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G = C23.413C24order 128 = 27

130th central stem extension by C23 of C24

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

Aliases: C24.18C23, C23.413C24, C22.2082+ 1+4, C428C434C2, (C2×C42).53C22, C4.42(C422C2), C23.11D434C2, (C22×C4).526C23, C24.C2270C2, (C22×D4).154C22, C23.65C2376C2, C24.3C22.39C2, C2.C42.490C22, C2.56(C23.36C23), C2.13(C22.53C24), C2.19(C22.34C24), C2.42(C22.47C24), C2.18(C22.49C24), (C4×C4⋊C4)⋊76C2, (C2×C4).134(C4○D4), (C2×C4⋊C4).861C22, C2.20(C2×C422C2), C22.290(C2×C4○D4), (C2×C22⋊C4).49C22, SmallGroup(128,1245)

Series: Derived Chief Lower central Upper central Jennings

C1C23 — C23.413C24
C1C2C22C23C22×C4C2×C42C4×C4⋊C4 — C23.413C24
C1C23 — C23.413C24
C1C23 — C23.413C24
C1C23 — C23.413C24

Generators and relations for C23.413C24
 G = < a,b,c,d,e,f,g | a2=b2=c2=1, d2=cb=bc, e2=b, f2=ba=ab, g2=a, ac=ca, ede-1=gdg-1=ad=da, ae=ea, af=fa, ag=ga, fdf-1=bd=db, be=eb, bf=fb, bg=gb, cd=dc, fef-1=ce=ec, cf=fc, cg=gc, eg=ge, fg=gf >

Subgroups: 436 in 220 conjugacy classes, 96 normal (42 characteristic)
C1, C2 [×7], C2 [×2], C4 [×4], C4 [×14], C22 [×7], C22 [×14], C2×C4 [×10], C2×C4 [×34], D4 [×4], C23, C23 [×14], C42 [×8], C22⋊C4 [×16], C4⋊C4 [×12], C22×C4 [×7], C22×C4 [×6], C2×D4 [×6], C24 [×2], C2.C42 [×2], C2.C42 [×8], C2×C42 [×3], C2×C42 [×2], C2×C22⋊C4 [×12], C2×C4⋊C4 [×5], C2×C4⋊C4 [×2], C22×D4, C4×C4⋊C4 [×2], C428C4, C24.C22 [×4], C23.65C23, C24.3C22 [×3], C23.11D4 [×4], C23.413C24
Quotients: C1, C2 [×15], C22 [×35], C23 [×15], C4○D4 [×10], C24, C422C2 [×4], C2×C4○D4 [×5], 2+ 1+4 [×2], C2×C422C2, C23.36C23, C22.34C24, C22.47C24, C22.49C24, C22.53C24 [×2], C23.413C24

Smallest permutation representation of C23.413C24
On 64 points
Generators in S64
(1 58)(2 59)(3 60)(4 57)(5 56)(6 53)(7 54)(8 55)(9 48)(10 45)(11 46)(12 47)(13 52)(14 49)(15 50)(16 51)(17 43)(18 44)(19 41)(20 42)(21 39)(22 40)(23 37)(24 38)(25 34)(26 35)(27 36)(28 33)(29 61)(30 62)(31 63)(32 64)
(1 61)(2 62)(3 63)(4 64)(5 33)(6 34)(7 35)(8 36)(9 43)(10 44)(11 41)(12 42)(13 39)(14 40)(15 37)(16 38)(17 48)(18 45)(19 46)(20 47)(21 52)(22 49)(23 50)(24 51)(25 53)(26 54)(27 55)(28 56)(29 58)(30 59)(31 60)(32 57)
(1 63)(2 64)(3 61)(4 62)(5 35)(6 36)(7 33)(8 34)(9 41)(10 42)(11 43)(12 44)(13 37)(14 38)(15 39)(16 40)(17 46)(18 47)(19 48)(20 45)(21 50)(22 51)(23 52)(24 49)(25 55)(26 56)(27 53)(28 54)(29 60)(30 57)(31 58)(32 59)
(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 61 46)(2 42 62 12)(3 17 63 48)(4 44 64 10)(5 22 33 49)(6 37 34 15)(7 24 35 51)(8 39 36 13)(9 60 43 31)(11 58 41 29)(14 56 40 28)(16 54 38 26)(18 32 45 57)(20 30 47 59)(21 27 52 55)(23 25 50 53)
(1 51 29 38)(2 21 30 13)(3 49 31 40)(4 23 32 15)(5 41 28 46)(6 12 25 20)(7 43 26 48)(8 10 27 18)(9 54 17 35)(11 56 19 33)(14 63 22 60)(16 61 24 58)(34 42 53 47)(36 44 55 45)(37 64 50 57)(39 62 52 59)
(1 6 58 53)(2 54 59 7)(3 8 60 55)(4 56 57 5)(9 52 48 13)(10 14 45 49)(11 50 46 15)(12 16 47 51)(17 39 43 21)(18 22 44 40)(19 37 41 23)(20 24 42 38)(25 61 34 29)(26 30 35 62)(27 63 36 31)(28 32 33 64)

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

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

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

38 conjugacy classes

class 1 2A···2G2H2I4A···4H4I···4Z4AA4AB
order12···2224···44···444
size11···1882···24···488

38 irreducible representations

dim111111124
type++++++++
imageC1C2C2C2C2C2C2C4○D42+ 1+4
kernelC23.413C24C4×C4⋊C4C428C4C24.C22C23.65C23C24.3C22C23.11D4C2×C4C22
# reps1214134202

Matrix representation of C23.413C24 in GL6(𝔽5)

100000
010000
001000
000100
000040
000004
,
100000
010000
004000
000400
000040
000004
,
400000
040000
004000
000400
000010
000001
,
200000
020000
004200
000100
000002
000020
,
030000
200000
003400
000200
000004
000010
,
010000
100000
002000
002300
000003
000020
,
400000
040000
004000
000400
000001
000040

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

C23.413C24 in GAP, Magma, Sage, TeX

C_2^3._{413}C_2^4
% in TeX

G:=Group("C2^3.413C2^4");
// GroupNames label

G:=SmallGroup(128,1245);
// by ID

G=gap.SmallGroup(128,1245);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,672,253,232,758,723,268,675,80]);
// Polycyclic

G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=1,d^2=c*b=b*c,e^2=b,f^2=b*a=a*b,g^2=a,a*c=c*a,e*d*e^-1=g*d*g^-1=a*d=d*a,a*e=e*a,a*f=f*a,a*g=g*a,f*d*f^-1=b*d=d*b,b*e=e*b,b*f=f*b,b*g=g*b,c*d=d*c,f*e*f^-1=c*e=e*c,c*f=f*c,c*g=g*c,e*g=g*e,f*g=g*f>;
// generators/relations

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