.233C23, (C22×C4)⋊17D4, C4○(C23⋊2D4), C23⋊1(C4○D4), C4○(C23⋊Q8), C4.128C22≀C2, C23⋊2D4⋊52C2, C23⋊Q8⋊67C2, C23.143(C2×D4), C4○(C23.10D4), (C2×C42).454C22, (C23×C4).319C22, (C22×C4).778C23, C22.171(C22×D4), C23.10D4⋊116C2, C4○(C23.78C23), C2.9(C22.19C24), (C22×D4).493C22, (C22×Q8).409C22, C23.78C23⋊72C2, C2.8(C22.26C24), C2.C42.530C22, (C2×C4×D4)⋊16C2, (C2×C4)⋊1(C4○D4), (C4×C22⋊C4)⋊46C2, C2.9(C2×C22≀C2), (C2×C4).289(C2×D4), (C22×C4○D4)⋊1C2, (C2×C4)○(C23⋊2D4), (C2×C4)○(C23⋊Q8), (C2×C4⋊C4).834C22, C22.168(C2×C4○D4), (C22×C4)○(C23⋊2D4), (C22×C4)○(C23⋊Q8), (C2×C22⋊C4).447C22, SmallGroup(128,1120)

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

Derived series
C1 — C23 — C23.288C24
Chief series
C1 — C2 — C22 — C23 — C22×C4 — C23×C4 — C2×C4×D4 — C23.288C24
Lower central
C1 — C23 — C23.288C24
Upper central
C1 — C22×C4 — C23.288C24
Jennings
C1 — C23 — C23.288C24

Generators and relations for C23.288C24
 G = < a,b,c,d,e,f,g | a2=b2=c2=d2=e2=f2=1, g2=a, ab=ba, ac=ca, ede=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, dg=gd, eg=ge, fg=gf >

Subgroups: 884 in 472 conjugacy classes, 124 normal (13 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C2×C4, C2×C4, D4, Q8, C23, C23, C23, C42, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C22×C4, C2×D4, C2×Q8, C4○D4, C24, C24, C2.C42, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C4×D4, C23×C4, C23×C4, C22×D4, C22×Q8, C2×C4○D4, C4×C22⋊C4, C23⋊2D4, C23⋊Q8, C23.10D4, C23.78C23, C2×C4×D4, C22×C4○D4, C23.288C24
Quotients: C1, C2, C22, D4, C23, C2×D4, C4○D4, C24, C22≀C2, C22×D4, C2×C4○D4, C2×C22≀C2, C22.19C24, C22.26C24, C23.288C24

Smallest permutation representation of C23.288C24
►On 64 points
Generators in S64
(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 17)(2 18)(3 19)(4 20)(5 15)(6 16)(7 13)(8 14)(9 64)(10 61)(11 62)(12 63)(21 27)(22 28)(23 25)(24 26)(29 33)(30 34)(31 35)(32 36)(37 43)(38 44)(39 41)(40 42)(45 49)(46 50)(47 51)(48 52)(53 59)(54 60)(55 57)(56 58)

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

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

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

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

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

44 conjugacy classes

class 1 2A···2G2H···2O4A···4H4I···4AB
order12···22···24···44···4
size11···14···41···14···4

44 irreducible representations

dim11111111222
type+++++++++
imageC1C2C2C2C2C2C2C2D4C4○D4C4○D4
kernelC23.288C24C4×C22⋊C4C23⋊2D4C23⋊Q8C23.10D4C23.78C23C2×C4×D4C22×C4○D4C22×C4C2×C4C23
# reps1331313112124

Matrix representation of C23.288C24 ►in GL6(𝔽5)

400000
040000
001000
000100
000040
000004
,
400000
040000
004000
000400
000040
000004
,
400000
040000
001000
000100
000010
000001
,
410000
010000
004000
000100
000032
000012
,
230000
430000
001000
000100
000041
000001
,
400000
310000
000100
001000
000014
000004
,
200000
020000
004000
000400
000030
000003

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

C23.288C24 in GAP, Magma, Sage, TeX 

C_2^3._{288}C_2^4
 
% in TeX
 
G:=Group("C2^3.288C2^4");
 
// GroupNames label
 
G:=SmallGroup(128,1120);
 
// by ID
 
G=gap.SmallGroup(128,1120);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,2,253,758,723,80]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=d^2=e^2=f^2=1,g^2=a,a*b=b*a,a*c=c*a,e*d*e=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,d*g=g*d,e*g=g*e,f*g=g*f>;
 
// generators/relations
 

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