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G = C24.432C23order 128 = 27

272nd non-split extension by C24 of C23 acting via C23/C2=C22

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

Aliases: C24.432C23, C23.646C24, C22.4192+ 1+4, C22.3182- 1+4, C23.88(C4○D4), (C2×C42).94C22, C23.4Q858C2, C23.34D454C2, (C23×C4).159C22, (C22×C4).205C23, C23.8Q8124C2, C23.7Q8103C2, C23.11D4107C2, C23.23D4.64C2, C23.10D4.56C2, (C22×D4).264C22, C24.C22156C2, C23.84C2310C2, C23.83C2398C2, C23.81C23109C2, C23.63C23162C2, C2.98(C22.45C24), C2.24(C22.54C24), C2.C42.350C22, C2.88(C22.33C24), C2.94(C22.36C24), C2.93(C22.47C24), (C2×C4).447(C4○D4), (C2×C4⋊C4).457C22, C22.507(C2×C4○D4), (C2×C22⋊C4).64C22, SmallGroup(128,1478)

Series: Derived Chief Lower central Upper central Jennings

C1C23 — C24.432C23
C1C2C22C23C24C23×C4C23.34D4 — C24.432C23
C1C23 — C24.432C23
C1C23 — C24.432C23
C1C23 — C24.432C23

Generators and relations for C24.432C23
 G = < a,b,c,d,e,f,g | a2=b2=c2=d2=1, e2=bcd, f2=cb=bc, g2=b, faf-1=ab=ba, ac=ca, ad=da, eae-1=abc, ag=ga, 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 >

Subgroups: 468 in 224 conjugacy classes, 88 normal (82 characteristic)
C1, C2 [×7], C2 [×4], C4 [×14], C22 [×7], C22 [×20], C2×C4 [×2], C2×C4 [×46], D4 [×4], C23, C23 [×4], C23 [×12], C42, C22⋊C4 [×14], C4⋊C4 [×8], C22×C4 [×13], C22×C4 [×8], C2×D4 [×5], C24 [×2], C2.C42 [×14], C2×C42, C2×C22⋊C4 [×10], C2×C4⋊C4 [×7], C23×C4 [×2], C22×D4, C23.7Q8, C23.34D4, C23.8Q8 [×2], C23.23D4 [×2], C23.63C23, C24.C22 [×2], C23.10D4, C23.11D4, C23.81C23, C23.4Q8, C23.83C23, C23.84C23, C24.432C23
Quotients: C1, C2 [×15], C22 [×35], C23 [×15], C4○D4 [×6], C24, C2×C4○D4 [×3], 2+ 1+4 [×3], 2- 1+4, C22.33C24 [×2], C22.36C24, C22.45C24 [×2], C22.47C24, C22.54C24, C24.432C23

Smallest permutation representation of C24.432C23
On 64 points
Generators in S64
(1 32)(2 54)(3 30)(4 56)(5 58)(6 62)(7 60)(8 64)(9 46)(10 22)(11 48)(12 24)(13 19)(14 49)(15 17)(16 51)(18 41)(20 43)(21 39)(23 37)(25 61)(26 59)(27 63)(28 57)(29 35)(31 33)(34 53)(36 55)(38 45)(40 47)(42 52)(44 50)
(1 61)(2 62)(3 63)(4 64)(5 53)(6 54)(7 55)(8 56)(9 16)(10 13)(11 14)(12 15)(17 24)(18 21)(19 22)(20 23)(25 32)(26 29)(27 30)(28 31)(33 57)(34 58)(35 59)(36 60)(37 43)(38 44)(39 41)(40 42)(45 50)(46 51)(47 52)(48 49)
(1 58)(2 59)(3 60)(4 57)(5 32)(6 29)(7 30)(8 31)(9 41)(10 42)(11 43)(12 44)(13 40)(14 37)(15 38)(16 39)(17 45)(18 46)(19 47)(20 48)(21 51)(22 52)(23 49)(24 50)(25 53)(26 54)(27 55)(28 56)(33 64)(34 61)(35 62)(36 63)
(1 36)(2 33)(3 34)(4 35)(5 27)(6 28)(7 25)(8 26)(9 37)(10 38)(11 39)(12 40)(13 44)(14 41)(15 42)(16 43)(17 52)(18 49)(19 50)(20 51)(21 48)(22 45)(23 46)(24 47)(29 56)(30 53)(31 54)(32 55)(57 62)(58 63)(59 64)(60 61)
(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 18 34 51)(2 22 35 47)(3 20 36 49)(4 24 33 45)(5 16 25 41)(6 10 26 40)(7 14 27 43)(8 12 28 38)(9 32 39 53)(11 30 37 55)(13 29 42 54)(15 31 44 56)(17 57 50 64)(19 59 52 62)(21 58 46 61)(23 60 48 63)
(1 13 61 10)(2 37 62 43)(3 15 63 12)(4 39 64 41)(5 47 53 52)(6 20 54 23)(7 45 55 50)(8 18 56 21)(9 57 16 33)(11 59 14 35)(17 27 24 30)(19 25 22 32)(26 49 29 48)(28 51 31 46)(34 42 58 40)(36 44 60 38)

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

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

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

32 conjugacy classes

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

32 irreducible representations

dim11111111111112244
type++++++++++++++-
imageC1C2C2C2C2C2C2C2C2C2C2C2C2C4○D4C4○D42+ 1+42- 1+4
kernelC24.432C23C23.7Q8C23.34D4C23.8Q8C23.23D4C23.63C23C24.C22C23.10D4C23.11D4C23.81C23C23.4Q8C23.83C23C23.84C23C2×C4C23C22C22
# reps11122121111114831

Matrix representation of C24.432C23 in GL6(𝔽5)

010000
100000
004000
000400
000042
000001
,
400000
040000
001000
000100
000010
000001
,
100000
010000
001000
000100
000040
000004
,
100000
010000
004000
000400
000010
000001
,
040000
100000
002000
000200
000030
000032
,
200000
030000
002100
002300
000030
000003
,
200000
020000
004200
000100
000013
000004

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

C24.432C23 in GAP, Magma, Sage, TeX

C_2^4._{432}C_2^3
% in TeX

G:=Group("C2^4.432C2^3");
// GroupNames label

G:=SmallGroup(128,1478);
// by ID

G=gap.SmallGroup(128,1478);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,784,253,344,758,723,100,1571,346]);
// Polycyclic

G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=d^2=1,e^2=b*c*d,f^2=c*b=b*c,g^2=b,f*a*f^-1=a*b=b*a,a*c=c*a,a*d=d*a,e*a*e^-1=a*b*c,a*g=g*a,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

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