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G = C23.262C24order 128 = 27

115th central extension by C23 of C24

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

Aliases: C23.262C24, C24.229C23, C22.932+ 1+4, C41D421C4, C4232(C2×C4), C425C48C2, C23.30(C22×C4), (C23×C4).63C22, C23.23D419C2, (C2×C42).450C22, (C22×C4).490C23, C22.153(C23×C4), (C22×D4).116C22, C2.42(C22.11C24), C2.4(C22.54C24), C2.C42.70C22, (C2×D4)⋊23(C2×C4), (C2×C41D4).14C2, (C2×C4).238(C22×C4), (C2×C22⋊C4).44C22, SmallGroup(128,1112)

Series: Derived Chief Lower central Upper central Jennings

C1C22 — C23.262C24
C1C2C22C23C22×C4C2×C42C2×C41D4 — C23.262C24
C1C22 — C23.262C24
C1C23 — C23.262C24
C1C23 — C23.262C24

Generators and relations for C23.262C24
 G = < a,b,c,d,e,f,g | a2=b2=c2=d2=1, e2=d, f2=cb=bc, g2=b, eae-1=gag-1=ab=ba, ac=ca, ad=da, faf-1=abc, bd=db, be=eb, bf=fb, bg=gb, cd=dc, fef-1=ce=ec, cf=fc, cg=gc, de=ed, df=fd, dg=gd, geg-1=bce, fg=gf >

Subgroups: 732 in 324 conjugacy classes, 132 normal (6 characteristic)
C1, C2 [×7], C2 [×8], C4 [×14], C22, C22 [×6], C22 [×40], C2×C4 [×6], C2×C4 [×46], D4 [×24], C23, C23 [×8], C23 [×24], C42 [×4], C22⋊C4 [×12], C22×C4 [×11], C22×C4 [×12], C2×D4 [×24], C2×D4 [×12], C24 [×4], C2.C42 [×12], C2×C42, C2×C22⋊C4 [×12], C41D4 [×8], C23×C4 [×4], C22×D4 [×6], C425C4 [×2], C23.23D4 [×12], C2×C41D4, C23.262C24
Quotients: C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], C23 [×15], C22×C4 [×14], C24, C23×C4, 2+ 1+4 [×6], C22.11C24 [×3], C22.54C24 [×4], C23.262C24

Smallest permutation representation of C23.262C24
On 64 points
Generators in S64
(1 6)(2 40)(3 8)(4 38)(5 12)(7 10)(9 39)(11 37)(13 17)(14 46)(15 19)(16 48)(18 42)(20 44)(21 35)(22 62)(23 33)(24 64)(25 29)(26 58)(27 31)(28 60)(30 54)(32 56)(34 52)(36 50)(41 45)(43 47)(49 61)(51 63)(53 57)(55 59)
(1 9)(2 10)(3 11)(4 12)(5 38)(6 39)(7 40)(8 37)(13 41)(14 42)(15 43)(16 44)(17 45)(18 46)(19 47)(20 48)(21 49)(22 50)(23 51)(24 52)(25 53)(26 54)(27 55)(28 56)(29 57)(30 58)(31 59)(32 60)(33 63)(34 64)(35 61)(36 62)
(1 51)(2 52)(3 49)(4 50)(5 62)(6 63)(7 64)(8 61)(9 23)(10 24)(11 21)(12 22)(13 27)(14 28)(15 25)(16 26)(17 31)(18 32)(19 29)(20 30)(33 39)(34 40)(35 37)(36 38)(41 55)(42 56)(43 53)(44 54)(45 59)(46 60)(47 57)(48 58)
(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 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 31 23 45)(2 18 24 60)(3 29 21 47)(4 20 22 58)(5 54 36 16)(6 41 33 27)(7 56 34 14)(8 43 35 25)(9 59 51 17)(10 46 52 32)(11 57 49 19)(12 48 50 30)(13 63 55 39)(15 61 53 37)(26 62 44 38)(28 64 42 40)
(1 13 9 41)(2 56 10 28)(3 15 11 43)(4 54 12 26)(5 30 38 58)(6 45 39 17)(7 32 40 60)(8 47 37 19)(14 52 42 24)(16 50 44 22)(18 34 46 64)(20 36 48 62)(21 53 49 25)(23 55 51 27)(29 61 57 35)(31 63 59 33)

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

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

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

38 conjugacy classes

class 1 2A···2G2H···2O4A···4V
order12···22···24···4
size11···14···44···4

38 irreducible representations

dim111114
type+++++
imageC1C2C2C2C42+ 1+4
kernelC23.262C24C425C4C23.23D4C2×C41D4C41D4C22
# reps12121166

Matrix representation of C23.262C24 in GL9(𝔽5)

100000000
040000000
041000000
020100000
000040000
000001412
000000010
000000100
000000004
,
100000000
040000000
004000000
000400000
000040000
000001000
000000100
000000010
000000001
,
100000000
010000000
001000000
000100000
000010000
000004000
000000400
000000040
000000004
,
400000000
040000000
004000000
000400000
000040000
000001000
000000100
000000010
000000001
,
200000000
030300000
000410000
000200000
004200000
000004000
000000100
000000010
000000144
,
100000000
013000000
014000000
002010000
032400000
000000010
000001412
000004000
000001401
,
400000000
013000000
014000000
032040000
002100000
000000100
000001000
000001412
000000004

G:=sub<GL(9,GF(5))| [1,0,0,0,0,0,0,0,0,0,4,4,2,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,1,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,2,0,0,4],[1,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,4],[4,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1],[2,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,3,4,2,2,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,1,0,1,0,0,0,0,0,0,0,1,4,0,0,0,0,0,0,0,0,4],[1,0,0,0,0,0,0,0,0,0,1,1,0,3,0,0,0,0,0,3,4,2,2,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,4,1,0,0,0,0,0,0,4,0,4,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,2,0,1],[4,0,0,0,0,0,0,0,0,0,1,1,3,0,0,0,0,0,0,3,4,2,2,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,1,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,2,4] >;

C23.262C24 in GAP, Magma, Sage, TeX

C_2^3._{262}C_2^4
% in TeX

G:=Group("C2^3.262C2^4");
// GroupNames label

G:=SmallGroup(128,1112);
// by ID

G=gap.SmallGroup(128,1112);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,448,253,758,555,268,1571,346,80]);
// Polycyclic

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

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