Copied to
clipboard

G = C23.395C24  order 128 = 27

112nd central stem extension by C23 of C24

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

Aliases: C23.395C24, C24.305C23, C22.1952+ (1+4), C22.1472- (1+4), C42⋊5C4⋊11C2, C42⋊8C4⋊28C2, C23.43(C4○D4), (C22×C4).75C23, (C2×C42).44C22, C4.27(C42⋊2C2), (C23×C4).379C22, C23.11D4.8C2, C23.7Q8.46C2, C23.83C23⋊22C2, C23.65C23⋊71C2, C24.C22.21C2, C2.C42.147C22, C2.39(C22.46C24), C2.31(C22.36C24), C2.34(C22.47C24), C2.51(C23.36C23), C2.14(C22.49C24), (C4×C4⋊C4)⋊72C2, (C4×C22⋊C4).48C2, (C2×C4).734(C4○D4), (C2×C4⋊C4).857C22, C2.14(C2×C42⋊2C2), C22.272(C2×C4○D4), (C2×C22⋊C4).159C22, SmallGroup(128,1227)

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

Derived series
C1 — C23 — C23.395C24
Chief series
C1 — C2 — C22 — C23 — C22×C4 — C23×C4 — C4×C22⋊C4 — C23.395C24
Lower central
C1 — C23 — C23.395C24
Upper central
C1 — C23 — C23.395C24
Jennings
C1 — C23 — C23.395C24

Subgroups: 372 in 206 conjugacy classes, 96 normal (42 characteristic)
C1, C2 [×7], C2 [×2], C4 [×4], C4 [×14], C22 [×7], C22 [×10], C2×C4 [×8], C2×C4 [×42], C23, C23 [×2], C23 [×6], C42 [×8], C22⋊C4 [×10], C4⋊C4 [×12], C22×C4 [×6], C22×C4 [×8], C22×C4 [×6], C24, C2.C42 [×2], C2.C42 [×14], C2×C42 [×2], C2×C42 [×2], C2×C22⋊C4 [×2], C2×C22⋊C4 [×4], C2×C4⋊C4 [×4], C2×C4⋊C4 [×4], C23×C4, C4×C22⋊C4, C4×C4⋊C4, C23.7Q8 [×2], C42⋊8C4, C42⋊5C4 [×2], C24.C22 [×2], C23.65C23 [×2], C23.11D4 [×2], C23.83C23 [×2], C23.395C24

Quotients:
C1, C2 [×15], C22 [×35], C23 [×15], C4○D4 [×10], C24, C42⋊2C2 [×4], C2×C4○D4 [×5], 2+ (1+4), 2- (1+4), C2×C42⋊2C2, C23.36C23, C22.36C24, C22.46C24 [×2], C22.47C24, C22.49C24, C23.395C24

Generators and relations
 G = < a,b,c,d,e,f,g | a2=b2=c2=f2=1, d2=abc, e2=b, g2=a, ab=ba, 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 55)(2 56)(3 53)(4 54)(5 61)(6 62)(7 63)(8 64)(9 43)(10 44)(11 41)(12 42)(13 45)(14 46)(15 47)(16 48)(17 49)(18 50)(19 51)(20 52)(21 58)(22 59)(23 60)(24 57)(25 32)(26 29)(27 30)(28 31)(33 40)(34 37)(35 38)(36 39)

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

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

(2 41)(4 43)(5 18)(6 64)(7 20)(8 62)(9 54)(11 56)(14 32)(16 30)(17 51)(19 49)(21 34)(22 57)(23 36)(24 59)(25 46)(27 48)(33 38)(35 40)(37 58)(39 60)(50 61)(52 63)

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

Matrix representation ►G ⊆ GL6(𝔽5)

100000
010000
004000
000400
000010
000001
,
400000
040000
004000
000400
000040
000004
,
100000
010000
001000
000100
000040
000004
,
430000
110000
000100
001000
000003
000020
,
200000
020000
003000
000200
000001
000040
,
100000
440000
001000
000400
000010
000004
,
400000
040000
002000
000300
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],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[4,1,0,0,0,0,3,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,2,0,0,0,0,3,0],[2,0,0,0,0,0,0,2,0,0,0,0,0,0,3,0,0,0,0,0,0,2,0,0,0,0,0,0,0,4,0,0,0,0,1,0],[1,4,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,4],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,2,0,0,0,0,0,0,3,0,0,0,0,0,0,4,0,0,0,0,0,0,4] >;
 

38 conjugacy classes

class 1 2A···2G2H2I4A···4H4I···4X4Y4Z4AA4AB
order12···2224···44···44444
size11···1442···24···48888

38 irreducible representations

dim11111111112244
type+++++++++++-
imageC1C2C2C2C2C2C2C2C2C2C4○D4C4○D42+ (1+4)2- (1+4)
kernelC23.395C24C4×C22⋊C4C4×C4⋊C4C23.7Q8C42⋊8C4C42⋊5C4C24.C22C23.65C23C23.11D4C23.83C23C2×C4C23C22C22
# reps111212222216411

In GAP, Magma, Sage, TeX 

C_2^3._{395}C_2^4
 
% in TeX
 
G:=Group("C2^3.395C2^4");
 
// GroupNames label
 
G:=SmallGroup(128,1227);
 
// by ID
 
G=gap.SmallGroup(128,1227);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,2,784,253,232,758,723,675,80]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=f^2=1,d^2=a*b*c,e^2=b,g^2=a,a*b=b*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
○
≀