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G = C23.527C24  order 128 = 27

244th central stem extension by C23 of C24

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

Aliases: C23.527C24, C24.591C23, C22.2222- 1+4, C22.3042+ 1+4, (C22×C4)⋊12Q8, (C22×C4).405D4, C23.627(C2×D4), C23.100(C2×Q8), C23.243(C4○D4), C2.26(C23⋊3D4), (C22×C4).137C23, (C23×C4).429C22, C22.352(C22×D4), C22.43(C22⋊Q8), C23.7Q8.57C2, C23.8Q8.41C2, C22.132(C22×Q8), C23.34D4.23C2, (C22×Q8).154C22, C23.81C23⋊60C2, C23.83C23⋊59C2, C23.78C23⋊26C2, C2.C42.252C22, C2.38(C23.38C23), C2.38(C22.33C24), C2.19(C23.41C23), (C2×C4).386(C2×D4), (C2×C4).129(C2×Q8), C2.42(C2×C22⋊Q8), (C22×C4⋊C4).40C2, (C2×C22⋊Q8).39C2, (C2×C4⋊C4).890C22, C22.399(C2×C4○D4), (C2×C22⋊C4).216C22, SmallGroup(128,1359)

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

Derived series
C1 — C23 — C23.527C24
Chief series
C1 — C2 — C22 — C23 — C24 — C2×C22⋊C4 — C23.34D4 — C23.527C24
Lower central
C1 — C23 — C23.527C24
Upper central
C1 — C23 — C23.527C24
Jennings
C1 — C23 — C23.527C24

Generators and relations for C23.527C24
 G = < a,b,c,d,e,f,g | a2=b2=c2=g2=1, d2=f2=b, e2=a, ab=ba, ac=ca, ede-1=ad=da, ae=ea, gfg=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: 468 in 254 conjugacy classes, 108 normal (20 characteristic)
C1, C2, C2, C2, C4, C22, C22, C22, C2×C4, C2×C4, Q8, C23, C23, C23, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C22×C4, C2×Q8, C24, C2.C42, C2×C22⋊C4, C2×C4⋊C4, C2×C4⋊C4, C2×C4⋊C4, C22⋊Q8, C23×C4, C22×Q8, C23.7Q8, C23.34D4, C23.8Q8, C23.78C23, C23.81C23, C23.83C23, C22×C4⋊C4, C2×C22⋊Q8, C23.527C24
Quotients: C1, C2, C22, D4, Q8, C23, C2×D4, C2×Q8, C4○D4, C24, C22⋊Q8, C22×D4, C22×Q8, C2×C4○D4, 2+ 1+4, 2- 1+4, C2×C22⋊Q8, C23⋊3D4, C23.38C23, C22.33C24, C23.41C23, C23.527C24

Smallest permutation representation of C23.527C24
►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)

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

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

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

32 conjugacy classes

class 1 2A···2G2H2I2J2K4A···4L4M···4T
order12···222224···44···4
size11···122224···48···8

32 irreducible representations

dim11111111122244
type++++++++++-+-
imageC1C2C2C2C2C2C2C2C2D4Q8C4○D42+ 1+42- 1+4
kernelC23.527C24C23.7Q8C23.34D4C23.8Q8C23.78C23C23.81C23C23.83C23C22×C4⋊C4C2×C22⋊Q8C22×C4C22×C4C23C22C22
# reps11222421144422

Matrix representation of C23.527C24 ►in GL8(𝔽5)

10000000
01000000
00100000
00010000
00004000
00000400
00000040
00000004
,
40000000
04000000
00100000
00010000
00004000
00000400
00000040
00000004
,
10000000
01000000
00400000
00040000
00001000
00000100
00000010
00000001
,
01000000
40000000
00400000
00040000
00004020
00001241
00004010
00003013
,
40000000
04000000
00100000
00040000
00002000
00000200
00002030
00003203
,
30000000
02000000
00010000
00100000
00000100
00004000
00002232
00003102
,
40000000
04000000
00100000
00010000
00001000
00000400
00000010
00003024

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],[4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4],[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],[0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,1,4,3,0,0,0,0,0,2,0,0,0,0,0,0,2,4,1,1,0,0,0,0,0,1,0,3],[4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,2,0,2,3,0,0,0,0,0,2,0,2,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,3],[3,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,4,2,3,0,0,0,0,1,0,2,1,0,0,0,0,0,0,3,0,0,0,0,0,0,0,2,2],[4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,3,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1,2,0,0,0,0,0,0,0,4] >;
 

C23.527C24 in GAP, Magma, Sage, TeX 

C_2^3._{527}C_2^4
 
% in TeX
 
G:=Group("C2^3.527C2^4");
 
// GroupNames label
 
G:=SmallGroup(128,1359);
 
// by ID
 
G=gap.SmallGroup(128,1359);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,2,224,253,120,758,723,184,185]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=g^2=1,d^2=f^2=b,e^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=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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