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G = C24.419C23  order 128 = 27

259th non-split extension by C24 of C23 acting via C23/C2=C22

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

Aliases: C24.419C23, C23.624C24, C22.3972+ (1+4), C22.2992- (1+4), (C2×D4).145D4, C23.221(C2×D4), C2.75(D4⋊6D4), C23.11D4⋊99C2, C2.51(C23⋊3D4), (C2×C42).675C22, (C22×C4).887C23, (C23×C4).157C22, C23.8Q8⋊120C2, C22.433(C22×D4), (C22×D4).253C22, C23.81C23⋊99C2, C24.3C22.64C2, C23.65C23⋊134C2, C2.C42.330C22, C2.36(C22.53C24), C2.26(C22.57C24), C2.76(C22.33C24), (C2×C4).119(C2×D4), (C2×C4).205(C4○D4), (C2×C4⋊C4).437C22, C22.486(C2×C4○D4), (C2×C22⋊C4).288C22, (C2×C22.D4).30C2, SmallGroup(128,1456)

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

Derived series
C1 — C23 — C24.419C23
Chief series
C1 — C2 — C22 — C23 — C24 — C23×C4 — C23.8Q8 — C24.419C23
Lower central
C1 — C23 — C24.419C23
Upper central
C1 — C23 — C24.419C23
Jennings
C1 — C23 — C24.419C23

Subgroups: 500 in 252 conjugacy classes, 96 normal (14 characteristic)
C1, C2 [×3], C2 [×4], C2 [×4], C4 [×16], C22 [×3], C22 [×4], C22 [×20], C2×C4 [×6], C2×C4 [×44], D4 [×4], C23, C23 [×4], C23 [×12], C42, C22⋊C4 [×20], C4⋊C4 [×17], C22×C4, C22×C4 [×12], C22×C4 [×8], C2×D4 [×4], C2×D4 [×2], C24 [×2], C2.C42 [×10], C2×C42, C2×C22⋊C4 [×10], C2×C4⋊C4, C2×C4⋊C4 [×10], C22.D4 [×8], C23×C4 [×2], C22×D4, C23.8Q8 [×4], C23.65C23 [×2], C24.3C22, C23.11D4 [×4], C23.81C23 [×2], C2×C22.D4 [×2], C24.419C23

Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], C2×D4 [×6], C4○D4 [×4], C24, C22×D4, C2×C4○D4 [×2], 2+ (1+4) [×2], 2- (1+4) [×2], C23⋊3D4, C22.33C24 [×2], D4⋊6D4 [×2], C22.53C24, C22.57C24, C24.419C23

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

Smallest permutation representation
►On 64 points
Generators in S64
(5 64)(6 61)(7 62)(8 63)(9 23)(10 24)(11 21)(12 22)(13 15)(14 16)(17 51)(18 52)(19 49)(20 50)(25 27)(26 28)(29 31)(30 32)(33 39)(34 40)(35 37)(36 38)(45 47)(46 48)

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

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

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

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

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

Matrix representation ►G ⊆ GL6(𝔽5)

100000
010000
001000
004400
000010
000044
,
100000
010000
004000
000400
000010
000001
,
100000
010000
001000
000100
000040
000004
,
400000
040000
001000
000100
000010
000001
,
020000
300000
002000
003300
000030
000003
,
010000
100000
003000
000300
000024
000003
,
100000
010000
004300
001100
000043
000011

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

32 conjugacy classes

class 1 2A···2G2H2I2J2K4A···4N4O···4T
order12···222224···44···4
size11···144444···48···8

32 irreducible representations

dim11111112244
type+++++++++-
imageC1C2C2C2C2C2C2D4C4○D42+ (1+4)2- (1+4)
kernelC24.419C23C23.8Q8C23.65C23C24.3C22C23.11D4C23.81C23C2×C22.D4C2×D4C2×C4C22C22
# reps14214224822

In GAP, Magma, Sage, TeX 

C_2^4._{419}C_2^3
 
% in TeX
 
G:=Group("C2^4.419C2^3");
 
// GroupNames label
 
G:=SmallGroup(128,1456);
 
// by ID
 
G=gap.SmallGroup(128,1456);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,2,336,253,344,758,723,268,1571,346]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=d^2=1,e^2=f^2=g^2=c*b=b*c,a*b=b*a,f*a*f^-1=a*c=c*a,a*d=d*a,a*e=e*a,g*a*g^-1=a*b*c,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,g*f*g^-1=c*f=f*c,c*g=g*c,f*e*f^-1=d*e=e*d,d*f=f*d,d*g=g*d>;
 
// generators/relations
 

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