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G = C23.234C24  order 128 = 27

87th central extension by C23 of C24

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

Aliases: C23.234C24, C24.210C23, C22.692+ 1+4, C22.512- 1+4, C4⋊C4⋊48D4, (C4×D4)⋊23C4, C42⋊25(C2×C4), C4.150(C4×D4), C42⋊8C4⋊18C2, C2.5(D4⋊5D4), C2.4(Q8⋊5D4), C23.90(C22×C4), C23.7Q8⋊25C2, C22⋊3(C42⋊C2), C23.290(C4○D4), (C2×C42).433C22, C22.125(C23×C4), (C23×C4).305C22, C23.23D4.7C2, C22.109(C22×D4), C24.C22⋊13C2, (C22×C4).1249C23, (C22×D4).481C22, C23.63C23⋊15C2, C2.C42.476C22, C2.3(C22.47C24), C2.4(C22.46C24), C2.29(C23.33C23), (C4×C4⋊C4)⋊38C2, C2.32(C2×C4×D4), C4⋊C4⋊45(C2×C4), (C2×C4×D4).35C2, C22⋊C4○5(C4⋊C4), (C4×C22⋊C4)⋊9C2, C22⋊C4⋊42(C2×C4), (C22×C4⋊C4)⋊10C2, (C22×C4)⋊33(C2×C4), (C2×D4).214(C2×C4), (C2×C4).1073(C2×D4), (C2×C42⋊C2)⋊12C2, (C2×C4).794(C4○D4), (C2×C4⋊C4).823C22, (C2×C4).235(C22×C4), C2.31(C2×C42⋊C2), C22.119(C2×C4○D4), (C2×C22⋊C4).35C22, C22⋊C4○(C2×C4⋊C4), C4⋊C4○2(C2×C22⋊C4), SmallGroup(128,1084)

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

Derived series
C1 — C22 — C23.234C24
Chief series
C1 — C2 — C22 — C23 — C24 — C23×C4 — C2×C4×D4 — C23.234C24
Lower central
C1 — C22 — C23.234C24
Upper central
C1 — C23 — C23.234C24
Jennings
C1 — C23 — C23.234C24

Generators and relations for C23.234C24
 G = < a,b,c,d,e,f,g | a2=b2=c2=d2=g2=1, e2=d, f2=c, gag=ab=ba, ac=ca, ad=da, eae-1=abc, af=fa, bc=cb, bd=db, fef-1=be=eb, gfg=bf=fb, bg=gb, cd=dc, ce=ec, cf=fc, cg=gc, de=ed, df=fd, dg=gd, eg=ge >

Subgroups: 556 in 330 conjugacy classes, 156 normal (42 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C2×C4, C2×C4, D4, C23, C23, C23, C42, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C22×C4, C2×D4, C2×D4, C24, C2.C42, C2×C42, C2×C42, C2×C22⋊C4, C2×C22⋊C4, C2×C4⋊C4, C2×C4⋊C4, C2×C4⋊C4, C42⋊C2, C4×D4, C23×C4, C23×C4, C22×D4, C4×C22⋊C4, C4×C4⋊C4, C23.7Q8, C42⋊8C4, C23.23D4, C23.63C23, C24.C22, C22×C4⋊C4, C2×C42⋊C2, C2×C4×D4, C23.234C24
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22×C4, C2×D4, C4○D4, C24, C42⋊C2, C4×D4, C23×C4, C22×D4, C2×C4○D4, 2+ 1+4, 2- 1+4, C2×C42⋊C2, C2×C4×D4, C23.33C23, D4⋊5D4, Q8⋊5D4, C22.46C24, C22.47C24, C23.234C24

Smallest permutation representation of C23.234C24
►On 64 points
Generators in S64
(2 24)(4 22)(5 36)(6 39)(7 34)(8 37)(10 50)(12 52)(13 41)(14 26)(15 43)(16 28)(18 58)(20 60)(25 53)(27 55)(30 46)(32 48)(33 63)(35 61)(38 62)(40 64)(42 54)(44 56)

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

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

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

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

50 conjugacy classes

class 1 2A···2G2H2I2J2K2L2M4A···4T4U···4AJ
order12···22222224···44···4
size11···12222442···24···4

50 irreducible representations

dim11111111111122244
type+++++++++++++-
imageC1C2C2C2C2C2C2C2C2C2C2C4D4C4○D4C4○D42+ 1+42- 1+4
kernelC23.234C24C4×C22⋊C4C4×C4⋊C4C23.7Q8C42⋊8C4C23.23D4C23.63C23C24.C22C22×C4⋊C4C2×C42⋊C2C2×C4×D4C4×D4C4⋊C4C2×C4C23C22C22
# reps121212221111644811

Matrix representation of C23.234C24 ►in GL6(𝔽5)

100000
240000
001000
000400
000010
000024
,
100000
010000
001000
000100
000040
000004
,
400000
040000
004000
000400
000010
000001
,
100000
010000
004000
000400
000040
000004
,
140000
040000
000100
004000
000032
000002
,
300000
030000
002000
000200
000010
000024
,
100000
010000
001000
000100
000041
000001

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

C23.234C24 in GAP, Magma, Sage, TeX 

C_2^3._{234}C_2^4
 
% in TeX
 
G:=Group("C2^3.234C2^4");
 
// GroupNames label
 
G:=SmallGroup(128,1084);
 
// by ID
 
G=gap.SmallGroup(128,1084);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,2,448,253,232,758,184,346]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=d^2=g^2=1,e^2=d,f^2=c,g*a*g=a*b=b*a,a*c=c*a,a*d=d*a,e*a*e^-1=a*b*c,a*f=f*a,b*c=c*b,b*d=d*b,f*e*f^-1=b*e=e*b,g*f*g=b*f=f*b,b*g=g*b,c*d=d*c,c*e=e*c,c*f=f*c,c*g=g*c,d*e=e*d,d*f=f*d,d*g=g*d,e*g=g*e>;
 
// generators/relations
 

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