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G = C23.593C24  order 128 = 27

310th central stem extension by C23 of C24

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

Aliases: C23.593C24, C24.400C23, C22.3672+ 1+4, C22.2732- 1+4, C22⋊C4⋊15D4, C23.67(C2×D4), C2.98(D4⋊5D4), C23⋊2D4.21C2, C23.Q8⋊58C2, C23.7Q8⋊87C2, C23.171(C4○D4), C23.11D4⋊82C2, C23.23D4⋊87C2, (C23×C4).149C22, (C22×C4).558C23, C23.8Q8⋊105C2, C22.402(C22×D4), (C22×D4).230C22, C23.83C23⋊78C2, C2.14(C22.54C24), C2.77(C22.45C24), C2.C42.300C22, C2.44(C22.31C24), C2.63(C22.33C24), (C2×C4).420(C2×D4), (C2×C4⋊C4).407C22, C22.455(C2×C4○D4), (C2×C22.D4)⋊34C2, (C2×C22⋊C4).260C22, SmallGroup(128,1425)

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

Derived series
C1 — C23 — C23.593C24
Chief series
C1 — C2 — C22 — C23 — C22×C4 — C23×C4 — C23.7Q8 — C23.593C24
Lower central
C1 — C23 — C23.593C24
Upper central
C1 — C23 — C23.593C24
Jennings
C1 — C23 — C23.593C24

Generators and relations for C23.593C24
 G = < a,b,c,d,e,f,g | a2=b2=c2=d2=1, e2=c, f2=g2=b, eae-1=ab=ba, faf-1=ac=ca, ad=da, gag-1=abc, bc=cb, bd=db, geg-1=be=eb, bf=fb, bg=gb, cd=dc, ce=ec, cf=fc, cg=gc, fef-1=de=ed, df=fd, dg=gd, fg=gf >

Subgroups: 612 in 284 conjugacy classes, 96 normal (22 characteristic)
C1, C2, C2, C2, C4, C22, C22, C22, C2×C4, C2×C4, D4, C23, C23, C23, C22⋊C4, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C22×C4, C2×D4, C24, C24, C2.C42, C2.C42, C2×C22⋊C4, C2×C22⋊C4, C2×C4⋊C4, C2×C4⋊C4, C22.D4, C23×C4, C23×C4, C22×D4, C22×D4, C23.7Q8, C23.8Q8, C23.23D4, C23⋊2D4, C23.Q8, C23.11D4, C23.83C23, C2×C22.D4, C23.593C24
Quotients: C1, C2, C22, D4, C23, C2×D4, C4○D4, C24, C22×D4, C2×C4○D4, 2+ 1+4, 2- 1+4, C22.31C24, C22.33C24, D4⋊5D4, C22.45C24, C22.54C24, C23.593C24

Smallest permutation representation of C23.593C24
►On 64 points
Generators in S64
(1 47)(2 35)(3 45)(4 33)(5 55)(6 51)(7 53)(8 49)(9 52)(10 54)(11 50)(12 56)(13 57)(14 42)(15 59)(16 44)(17 38)(18 31)(19 40)(20 29)(21 36)(22 46)(23 34)(24 48)(25 41)(26 58)(27 43)(28 60)(30 62)(32 64)(37 61)(39 63)

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

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

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

32 conjugacy classes

class 1 2A···2G2H···2M4A···4L4M···4R
order12···22···24···44···4
size11···14···44···48···8

32 irreducible representations

dim1111111112244
type+++++++++++-
imageC1C2C2C2C2C2C2C2C2D4C4○D42+ 1+42- 1+4
kernelC23.593C24C23.7Q8C23.8Q8C23.23D4C23⋊2D4C23.Q8C23.11D4C23.83C23C2×C22.D4C22⋊C4C23C22C22
# reps1214121224831

Matrix representation of C23.593C24 ►in GL6(𝔽5)

100000
010000
001000
001400
000011
000004
,
100000
010000
001000
000100
000040
000004
,
100000
010000
004000
000400
000010
000001
,
400000
040000
001000
000100
000010
000001
,
010000
100000
002000
000200
000022
000013
,
100000
040000
002100
002300
000020
000002
,
100000
010000
002100
002300
000030
000042

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

C23.593C24 in GAP, Magma, Sage, TeX 

C_2^3._{593}C_2^4
 
% in TeX
 
G:=Group("C2^3.593C2^4");
 
// GroupNames label
 
G:=SmallGroup(128,1425);
 
// by ID
 
G=gap.SmallGroup(128,1425);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,2,253,232,758,723,100,1571,346,80]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=d^2=1,e^2=c,f^2=g^2=b,e*a*e^-1=a*b=b*a,f*a*f^-1=a*c=c*a,a*d=d*a,g*a*g^-1=a*b*c,b*c=c*b,b*d=d*b,g*e*g^-1=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,f*g=g*f>;
 
// generators/relations
 

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