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G = C23.216C24  order 128 = 27

69th central extension by C23 of C24

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

Aliases: C23.216C24, C24.203C23, C22.542+ (1+4), C22.372- (1+4), C42⋊20(C2×C4), C42⋊2C2⋊1C4, C42⋊8C4⋊14C2, C23.15(C22×C4), (C23×C4).50C22, C23.8Q8.5C2, C22.107(C23×C4), (C2×C42).421C22, (C22×C4).481C23, C2.7(C22.32C24), C24.C22.4C2, C23.65C23⋊16C2, C23.63C23⋊10C2, C2.C42.51C22, C2.7(C22.33C24), C2.6(C22.35C24), C2.9(C22.36C24), C2.18(C23.33C23), (C4×C4⋊C4)⋊28C2, C4⋊C4⋊13(C2×C4), C2.19(C4×C4○D4), C22⋊C4.10(C2×C4), (C4×C22⋊C4).23C2, (C2×C4).36(C22×C4), (C2×C4).518(C4○D4), (C2×C4⋊C4).811C22, (C2×C42⋊2C2).2C2, C22.101(C2×C4○D4), (C2×C22⋊C4).431C22, SmallGroup(128,1066)

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

Derived series
C1 — C22 — C23.216C24
Chief series
C1 — C2 — C22 — C23 — C22×C4 — C2×C42 — C4×C4⋊C4 — C23.216C24
Lower central
C1 — C22 — C23.216C24
Upper central
C1 — C23 — C23.216C24
Jennings
C1 — C23 — C23.216C24

Subgroups: 396 in 238 conjugacy classes, 136 normal (82 characteristic)
C1, C2 [×7], C2 [×2], C4 [×22], C22 [×7], C22 [×10], C2×C4 [×16], C2×C4 [×38], C23, C23 [×2], C23 [×6], C42 [×4], C42 [×6], C22⋊C4 [×12], C22⋊C4 [×4], C4⋊C4 [×12], C4⋊C4 [×10], C22×C4 [×14], C22×C4 [×4], C24, C2.C42 [×12], C2×C42 [×6], C2×C22⋊C4 [×6], C2×C4⋊C4 [×10], C42⋊2C2 [×8], C23×C4, C4×C22⋊C4, C4×C4⋊C4 [×2], C42⋊8C4, C23.8Q8 [×2], C23.63C23 [×3], C24.C22 [×3], C23.65C23 [×2], C2×C42⋊2C2, C23.216C24

Quotients:
C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], C23 [×15], C22×C4 [×14], C4○D4 [×4], C24, C23×C4, C2×C4○D4 [×2], 2+ (1+4) [×2], 2- (1+4) [×2], C4×C4○D4, C23.33C23 [×2], C22.32C24, C22.33C24, C22.35C24, C22.36C24, C23.216C24

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

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

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

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

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

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

Matrix representation ►G ⊆ GL8(𝔽5)

10000000
01000000
00100000
00010000
00004000
00000400
00000040
00000004
,
40000000
04000000
00400000
00040000
00004000
00000400
00000040
00000004
,
40000000
04000000
00100000
00010000
00004000
00000400
00000040
00000004
,
04000000
10000000
00240000
00330000
00000300
00003000
00000002
00000020
,
30000000
03000000
00100000
00010000
00000020
00000002
00002000
00000200
,
01000000
10000000
00430000
00010000
00000010
00000001
00004000
00000400
,
20000000
02000000
00300000
00030000
00000100
00004000
00000001
00000040

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

44 conjugacy classes

class 1 2A···2G2H2I4A···4L4M···4AH
order12···2224···44···4
size11···1442···24···4

44 irreducible representations

dim1111111111244
type++++++++++-
imageC1C2C2C2C2C2C2C2C2C4C4○D42+ (1+4)2- (1+4)
kernelC23.216C24C4×C22⋊C4C4×C4⋊C4C42⋊8C4C23.8Q8C23.63C23C24.C22C23.65C23C2×C42⋊2C2C42⋊2C2C2×C4C22C22
# reps11212332116822

In GAP, Magma, Sage, TeX 

C_2^3._{216}C_2^4
 
% in TeX
 
G:=Group("C2^3.216C2^4");
 
// GroupNames label
 
G:=SmallGroup(128,1066);
 
// by ID
 
G=gap.SmallGroup(128,1066);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,2,448,253,456,758,219,100,675,136]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=1,d^2=e^2=c,f^2=a,g^2=b,a*b=b*a,a*c=c*a,e*d*e^-1=g*d*g^-1=a*d=d*a,f*e*f^-1=a*e=e*a,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,c*e=e*c,c*f=f*c,c*g=g*c,e*g=g*e,f*g=g*f>;
 
// generators/relations
 

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