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G = C23.211C24  order 128 = 27

64th central extension by C23 of C24

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

Aliases: C23.211C24, C24.548C23, C22.332- 1+4, C22.492+ 1+4, (C22×C4)⋊4Q8, C22⋊Q8⋊17C4, C22.5(C4×Q8), C23.94(C2×Q8), (C23×C4).49C22, C2.2(C23⋊2Q8), C23.8Q8.4C2, C23.226(C4○D4), C22.33(C22×Q8), C22.102(C23×C4), (C22×C4).476C23, (C2×C42).418C22, C23.126(C22×C4), C23.7Q8.27C2, (C22×Q8).87C22, C23.63C23⋊7C2, C2.5(C22.32C24), C23.65C23⋊15C2, C23.67C23⋊15C2, C2.17(C22.11C24), C2.C42.47C22, C2.3(C23.41C23), C2.5(C22.33C24), C2.16(C23.33C23), C4⋊C4⋊11(C2×C4), C2.11(C2×C4×Q8), (C2×Q8)⋊13(C2×C4), (C2×C4).161(C2×Q8), (C4×C22⋊C4).21C2, C22⋊C4.31(C2×C4), (C2×C4).31(C22×C4), C22.96(C2×C4○D4), (C2×C22⋊Q8).16C2, (C2×C4⋊C4).181C22, (C22×C4).304(C2×C4), (C2×C22⋊C4).429C22, (C2×C2.C42).19C2, SmallGroup(128,1061)

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

Derived series
C1 — C22 — C23.211C24
Chief series
C1 — C2 — C22 — C23 — C24 — C23×C4 — C2×C2.C42 — C23.211C24
Lower central
C1 — C22 — C23.211C24
Upper central
C1 — C23 — C23.211C24
Jennings
C1 — C23 — C23.211C24

Generators and relations for C23.211C24
 G = < a,b,c,d,e,f,g | a2=b2=c2=d2=1, e2=d, f2=dc=cd, g2=c, eae-1=ab=ba, ac=ca, ad=da, af=fa, ag=ga, bc=cb, bd=db, fef-1=be=eb, gfg-1=bf=fb, bg=gb, geg-1=ce=ec, cf=fc, cg=gc, de=ed, df=fd, dg=gd >

Subgroups: 460 in 264 conjugacy classes, 148 normal (30 characteristic)
C1, C2, C2, C4, C22, C22, C22, C2×C4, C2×C4, Q8, C23, C23, C23, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C22×C4, C2×Q8, C2×Q8, C24, C2.C42, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C2×C4⋊C4, C22⋊Q8, C23×C4, C22×Q8, C2×C2.C42, C4×C22⋊C4, C23.7Q8, C23.8Q8, C23.63C23, C23.65C23, C23.67C23, C2×C22⋊Q8, C23.211C24
Quotients: C1, C2, C4, C22, C2×C4, Q8, C23, C22×C4, C2×Q8, C4○D4, C24, C4×Q8, C23×C4, C22×Q8, C2×C4○D4, 2+ 1+4, 2- 1+4, C2×C4×Q8, C22.11C24, C23.33C23, C22.32C24, C22.33C24, C23⋊2Q8, C23.41C23, C23.211C24

Smallest permutation representation of C23.211C24
►On 64 points
Generators in S64
(1 3)(2 10)(4 12)(5 40)(6 8)(7 38)(9 11)(13 15)(14 44)(16 42)(17 19)(18 48)(20 46)(21 23)(22 52)(24 50)(25 27)(26 56)(28 54)(29 31)(30 60)(32 58)(33 35)(34 62)(36 64)(37 39)(41 43)(45 47)(49 51)(53 55)(57 59)(61 63)

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

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

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

44 conjugacy classes

class 1 2A···2G2H2I2J2K4A···4H4I···4AF
order12···222224···44···4
size11···122222···24···4

44 irreducible representations

dim11111111112244
type+++++++++-+-
imageC1C2C2C2C2C2C2C2C2C4Q8C4○D42+ 1+42- 1+4
kernelC23.211C24C2×C2.C42C4×C22⋊C4C23.7Q8C23.8Q8C23.63C23C23.65C23C23.67C23C2×C22⋊Q8C22⋊Q8C22×C4C23C22C22
# reps112124221164431

Matrix representation of C23.211C24 ►in GL8(𝔽5)

40000000
04000000
00100000
00010000
00004000
00000400
00000010
00000001
,
10000000
01000000
00100000
00010000
00004000
00000400
00000040
00000004
,
40000000
04000000
00400000
00040000
00001000
00000100
00000010
00000001
,
10000000
01000000
00400000
00040000
00004000
00000400
00000040
00000004
,
01000000
10000000
00030000
00300000
00000010
00000001
00004000
00000400
,
20000000
02000000
00400000
00040000
00003000
00000200
00000020
00000003
,
01000000
40000000
00010000
00400000
00000100
00001000
00000001
00000010

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

C23.211C24 in GAP, Magma, Sage, TeX 

C_2^3._{211}C_2^4
 
% in TeX
 
G:=Group("C2^3.211C2^4");
 
// GroupNames label
 
G:=SmallGroup(128,1061);
 
// by ID
 
G=gap.SmallGroup(128,1061);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,2,448,253,680,758,219,184,675]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=d^2=1,e^2=d,f^2=d*c=c*d,g^2=c,e*a*e^-1=a*b=b*a,a*c=c*a,a*d=d*a,a*f=f*a,a*g=g*a,b*c=c*b,b*d=d*b,f*e*f^-1=b*e=e*b,g*f*g^-1=b*f=f*b,b*g=g*b,g*e*g^-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>;
 
// generators/relations
 

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