Copied to
clipboard

G = C23.391C24  order 128 = 27

108th central stem extension by C23 of C24

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

Aliases: C23.391C24, C24.302C23, C22.1912+ (1+4), C22.1432- (1+4), C2.27(D42), C4⋊C4.229D4, C22⋊C4⋊25D4, C4⋊2(C4.4D4), C23.42(C2×D4), C23⋊Q8⋊16C2, C2.42(D4⋊6D4), C2.34(Q8⋊5D4), C23.7Q8⋊57C2, C23.10D4⋊36C2, (C2×C42).519C22, (C23×C4).376C22, (C22×C4).826C23, C22.271(C22×D4), C24.3C22⋊48C2, C24.C22⋊65C2, (C22×D4).147C22, (C22×Q8).116C22, C2.C42.540C22, C2.21(C22.26C24), C2.13(C22.49C24), C2.28(C22.36C24), (C4×C4⋊C4)⋊69C2, (C2×C4⋊Q8)⋊10C2, (C2×C4).62(C2×D4), (C4×C22⋊C4)⋊74C2, (C2×C4⋊D4).32C2, (C2×C4.4D4)⋊13C2, C2.17(C2×C4.4D4), (C2×C4).122(C4○D4), (C2×C4⋊C4).261C22, C22.268(C2×C4○D4), (C2×C22⋊C4).156C22, SmallGroup(128,1223)

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

Derived series
C1 — C23 — C23.391C24
Chief series
C1 — C2 — C22 — C23 — C22×C4 — C23×C4 — C4×C22⋊C4 — C23.391C24
Lower central
C1 — C23 — C23.391C24
Upper central
C1 — C23 — C23.391C24
Jennings
C1 — C23 — C23.391C24

Subgroups: 644 in 316 conjugacy classes, 112 normal (42 characteristic)
C1, C2 [×7], C2 [×4], C4 [×4], C4 [×16], C22 [×7], C22 [×24], C2×C4 [×16], C2×C4 [×32], D4 [×12], Q8 [×8], C23, C23 [×2], C23 [×20], C42 [×8], C22⋊C4 [×4], C22⋊C4 [×24], C4⋊C4 [×4], C4⋊C4 [×8], C22×C4 [×6], C22×C4 [×6], C22×C4 [×6], C2×D4 [×16], C2×Q8 [×10], C24, C24 [×2], C2.C42 [×2], C2.C42 [×4], C2×C42 [×2], C2×C42 [×2], C2×C22⋊C4 [×2], C2×C22⋊C4 [×12], C2×C4⋊C4 [×3], C2×C4⋊C4 [×2], C4⋊D4 [×4], C4.4D4 [×8], C4⋊Q8 [×4], C23×C4, C22×D4, C22×D4 [×2], C22×Q8 [×2], C4×C22⋊C4, C4×C4⋊C4, C23.7Q8, C24.C22 [×2], C24.3C22 [×2], C23⋊Q8 [×2], C23.10D4 [×2], C2×C4⋊D4, C2×C4.4D4 [×2], C2×C4⋊Q8, C23.391C24

Quotients:
C1, C2 [×15], C22 [×35], D4 [×8], C23 [×15], C2×D4 [×12], C4○D4 [×6], C24, C4.4D4 [×4], C22×D4 [×2], C2×C4○D4 [×3], 2+ (1+4), 2- (1+4), C2×C4.4D4, C22.26C24, C22.36C24, D42, D4⋊6D4, Q8⋊5D4, C22.49C24, C23.391C24

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

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

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

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

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

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

Matrix representation ►G ⊆ GL6(𝔽5)

100000
010000
004000
000400
000010
000001
,
400000
040000
004000
000400
000010
000001
,
400000
040000
001000
000100
000040
000004
,
330000
020000
004000
000100
000020
000002
,
330000
020000
000200
003000
000034
000032
,
400000
210000
000200
003000
000042
000001
,
400000
040000
000100
004000
000040
000004

G:=sub<GL(6,GF(5))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[3,0,0,0,0,0,3,2,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,2,0,0,0,0,0,0,2],[3,0,0,0,0,0,3,2,0,0,0,0,0,0,0,3,0,0,0,0,2,0,0,0,0,0,0,0,3,3,0,0,0,0,4,2],[4,2,0,0,0,0,0,1,0,0,0,0,0,0,0,3,0,0,0,0,2,0,0,0,0,0,0,0,4,0,0,0,0,0,2,1],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,0,4,0,0,0,0,1,0,0,0,0,0,0,0,4,0,0,0,0,0,0,4] >;
 

38 conjugacy classes

class 1 2A···2G2H2I2J2K4A···4H4I···4X4Y4Z
order12···222224···44···444
size11···144882···24···488

38 irreducible representations

dim1111111111122244
type++++++++++++++-
imageC1C2C2C2C2C2C2C2C2C2C2D4D4C4○D42+ (1+4)2- (1+4)
kernelC23.391C24C4×C22⋊C4C4×C4⋊C4C23.7Q8C24.C22C24.3C22C23⋊Q8C23.10D4C2×C4⋊D4C2×C4.4D4C2×C4⋊Q8C22⋊C4C4⋊C4C2×C4C22C22
# reps11112222121441211

In GAP, Magma, Sage, TeX 

C_2^3._{391}C_2^4
 
% in TeX
 
G:=Group("C2^3.391C2^4");
 
// GroupNames label
 
G:=SmallGroup(128,1223);
 
// by ID
 
G=gap.SmallGroup(128,1223);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,2,448,253,344,758,723,675,80]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=f^2=1,d^2=c,e^2=b*a=a*b,g^2=a,a*c=c*a,e*d*e^-1=g*d*g^-1=a*d=d*a,a*e=e*a,a*f=f*a,a*g=g*a,b*c=c*b,f*d*f=b*d=d*b,b*e=e*b,b*f=f*b,b*g=g*b,c*d=d*c,f*e*f=c*e=e*c,c*f=f*c,c*g=g*c,e*g=g*e,f*g=g*f>;
 
// generators/relations
 

׿
×
⋊
ℤ
𝔽
○
≀