Copied to
clipboard

G = C24.401C23  order 128 = 27

241st non-split extension by C24 of C23 acting via C23/C2=C22

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

Aliases: C24.401C23, C23.594C24, C22.3682+ (1+4), C22.2742- (1+4), C4⋊C4.118D4, C2.99(D4⋊5D4), C2.29(Q8⋊6D4), C23.7Q8⋊88C2, C23.Q8⋊59C2, C23.4Q8⋊44C2, C23.172(C4○D4), C23.11D4⋊83C2, (C2×C42).647C22, (C22×C4).873C23, (C23×C4).457C22, C23.8Q8⋊106C2, C22.403(C22×D4), C23.10D4.41C2, (C22×D4).231C22, C23.81C23⋊84C2, C24.C22⋊126C2, C24.3C22.61C2, C23.65C23⋊119C2, C23.63C23⋊133C2, C2.C42.301C22, C2.82(C22.46C24), C2.59(C23.38C23), C2.14(C22.56C24), C2.64(C22.33C24), (C2×C4).97(C2×D4), (C2×C42.C2)⋊21C2, (C2×C4).192(C4○D4), (C2×C4⋊C4).408C22, C22.456(C2×C4○D4), (C2×C22⋊C4).261C22, (C2×C22.D4).25C2, SmallGroup(128,1426)

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

Derived series
C1 — C23 — C24.401C23
Chief series
C1 — C2 — C22 — C23 — C22×C4 — C23×C4 — C23.7Q8 — C24.401C23
Lower central
C1 — C23 — C24.401C23
Upper central
C1 — C23 — C24.401C23
Jennings
C1 — C23 — C24.401C23

Subgroups: 484 in 245 conjugacy classes, 96 normal (82 characteristic)
C1, C2 [×7], C2 [×3], C4 [×17], C22 [×7], C22 [×17], C2×C4 [×8], C2×C4 [×39], D4 [×4], C23, C23 [×2], C23 [×13], C42 [×3], C22⋊C4 [×17], C4⋊C4 [×4], C4⋊C4 [×18], C22×C4 [×13], C22×C4 [×5], C2×D4 [×5], C24 [×2], C2.C42 [×8], C2×C42 [×2], C2×C22⋊C4 [×11], C2×C4⋊C4 [×12], C22.D4 [×4], C42.C2 [×4], C23×C4, C22×D4, C23.7Q8, C23.8Q8, C23.63C23, C24.C22 [×2], C23.65C23, C24.3C22, C23.10D4, C23.Q8, C23.11D4, C23.81C23, C23.4Q8 [×2], C2×C22.D4, C2×C42.C2, C24.401C23

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], C23.38C23, C22.33C24 [×2], D4⋊5D4, Q8⋊6D4, C22.46C24, C22.56C24, C24.401C23

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

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

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

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

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

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

Matrix representation ►G ⊆ GL6(𝔽5)

100000
010000
004000
001100
000002
000030
,
100000
010000
001000
000100
000040
000004
,
400000
040000
001000
000100
000010
000001
,
100000
010000
004000
000400
000010
000001
,
100000
040000
004000
000400
000003
000030
,
200000
020000
004300
000100
000040
000001
,
010000
100000
004000
001100
000020
000002

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

32 conjugacy classes

class 1 2A···2G2H2I2J4A···4P4Q···4U
order12···22224···44···4
size11···14484···48···8

32 irreducible representations

dim1111111111111122244
type++++++++++++++++-
imageC1C2C2C2C2C2C2C2C2C2C2C2C2C2D4C4○D4C4○D42+ (1+4)2- (1+4)
kernelC24.401C23C23.7Q8C23.8Q8C23.63C23C24.C22C23.65C23C24.3C22C23.10D4C23.Q8C23.11D4C23.81C23C23.4Q8C2×C22.D4C2×C42.C2C4⋊C4C2×C4C23C22C22
# reps1111211111121144422

In GAP, Magma, Sage, TeX 

C_2^4._{401}C_2^3
 
% in TeX
 
G:=Group("C2^4.401C2^3");
 
// GroupNames label
 
G:=SmallGroup(128,1426);
 
// by ID
 
G=gap.SmallGroup(128,1426);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,2,112,253,232,758,723,100,1571,346,80]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=d^2=1,e^2=g^2=b,f^2=c,e*a*e^-1=a*b=b*a,a*c=c*a,a*d=d*a,f*a*f^-1=a*b*d,a*g=g*a,b*c=c*b,b*d=d*b,f*e*f^-1=b*e=e*b,b*f=f*b,b*g=g*b,c*d=d*c,g*e*g^-1=c*e=e*c,c*f=f*c,c*g=g*c,d*e=e*d,g*f*g^-1=d*f=f*d,d*g=g*d>;
 
// generators/relations
 

׿
×
⋊
ℤ
𝔽
○
≀