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G = C24.583C23  order 128 = 27

64th non-split extension by C24 of C23 acting via C23/C22=C2

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

Aliases: C24.583C23, C23.464C24, C22.2492+ 1+4, (C2×D4)⋊18Q8, C23.24(C2×Q8), C23⋊Q8⋊21C2, C23.369(C2×D4), (C22×C4).393D4, C2.35(D4⋊3Q8), C23.8Q8⋊68C2, C23.4Q8⋊23C2, C23.7Q8⋊71C2, C23.314(C4○D4), C2.15(C23⋊3D4), (C22×C4).101C23, (C2×C42).565C22, (C23×C4).405C22, C22.315(C22×D4), C22.18(C22⋊Q8), C22.105(C22×Q8), C23.23D4.36C2, (C22×D4).533C22, (C22×Q8).140C22, C23.78C23⋊18C2, C23.83C23⋊40C2, C23.63C23⋊88C2, C2.51(C22.45C24), C2.C42.200C22, C2.41(C22.26C24), (C2×C4×D4).64C2, (C2×C4).53(C2×Q8), (C2×C4).358(C2×D4), (C2×C22⋊Q8)⋊25C2, C2.32(C2×C22⋊Q8), (C2×C4).392(C4○D4), (C2×C4⋊C4).312C22, C22.340(C2×C4○D4), (C2×C2.C42)⋊37C2, (C2×C22⋊C4).187C22, SmallGroup(128,1296)

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

Derived series
C1 — C23 — C24.583C23
Chief series
C1 — C2 — C22 — C23 — C24 — C23×C4 — C2×C2.C42 — C24.583C23
Lower central
C1 — C23 — C24.583C23
Upper central
C1 — C23 — C24.583C23
Jennings
C1 — C23 — C24.583C23

Generators and relations for C24.583C23
 G = < a,b,c,d,e,f,g | a2=b2=c2=g2=1, d2=c, e2=ca=ac, f2=b, ab=ba, ede-1=gdg=ad=da, ae=ea, af=fa, ag=ga, bc=cb, fdf-1=bd=db, be=eb, bf=fb, bg=gb, cd=dc, fef-1=ce=ec, cf=fc, cg=gc, eg=ge, fg=gf >

Subgroups: 564 in 292 conjugacy classes, 112 normal (34 characteristic)
C1, C2, C2, C2, 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×D4, C2×Q8, C24, C2.C42, C2.C42, C2×C42, C2×C22⋊C4, C2×C22⋊C4, C2×C4⋊C4, C2×C4⋊C4, C4×D4, C22⋊Q8, C23×C4, C23×C4, C22×D4, C22×Q8, C2×C2.C42, C23.7Q8, C23.8Q8, C23.23D4, C23.63C23, C23⋊Q8, C23.78C23, C23.4Q8, C23.83C23, C2×C4×D4, C2×C22⋊Q8, C24.583C23
Quotients: C1, C2, C22, D4, Q8, C23, C2×D4, C2×Q8, C4○D4, C24, C22⋊Q8, C22×D4, C22×Q8, C2×C4○D4, 2+ 1+4, C2×C22⋊Q8, C22.26C24, C23⋊3D4, C22.45C24, D4⋊3Q8, C24.583C23

Smallest permutation representation of C24.583C23
►On 64 points
Generators in S64
(1 14)(2 15)(3 16)(4 13)(5 40)(6 37)(7 38)(8 39)(9 42)(10 43)(11 44)(12 41)(17 47)(18 48)(19 45)(20 46)(21 51)(22 52)(23 49)(24 50)(25 59)(26 60)(27 57)(28 58)(29 55)(30 56)(31 53)(32 54)(33 64)(34 61)(35 62)(36 63)

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

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

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

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

38 conjugacy classes

class 1 2A···2G2H2I2J2K2L2M4A4B4C4D4E···4T4U4V4W4X
order12···222222244444···44444
size11···122224422224···48888

38 irreducible representations

dim11111111111122224
type+++++++++++++-+
imageC1C2C2C2C2C2C2C2C2C2C2C2D4Q8C4○D4C4○D42+ 1+4
kernelC24.583C23C2×C2.C42C23.7Q8C23.8Q8C23.23D4C23.63C23C23⋊Q8C23.78C23C23.4Q8C23.83C23C2×C4×D4C2×C22⋊Q8C22×C4C2×D4C2×C4C23C22
# reps11222211111144842

Matrix representation of C24.583C23 ►in GL6(𝔽5)

400000
040000
001000
000100
000010
000001
,
100000
010000
001000
000100
000040
000004
,
100000
010000
004000
000400
000040
000004
,
030000
200000
002000
000200
000022
000003
,
030000
300000
003000
003200
000033
000002
,
400000
040000
001300
000400
000020
000013
,
010000
100000
004000
000400
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],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[0,2,0,0,0,0,3,0,0,0,0,0,0,0,2,0,0,0,0,0,0,2,0,0,0,0,0,0,2,0,0,0,0,0,2,3],[0,3,0,0,0,0,3,0,0,0,0,0,0,0,3,3,0,0,0,0,0,2,0,0,0,0,0,0,3,0,0,0,0,0,3,2],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,3,4,0,0,0,0,0,0,2,1,0,0,0,0,0,3],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;
 

C24.583C23 in GAP, Magma, Sage, TeX 

C_2^4._{583}C_2^3
 
% in TeX
 
G:=Group("C2^4.583C2^3");
 
// GroupNames label
 
G:=SmallGroup(128,1296);
 
// by ID
 
G=gap.SmallGroup(128,1296);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,2,448,253,568,758,723,184,675]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=g^2=1,d^2=c,e^2=c*a=a*c,f^2=b,a*b=b*a,e*d*e^-1=g*d*g=a*d=d*a,a*e=e*a,a*f=f*a,a*g=g*a,b*c=c*b,f*d*f^-1=b*d=d*b,b*e=e*b,b*f=f*b,b*g=g*b,c*d=d*c,f*e*f^-1=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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