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G = C24.219C23  order 128 = 27

59th non-split extension by C24 of C23 acting via C23/C2=C22

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

Aliases: C24.219C23, C23.248C24, C22.802+ 1+4, C2.5D42, C4⋊3(C4×D4), C4⋊C4⋊52D4, C4⋊1D4⋊20C4, C42⋊27(C2×C4), C2.5(Q8⋊6D4), (C23×C4).57C22, C23.21(C22×C4), C23.23D4⋊16C2, (C2×C42).439C22, C22.139(C23×C4), C22.119(C22×D4), (C22×C4).1253C23, C24.3C22⋊22C2, (C22×D4).490C22, C2.32(C22.11C24), C2.C42.525C22, C2.3(C22.53C24), C4⋊C4○6(C4⋊C4), (C2×C4×D4)⋊15C2, (C4×C4⋊C4)⋊46C2, C2.42(C2×C4×D4), (C2×D4)⋊21(C2×C4), (C2×C4⋊1D4).13C2, (C2×C4).1075(C2×D4), (C2×C4).891(C4○D4), (C2×C4⋊C4).978C22, (C2×C4).453(C22×C4), C22.133(C2×C4○D4), (C2×C22⋊C4).444C22, C4⋊C4○3(C2×C4⋊C4), SmallGroup(128,1098)

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

Derived series
C1 — C22 — C24.219C23
Chief series
C1 — C2 — C22 — C23 — C22×C4 — C2×C42 — C4×C4⋊C4 — C24.219C23
Lower central
C1 — C22 — C24.219C23
Upper central
C1 — C23 — C24.219C23
Jennings
C1 — C23 — C24.219C23

Generators and relations for C24.219C23
 G = < a,b,c,d,e,f,g | a2=b2=c2=1, d2=c, e2=ca=ac, f2=b, g2=a, ab=ba, ede-1=ad=da, geg-1=ae=ea, af=fa, ag=ga, bc=cb, fdf-1=bd=db, be=eb, bf=fb, bg=gb, cd=dc, ce=ec, cf=fc, cg=gc, dg=gd, ef=fe, fg=gf >

Subgroups: 796 in 416 conjugacy classes, 164 normal (12 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C2×C4, C2×C4, D4, C23, C23, C23, C42, C42, 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×C4⋊C4, C4×D4, C4⋊1D4, C23×C4, C22×D4, C4×C4⋊C4, C23.23D4, C24.3C22, C2×C4×D4, C2×C4⋊1D4, C24.219C23
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22×C4, C2×D4, C4○D4, C24, C4×D4, C23×C4, C22×D4, C2×C4○D4, 2+ 1+4, C2×C4×D4, C22.11C24, D42, Q8⋊6D4, C22.53C24, C24.219C23

Smallest permutation representation of C24.219C23
►On 64 points
Generators in S64
(1 28)(2 25)(3 26)(4 27)(5 54)(6 55)(7 56)(8 53)(9 46)(10 47)(11 48)(12 45)(13 50)(14 51)(15 52)(16 49)(17 39)(18 40)(19 37)(20 38)(21 43)(22 44)(23 41)(24 42)(29 35)(30 36)(31 33)(32 34)(57 64)(58 61)(59 62)(60 63)

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

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

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

50 conjugacy classes

class 1 2A···2G2H···2O4A···4X4Y···4AH
order12···22···24···44···4
size11···14···42···24···4

50 irreducible representations

dim1111111224
type++++++++
imageC1C2C2C2C2C2C4D4C4○D42+ 1+4
kernelC24.219C23C4×C4⋊C4C23.23D4C24.3C22C2×C4×D4C2×C4⋊1D4C4⋊1D4C4⋊C4C2×C4C22
# reps12444116882

Matrix representation of C24.219C23 ►in GL5(𝔽5)

10000
04000
00400
00010
00001
,
10000
01000
00100
00040
00004
,
40000
01000
00100
00040
00004
,
20000
03400
03200
00031
00002
,
20000
03000
03200
00020
00002
,
10000
01000
00100
00043
00011
,
10000
04200
04100
00040
00004

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

C24.219C23 in GAP, Magma, Sage, TeX 

C_2^4._{219}C_2^3
 
% in TeX
 
G:=Group("C2^4.219C2^3");
 
// GroupNames label
 
G:=SmallGroup(128,1098);
 
// by ID
 
G=gap.SmallGroup(128,1098);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,2,448,253,568,758,184,346,80]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=1,d^2=c,e^2=c*a=a*c,f^2=b,g^2=a,a*b=b*a,e*d*e^-1=a*d=d*a,g*e*g^-1=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,c*e=e*c,c*f=f*c,c*g=g*c,d*g=g*d,e*f=f*e,f*g=g*f>;
 
// generators/relations
 

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