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G = C24.416C23  order 128 = 27

256th non-split extension by C24 of C23 acting via C23/C2=C22

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

Aliases: C24.416C23, C23.620C24, C22.2952- (1+4), C22.3932+ (1+4), C22⋊C4⋊4Q8, C2.31(D4×Q8), C4⋊C4.125D4, C42⋊9C4⋊34C2, C23.35(C2×Q8), C2.71(D4⋊6D4), C2.51(D4⋊3Q8), (C23×C4).472C22, (C2×C42).671C22, (C22×C4).884C23, C22.429(C22×D4), C23.8Q8.49C2, C23.Q8.29C2, C23.4Q8.20C2, C22.147(C22×Q8), (C22×Q8).195C22, C23.83C23⋊87C2, C23.81C23⋊96C2, C23.78C23⋊50C2, C2.65(C22.29C24), C24.C22.53C2, C23.63C23⋊145C2, C23.65C23⋊130C2, C2.C42.326C22, C2.32(C23.41C23), C2.73(C22.33C24), C2.22(C22.57C24), (C2×C4).71(C2×Q8), (C2×C4).116(C2×D4), (C2×C42.C2)⋊25C2, (C2×C22⋊Q8).46C2, (C2×C4).436(C4○D4), (C2×C4⋊C4).433C22, C22.482(C2×C4○D4), (C2×C22⋊C4).284C22, SmallGroup(128,1452)

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

Derived series
C1 — C23 — C24.416C23
Chief series
C1 — C2 — C22 — C23 — C22×C4 — C2×C42 — C23.65C23 — C24.416C23
Lower central
C1 — C23 — C24.416C23
Upper central
C1 — C23 — C24.416C23
Jennings
C1 — C23 — C24.416C23

Subgroups: 436 in 236 conjugacy classes, 104 normal (82 characteristic)
C1, C2 [×7], C2 [×2], C4 [×20], C22 [×7], C22 [×10], C2×C4 [×12], C2×C4 [×40], Q8 [×4], C23, C23 [×2], C23 [×6], C42 [×4], C22⋊C4 [×4], C22⋊C4 [×8], C4⋊C4 [×4], C4⋊C4 [×22], C22×C4 [×14], C22×C4 [×4], C2×Q8 [×4], C24, C2.C42 [×10], C2×C42 [×2], C2×C22⋊C4 [×6], C2×C4⋊C4 [×15], C22⋊Q8 [×4], C42.C2 [×4], C23×C4, C22×Q8, C42⋊9C4, C23.8Q8 [×2], C23.63C23, C24.C22 [×2], C23.65C23, C23.78C23 [×2], C23.Q8, C23.81C23, C23.4Q8, C23.83C23, C2×C22⋊Q8, C2×C42.C2, C24.416C23

Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], Q8 [×4], C23 [×15], C2×D4 [×6], C2×Q8 [×6], C4○D4 [×2], C24, C22×D4, C22×Q8, C2×C4○D4, 2+ (1+4) [×2], 2- (1+4) [×2], C22.29C24, C22.33C24, C23.41C23, D4⋊6D4, D4×Q8, D4⋊3Q8, C22.57C24, C24.416C23

Generators and relations
 G = < a,b,c,d,e,f,g | a2=b2=c2=d2=1, e2=g2=cb=bc, f2=c, ab=ba, gag-1=ac=ca, ad=da, ae=ea, faf-1=acd, 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, gfg-1=cdf >

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

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

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

Matrix representation ►G ⊆ GL6(𝔽5)

100000
010000
001000
001400
000010
000034
,
400000
040000
001000
000100
000010
000001
,
100000
010000
004000
000400
000010
000001
,
100000
010000
001000
000100
000040
000004
,
200000
230000
003000
000300
000010
000034
,
100000
010000
003400
000200
000022
000013
,
420000
410000
004200
004100
000040
000021

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

32 conjugacy classes

class 1 2A···2G2H2I4A···4P4Q···4V
order12···2224···44···4
size11···1444···48···8

32 irreducible representations

dim111111111111122244
type+++++++++++++-++-
imageC1C2C2C2C2C2C2C2C2C2C2C2C2Q8D4C4○D42+ (1+4)2- (1+4)
kernelC24.416C23C42⋊9C4C23.8Q8C23.63C23C24.C22C23.65C23C23.78C23C23.Q8C23.81C23C23.4Q8C23.83C23C2×C22⋊Q8C2×C42.C2C22⋊C4C4⋊C4C2×C4C22C22
# reps112121211111144422

In GAP, Magma, Sage, TeX 

C_2^4._{416}C_2^3
 
% in TeX
 
G:=Group("C2^4.416C2^3");
 
// GroupNames label
 
G:=SmallGroup(128,1452);
 
// by ID
 
G=gap.SmallGroup(128,1452);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,2,112,253,344,758,723,184,1571,346,80]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=d^2=1,e^2=g^2=c*b=b*c,f^2=c,a*b=b*a,g*a*g^-1=a*c=c*a,a*d=d*a,a*e=e*a,f*a*f^-1=a*c*d,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,g*f*g^-1=c*d*f>;
 
// generators/relations
 

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