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G = C24.403C23  order 128 = 27

243rd non-split extension by C24 of C23 acting via C23/C2=C22

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

Aliases: C24.403C23, C23.596C24, C22.3702+ (1+4), C22.2762- (1+4), C4⋊C4⋊14D4, C23⋊Q8⋊45C2, C2.46(Q8⋊5D4), C23⋊2D4.22C2, C2.101(D4⋊5D4), C23.7Q8⋊90C2, C23.174(C4○D4), C23.23D4⋊88C2, C23.10D4⋊84C2, C23.11D4⋊85C2, (C2×C42).649C22, (C22×C4).874C23, (C23×C4).459C22, C22.405(C22×D4), C24.3C22⋊79C2, (C22×D4).233C22, (C22×Q8).184C22, C23.83C23⋊80C2, C24.C22⋊128C2, C2.66(C22.32C24), C2.58(C22.29C24), C23.63C23⋊134C2, C2.78(C22.45C24), C2.C42.303C22, C2.66(C22.33C24), C2.15(C22.56C24), (C2×C4).98(C2×D4), (C2×C22⋊Q8)⋊40C2, (C2×C4⋊C4).410C22, C22.458(C2×C4○D4), (C2×C22.D4)⋊35C2, (C2×C22⋊C4).263C22, SmallGroup(128,1428)

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

Derived series
C1 — C23 — C24.403C23
Chief series
C1 — C2 — C22 — C23 — C22×C4 — C23×C4 — C23.7Q8 — C24.403C23
Lower central
C1 — C23 — C24.403C23
Upper central
C1 — C23 — C24.403C23
Jennings
C1 — C23 — C24.403C23

Subgroups: 612 in 281 conjugacy classes, 96 normal (82 characteristic)
C1, C2 [×7], C2 [×5], C4 [×15], C22 [×7], C22 [×27], C2×C4 [×6], C2×C4 [×41], D4 [×12], Q8 [×4], C23, C23 [×4], C23 [×19], C42, C22⋊C4 [×19], C4⋊C4 [×4], C4⋊C4 [×7], C22×C4 [×12], C22×C4 [×9], C2×D4 [×14], C2×Q8 [×4], C24 [×3], C2.C42 [×10], C2×C42, C2×C22⋊C4 [×13], C2×C4⋊C4 [×5], C22⋊Q8 [×4], C22.D4 [×4], C23×C4 [×2], C22×D4 [×3], C22×Q8, C23.7Q8, C23.23D4 [×3], C23.63C23, C24.C22, C24.3C22, C23⋊2D4, C23⋊Q8 [×2], C23.10D4, C23.11D4, C23.83C23, C2×C22⋊Q8, C2×C22.D4, C24.403C23

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) [×3], 2- (1+4), C22.29C24, C22.32C24, C22.33C24, D4⋊5D4, Q8⋊5D4, C22.45C24, C22.56C24, C24.403C23

Generators and relations
 G = < a,b,c,d,e,f,g | a2=b2=c2=d2=1, e2=g2=cb=bc, f2=b, 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
(1 50)(2 51)(3 52)(4 49)(5 53)(6 54)(7 55)(8 56)(9 16)(10 13)(11 14)(12 15)(17 44)(18 41)(19 42)(20 43)(21 46)(22 47)(23 48)(24 45)(25 37)(26 38)(27 39)(28 40)(29 33)(30 34)(31 35)(32 36)(57 64)(58 61)(59 62)(60 63)

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

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

Matrix representation ►G ⊆ GL6(𝔽5)

400000
040000
000300
002000
000004
000040
,
400000
040000
001000
000100
000010
000001
,
100000
010000
004000
000400
000010
000001
,
100000
010000
001000
000100
000040
000004
,
300000
020000
003000
000300
000001
000010
,
300000
030000
000100
001000
000040
000001
,
010000
400000
002000
000300
000001
000010

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

32 conjugacy classes

class 1 2A···2G2H2I2J2K2L4A···4N4O···4S
order12···2222224···44···4
size11···1444484···48···8

32 irreducible representations

dim11111111111112244
type+++++++++++++++-
imageC1C2C2C2C2C2C2C2C2C2C2C2C2D4C4○D42+ (1+4)2- (1+4)
kernelC24.403C23C23.7Q8C23.23D4C23.63C23C24.C22C24.3C22C23⋊2D4C23⋊Q8C23.10D4C23.11D4C23.83C23C2×C22⋊Q8C2×C22.D4C4⋊C4C23C22C22
# reps11311112111114831

In GAP, Magma, Sage, TeX 

C_2^4._{403}C_2^3
 
% in TeX
 
G:=Group("C2^4.403C2^3");
 
// GroupNames label
 
G:=SmallGroup(128,1428);
 
// by ID
 
G=gap.SmallGroup(128,1428);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,2,112,253,344,758,723,100,1571,346,80]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=d^2=1,e^2=g^2=c*b=b*c,f^2=b,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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