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G = C24.598C23order 128 = 27

79th non-split extension by C24 of C23 acting via C23/C22=C2

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

Aliases: C24.598C23, C23.754C24, (C22×C4)⋊47D4, (C22×C42)⋊16C2, C23.633(C2×D4), C44(C22.D4), C23.254(C4○D4), C22.31(C4⋊D4), (C22×C4).262C23, (C23×C4).654C22, C23.7Q8117C2, C22.464(C22×D4), C23.23D4114C2, (C2×C42).1015C22, (C22×D4).312C22, C24.C22185C2, C24.3C22100C2, C2.97(C22.19C24), C23.65C23168C2, C2.C42.451C22, C2.55(C22.26C24), C2.112(C23.36C23), (C2×C4).686(C2×D4), C2.48(C2×C4⋊D4), (C2×C4⋊D4).51C2, (C2×C4).670(C4○D4), (C2×C4⋊C4).557C22, C22.595(C2×C4○D4), (C2×C22.D4)⋊46C2, C2.46(C2×C22.D4), (C2×C22⋊C4).364C22, SmallGroup(128,1586)

Series: Derived Chief Lower central Upper central Jennings

C1C23 — C24.598C23
C1C2C22C23C22×C4C23×C4C22×C42 — C24.598C23
C1C23 — C24.598C23
C1C23 — C24.598C23
C1C23 — C24.598C23

Generators and relations for C24.598C23
 G = < a,b,c,d,e,f,g | a2=b2=c2=d2=e2=1, f2=cb=bc, g2=b, ab=ba, eae=ac=ca, ad=da, af=fa, ag=ga, bd=db, fef-1=be=eb, bf=fb, bg=gb, cd=dc, ce=ec, cf=fc, cg=gc, geg-1=de=ed, df=fd, dg=gd, fg=gf >

Subgroups: 644 in 344 conjugacy classes, 120 normal (20 characteristic)
C1, C2 [×3], C2 [×4], C2 [×6], C4 [×4], C4 [×16], C22 [×3], C22 [×8], C22 [×26], C2×C4 [×16], C2×C4 [×52], D4 [×12], C23, C23 [×6], C23 [×18], C42 [×8], C22⋊C4 [×24], C4⋊C4 [×14], C22×C4 [×2], C22×C4 [×18], C22×C4 [×14], C2×D4 [×18], C24, C24 [×2], C2.C42 [×6], C2×C42 [×4], C2×C42 [×4], C2×C22⋊C4 [×12], C2×C4⋊C4, C2×C4⋊C4 [×6], C4⋊D4 [×4], C22.D4 [×8], C23×C4 [×3], C22×D4, C22×D4 [×2], C23.7Q8, C23.23D4 [×2], C24.C22 [×4], C23.65C23 [×2], C24.3C22 [×2], C22×C42, C2×C4⋊D4, C2×C22.D4 [×2], C24.598C23
Quotients: C1, C2 [×15], C22 [×35], D4 [×8], C23 [×15], C2×D4 [×12], C4○D4 [×10], C24, C4⋊D4 [×4], C22.D4 [×4], C22×D4 [×2], C2×C4○D4 [×5], C2×C4⋊D4, C2×C22.D4, C22.19C24, C23.36C23 [×2], C22.26C24 [×2], C24.598C23

Smallest permutation representation of C24.598C23
On 64 points
Generators in S64
(1 17)(2 18)(3 19)(4 20)(5 60)(6 57)(7 58)(8 59)(9 47)(10 48)(11 45)(12 46)(13 21)(14 22)(15 23)(16 24)(25 36)(26 33)(27 34)(28 35)(29 40)(30 37)(31 38)(32 39)(41 49)(42 50)(43 51)(44 52)(53 61)(54 62)(55 63)(56 64)
(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 20)(2 49)(3 18)(4 51)(5 57)(6 56)(7 59)(8 54)(9 26)(10 29)(11 28)(12 31)(13 33)(14 40)(15 35)(16 38)(17 42)(19 44)(21 30)(22 25)(23 32)(24 27)(34 46)(36 48)(37 47)(39 45)(41 52)(43 50)(53 64)(55 62)(58 63)(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 15 41 47)(2 16 42 48)(3 13 43 45)(4 14 44 46)(5 35 62 37)(6 36 63 38)(7 33 64 39)(8 34 61 40)(9 17 23 49)(10 18 24 50)(11 19 21 51)(12 20 22 52)(25 55 31 57)(26 56 32 58)(27 53 29 59)(28 54 30 60)

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

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

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

44 conjugacy classes

class 1 2A···2G2H2I2J2K2L2M4A···4X4Y···4AD
order12···22222224···44···4
size11···12222882···28···8

44 irreducible representations

dim111111111222
type++++++++++
imageC1C2C2C2C2C2C2C2C2D4C4○D4C4○D4
kernelC24.598C23C23.7Q8C23.23D4C24.C22C23.65C23C24.3C22C22×C42C2×C4⋊D4C2×C22.D4C22×C4C2×C4C23
# reps1124221128164

Matrix representation of C24.598C23 in GL6(𝔽5)

020000
300000
001000
000100
000001
000010
,
400000
040000
001000
000100
000010
000001
,
400000
040000
001000
000100
000040
000004
,
400000
040000
004000
000400
000010
000001
,
400000
010000
001000
000400
000002
000030
,
030000
200000
001000
000100
000020
000002
,
010000
400000
000100
001000
000040
000004

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

C24.598C23 in GAP, Magma, Sage, TeX

C_2^4._{598}C_2^3
% in TeX

G:=Group("C2^4.598C2^3");
// GroupNames label

G:=SmallGroup(128,1586);
// by ID

G=gap.SmallGroup(128,1586);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,253,344,758,2019,80]);
// Polycyclic

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

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