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G = C23.261C24  order 128 = 27

114th central extension by C23 of C24

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

Aliases: C23.261C24, C24.228C23, C22.672- 1+4, C22.922+ 1+4, C42⋊31(C2×C4), C4.4D4⋊26C4, C42⋊8C4⋊20C2, C23.29(C22×C4), (C23×C4).62C22, C23.8Q8⋊21C2, (C22×C4).489C23, (C2×C42).449C22, C22.152(C23×C4), (C22×Q8).95C22, C24.C22⋊28C2, C2.2(C24⋊C22), C23.23D4.13C2, (C22×D4).115C22, C23.67C23⋊26C2, C2.41(C22.11C24), C2.C42.69C22, C2.4(C22.56C24), C2.3(C22.57C24), C2.41(C23.33C23), (C2×Q8)⋊17(C2×C4), C22⋊C4⋊22(C2×C4), (C2×D4).135(C2×C4), (C2×C4).58(C22×C4), (C2×C4⋊C4).197C22, (C2×C4.4D4).19C2, (C2×C22⋊C4).43C22, SmallGroup(128,1111)

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

Derived series
C1 — C22 — C23.261C24
Chief series
C1 — C2 — C22 — C23 — C22×C4 — C2×C42 — C2×C4.4D4 — C23.261C24
Lower central
C1 — C22 — C23.261C24
Upper central
C1 — C23 — C23.261C24
Jennings
C1 — C23 — C23.261C24

Generators and relations for C23.261C24
 G = < a,b,c,d,e,f,g | a2=b2=c2=d2=1, e2=d, f2=c, g2=b, eae-1=gag-1=ab=ba, ac=ca, ad=da, faf-1=abc, bc=cb, 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: 508 in 260 conjugacy classes, 132 normal (14 characteristic)
C1, C2, C2, C2, C4, C22, C22, C22, C2×C4, C2×C4, D4, Q8, C23, C23, C23, C42, C42, C22⋊C4, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C2×Q8, C24, C2.C42, C2×C42, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C4.4D4, C23×C4, C22×D4, C22×Q8, C42⋊8C4, C23.8Q8, C23.23D4, C24.C22, C23.67C23, C2×C4.4D4, C23.261C24
Quotients: C1, C2, C4, C22, C2×C4, C23, C22×C4, C24, C23×C4, 2+ 1+4, 2- 1+4, C22.11C24, C23.33C23, C24⋊C22, C22.56C24, C22.57C24, C23.261C24

Smallest permutation representation of C23.261C24
►On 64 points
Generators in S64
(1 41)(2 14)(3 43)(4 16)(5 58)(6 31)(7 60)(8 29)(9 13)(10 42)(11 15)(12 44)(17 63)(18 34)(19 61)(20 36)(21 25)(22 54)(23 27)(24 56)(26 50)(28 52)(30 38)(32 40)(33 45)(35 47)(37 57)(39 59)(46 64)(48 62)(49 53)(51 55)

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

(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)| (1,41)(2,14)(3,43)(4,16)(5,58)(6,31)(7,60)(8,29)(9,13)(10,42)(11,15)(12,44)(17,63)(18,34)(19,61)(20,36)(21,25)(22,54)(23,27)(24,56)(26,50)(28,52)(30,38)(32,40)(33,45)(35,47)(37,57)(39,59)(46,64)(48,62)(49,53)(51,55), (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), (1,59,51,45)(2,46,52,60)(3,57,49,47)(4,48,50,58)(5,54,62,44)(6,41,63,55)(7,56,64,42)(8,43,61,53)(9,31,23,17)(10,18,24,32)(11,29,21,19)(12,20,22,30)(13,33,27,39)(14,40,28,34)(15,35,25,37)(16,38,26,36), (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( (1,41)(2,14)(3,43)(4,16)(5,58)(6,31)(7,60)(8,29)(9,13)(10,42)(11,15)(12,44)(17,63)(18,34)(19,61)(20,36)(21,25)(22,54)(23,27)(24,56)(26,50)(28,52)(30,38)(32,40)(33,45)(35,47)(37,57)(39,59)(46,64)(48,62)(49,53)(51,55), (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), (1,59,51,45)(2,46,52,60)(3,57,49,47)(4,48,50,58)(5,54,62,44)(6,41,63,55)(7,56,64,42)(8,43,61,53)(9,31,23,17)(10,18,24,32)(11,29,21,19)(12,20,22,30)(13,33,27,39)(14,40,28,34)(15,35,25,37)(16,38,26,36), (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([[(1,41),(2,14),(3,43),(4,16),(5,58),(6,31),(7,60),(8,29),(9,13),(10,42),(11,15),(12,44),(17,63),(18,34),(19,61),(20,36),(21,25),(22,54),(23,27),(24,56),(26,50),(28,52),(30,38),(32,40),(33,45),(35,47),(37,57),(39,59),(46,64),(48,62),(49,53),(51,55)], [(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)], [(1,59,51,45),(2,46,52,60),(3,57,49,47),(4,48,50,58),(5,54,62,44),(6,41,63,55),(7,56,64,42),(8,43,61,53),(9,31,23,17),(10,18,24,32),(11,29,21,19),(12,20,22,30),(13,33,27,39),(14,40,28,34),(15,35,25,37),(16,38,26,36)], [(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···2G2H2I2J2K4A···4Z
order12···222224···4
size11···144444···4

38 irreducible representations

dim1111111144
type++++++++-
imageC1C2C2C2C2C2C2C42+ 1+42- 1+4
kernelC23.261C24C42⋊8C4C23.8Q8C23.23D4C24.C22C23.67C23C2×C4.4D4C4.4D4C22C22
# reps12424211642

Matrix representation of C23.261C24 ►in GL12(𝔽5)

010000000000
100000000000
000400000000
004000000000
000001000000
000010000000
000000040000
000000400000
000000000100
000000001000
000000000004
000000000040
,
400000000000
040000000000
004000000000
000400000000
000040000000
000004000000
000000400000
000000040000
000000001000
000000000100
000000000010
000000000001
,
400000000000
040000000000
004000000000
000400000000
000040000000
000004000000
000000400000
000000040000
000000004000
000000000400
000000000040
000000000004
,
100000000000
010000000000
001000000000
000100000000
000040000000
000004000000
000000400000
000000040000
000000001000
000000000100
000000000010
000000000001
,
001000000000
000100000000
100000000000
010000000000
000000100000
000000010000
000040000000
000004000000
000000000002
000000000030
000000000200
000000003000
,
200000000000
020000000000
003000000000
000300000000
000020000000
000002000000
000000300000
000000030000
000000000010
000000000001
000000004000
000000000400
,
010000000000
400000000000
000100000000
004000000000
000001000000
000040000000
000000010000
000000400000
000000000100
000000001000
000000000001
000000000010

G:=sub<GL(12,GF(5))| [0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,4,0],[4,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1],[4,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,4],[1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1],[0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0,0,2,0,0,0],[2,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0],[0,4,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0] >;
 

C23.261C24 in GAP, Magma, Sage, TeX 

C_2^3._{261}C_2^4
 
% in TeX
 
G:=Group("C2^3.261C2^4");
 
// GroupNames label
 
G:=SmallGroup(128,1111);
 
// by ID
 
G=gap.SmallGroup(128,1111);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,2,448,253,758,555,184,1571,346,80]);
 
// 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,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*c=c*b,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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