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G = C23.525C24  order 128 = 27

242nd central stem extension by C23 of C24

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

Aliases: C23.525C24, C24.366C23, C22.2212- 1+4, C22.3022+ 1+4, (C22×C4).403D4, C23.196(C2×D4), (C23×C4).427C22, (C2×C42).604C22, (C22×C4).135C23, C22.350(C22×D4), C23.7Q8.56C2, C23.11D4.27C2, C4.97(C22.D4), (C22×Q8).153C22, C23.81C23⋊59C2, C23.83C23⋊58C2, C23.67C23⋊72C2, C23.65C23⋊103C2, C2.C42.251C22, C2.46(C22.36C24), C2.25(C22.35C24), C2.37(C23.38C23), C2.26(C22.31C24), (C2×C4).384(C2×D4), (C2×C22⋊Q8).38C2, (C2×C4).659(C4○D4), (C2×C4⋊C4).356C22, C22.397(C2×C4○D4), (C2×C42⋊C2).46C2, C2.43(C2×C22.D4), (C2×C22⋊C4).470C22, SmallGroup(128,1357)

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

Derived series
C1 — C23 — C23.525C24
Chief series
C1 — C2 — C22 — C23 — C22×C4 — C22×Q8 — C23.67C23 — C23.525C24
Lower central
C1 — C23 — C23.525C24
Upper central
C1 — C23 — C23.525C24
Jennings
C1 — C23 — C23.525C24

Generators and relations for C23.525C24
 G = < a,b,c,d,e,f,g | a2=b2=c2=d2=1, e2=b, f2=g2=a, ab=ba, ac=ca, ede-1=ad=da, ae=ea, gfg-1=af=fa, ag=ga, bc=cb, fdf-1=bd=db, be=eb, bf=fb, bg=gb, cd=dc, fef-1=ce=ec, cf=fc, cg=gc, dg=gd, eg=ge >

Subgroups: 420 in 232 conjugacy classes, 100 normal (20 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C2×C4, C2×C4, Q8, C23, C23, C23, C42, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C22×C4, C2×Q8, C24, C2.C42, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C2×C4⋊C4, C42⋊C2, C22⋊Q8, C23×C4, C22×Q8, C23.7Q8, C23.65C23, C23.67C23, C23.11D4, C23.81C23, C23.83C23, C2×C42⋊C2, C2×C22⋊Q8, C23.525C24
Quotients: C1, C2, C22, D4, C23, C2×D4, C4○D4, C24, C22.D4, C22×D4, C2×C4○D4, 2+ 1+4, 2- 1+4, C2×C22.D4, C23.38C23, C22.31C24, C22.35C24, C22.36C24, C23.525C24

Smallest permutation representation of C23.525C24
►On 64 points
Generators in S64
(1 39)(2 40)(3 37)(4 38)(5 17)(6 18)(7 19)(8 20)(9 41)(10 42)(11 43)(12 44)(13 45)(14 46)(15 47)(16 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 62)(34 63)(35 64)(36 61)

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

(2 40)(4 38)(5 7)(6 20)(8 18)(10 42)(12 44)(13 15)(14 48)(16 46)(17 19)(22 50)(24 52)(26 54)(28 56)(29 31)(30 60)(32 58)(33 64)(34 36)(35 62)(45 47)(57 59)(61 63)

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

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

32 conjugacy classes

class 1 2A···2G2H2I4A4B4C4D4E···4N4O···4V
order12···22244444···44···4
size11···14422224···48···8

32 irreducible representations

dim1111111112244
type+++++++++++-
imageC1C2C2C2C2C2C2C2C2D4C4○D42+ 1+42- 1+4
kernelC23.525C24C23.7Q8C23.65C23C23.67C23C23.11D4C23.81C23C23.83C23C2×C42⋊C2C2×C22⋊Q8C22×C4C2×C4C22C22
# reps1122422114813

Matrix representation of C23.525C24 ►in GL8(𝔽5)

10000000
01000000
00100000
00010000
00004000
00000400
00000040
00000004
,
40000000
04000000
00100000
00010000
00001000
00000100
00000010
00000001
,
10000000
01000000
00400000
00040000
00001000
00000100
00000010
00000001
,
10000000
34000000
00400000
00040000
00001000
00000100
00000040
00003304
,
30000000
03000000
00430000
00010000
00000010
00003343
00001000
00004402
,
44000000
01000000
00100000
00440000
00001400
00002400
00002222
00000003
,
40000000
04000000
00400000
00040000
00003200
00000200
00001111
00001034

G:=sub<GL(8,GF(5))| [1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4],[4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[1,3,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1,0,0,3,0,0,0,0,0,1,0,3,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4],[3,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,3,1,0,0,0,0,0,0,0,0,0,3,1,4,0,0,0,0,0,3,0,4,0,0,0,0,1,4,0,0,0,0,0,0,0,3,0,2],[4,0,0,0,0,0,0,0,4,1,0,0,0,0,0,0,0,0,1,4,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1,2,2,0,0,0,0,0,4,4,2,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,2,3],[4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,3,0,1,1,0,0,0,0,2,2,1,0,0,0,0,0,0,0,1,3,0,0,0,0,0,0,1,4] >;
 

C23.525C24 in GAP, Magma, Sage, TeX 

C_2^3._{525}C_2^4
 
% in TeX
 
G:=Group("C2^3.525C2^4");
 
// GroupNames label
 
G:=SmallGroup(128,1357);
 
// by ID
 
G=gap.SmallGroup(128,1357);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,2,253,232,758,723,100,185,80]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=d^2=1,e^2=b,f^2=g^2=a,a*b=b*a,a*c=c*a,e*d*e^-1=a*d=d*a,a*e=e*a,g*f*g^-1=a*f=f*a,a*g=g*a,b*c=c*b,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,d*g=g*d,e*g=g*e>;
 
// generators/relations
 

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