Copied to
clipboard

G = C23.595C24order 128 = 27

312nd central stem extension by C23 of C24

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

Aliases: C23.595C24, C24.402C23, C22.3692+ 1+4, C22.2752- 1+4, C22⋊C4.15D4, C23.68(C2×D4), C2.61(D46D4), C2.100(D45D4), C23.Q860C2, C23.7Q889C2, C23.173(C4○D4), C23.11D484C2, (C23×C4).458C22, (C2×C42).648C22, (C22×C4).559C23, C23.8Q8107C2, C22.404(C22×D4), C23.23D4.52C2, C23.10D4.42C2, (C22×D4).232C22, C23.81C2385C2, C23.83C2379C2, C24.C22127C2, C2.65(C22.32C24), C23.65C23120C2, C2.C42.302C22, C2.45(C22.31C24), C2.65(C22.33C24), C2.13(C22.57C24), C2.83(C22.46C24), (C2×C4).421(C2×D4), (C2×C422C2)⋊19C2, (C2×C4).424(C4○D4), (C2×C4⋊C4).409C22, C22.457(C2×C4○D4), (C2×C22⋊C4).262C22, (C2×C22.D4).26C2, SmallGroup(128,1427)

Series: Derived Chief Lower central Upper central Jennings

C1C23 — C23.595C24
C1C2C22C23C22×C4C23×C4C23.7Q8 — C23.595C24
C1C23 — C23.595C24
C1C23 — C23.595C24
C1C23 — C23.595C24

Generators and relations for C23.595C24
 G = < a,b,c,d,e,f,g | a2=b2=c2=d2=1, e2=ba=ab, f2=g2=a, ac=ca, ede-1=ad=da, geg-1=ae=ea, 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, gdg-1=abd, fg=gf >

Subgroups: 500 in 252 conjugacy classes, 96 normal (82 characteristic)
C1, C2 [×7], C2 [×4], C4 [×16], C22 [×7], C22 [×20], C2×C4 [×6], C2×C4 [×44], D4 [×4], C23, C23 [×4], C23 [×12], C42 [×2], C22⋊C4 [×4], C22⋊C4 [×15], C4⋊C4 [×17], C22×C4 [×13], C22×C4 [×9], C2×D4 [×5], C24 [×2], C2.C42 [×10], C2×C42, C2×C22⋊C4 [×10], C2×C4⋊C4 [×11], C22.D4 [×4], C422C2 [×4], C23×C4 [×2], C22×D4, C23.7Q8 [×2], C23.8Q8 [×2], C23.23D4, C24.C22, C23.65C23, C23.10D4, C23.Q8, C23.11D4, C23.81C23 [×2], C23.83C23, C2×C22.D4, C2×C422C2, C23.595C24
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], C2×D4 [×6], C4○D4 [×4], C24, C22×D4, C2×C4○D4 [×2], 2+ 1+4 [×2], 2- 1+4 [×2], C22.31C24, C22.32C24, C22.33C24, D45D4, D46D4, C22.46C24, C22.57C24, C23.595C24

Smallest permutation representation of C23.595C24
On 64 points
Generators in S64
(1 9)(2 10)(3 11)(4 12)(5 21)(6 22)(7 23)(8 24)(13 27)(14 28)(15 25)(16 26)(17 64)(18 61)(19 62)(20 63)(29 39)(30 40)(31 37)(32 38)(33 51)(34 52)(35 49)(36 50)(41 46)(42 47)(43 48)(44 45)(53 58)(54 59)(55 60)(56 57)
(1 11)(2 12)(3 9)(4 10)(5 23)(6 24)(7 21)(8 22)(13 25)(14 26)(15 27)(16 28)(17 62)(18 63)(19 64)(20 61)(29 37)(30 38)(31 39)(32 40)(33 49)(34 50)(35 51)(36 52)(41 48)(42 45)(43 46)(44 47)(53 60)(54 57)(55 58)(56 59)
(1 37)(2 38)(3 39)(4 40)(5 60)(6 57)(7 58)(8 59)(9 31)(10 32)(11 29)(12 30)(13 51)(14 52)(15 49)(16 50)(17 44)(18 41)(19 42)(20 43)(21 55)(22 56)(23 53)(24 54)(25 35)(26 36)(27 33)(28 34)(45 64)(46 61)(47 62)(48 63)
(1 48)(2 44)(3 46)(4 42)(5 52)(6 35)(7 50)(8 33)(9 43)(10 45)(11 41)(12 47)(13 54)(14 60)(15 56)(16 58)(17 38)(18 29)(19 40)(20 31)(21 34)(22 49)(23 36)(24 51)(25 57)(26 53)(27 59)(28 55)(30 62)(32 64)(37 63)(39 61)
(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 25 9 15)(2 36 10 50)(3 27 11 13)(4 34 12 52)(5 45 21 44)(6 61 22 18)(7 47 23 42)(8 63 24 20)(14 40 28 30)(16 38 26 32)(17 60 64 55)(19 58 62 53)(29 51 39 33)(31 49 37 35)(41 57 46 56)(43 59 48 54)
(1 15 9 25)(2 26 10 16)(3 13 11 27)(4 28 12 14)(5 17 21 64)(6 61 22 18)(7 19 23 62)(8 63 24 20)(29 33 39 51)(30 52 40 34)(31 35 37 49)(32 50 38 36)(41 57 46 56)(42 53 47 58)(43 59 48 54)(44 55 45 60)

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

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

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

32 conjugacy classes

class 1 2A···2G2H2I2J2K4A···4N4O···4T
order12···222224···44···4
size11···144444···48···8

32 irreducible representations

dim111111111111122244
type+++++++++++++++-
imageC1C2C2C2C2C2C2C2C2C2C2C2C2D4C4○D4C4○D42+ 1+42- 1+4
kernelC23.595C24C23.7Q8C23.8Q8C23.23D4C24.C22C23.65C23C23.10D4C23.Q8C23.11D4C23.81C23C23.83C23C2×C22.D4C2×C422C2C22⋊C4C2×C4C23C22C22
# reps122111111211144422

Matrix representation of C23.595C24 in GL6(𝔽5)

100000
010000
004000
000400
000010
000001
,
100000
010000
001000
000100
000040
000004
,
400000
040000
001000
000100
000010
000001
,
400000
040000
000400
004000
000023
000043
,
010000
100000
000100
004000
000030
000003
,
400000
010000
002000
000200
000010
000024
,
400000
040000
003000
000200
000010
000024

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

C23.595C24 in GAP, Magma, Sage, TeX

C_2^3._{595}C_2^4
% in TeX

G:=Group("C2^3.595C2^4");
// GroupNames label

G:=SmallGroup(128,1427);
// by ID

G=gap.SmallGroup(128,1427);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,253,344,758,723,100,1571,346,80]);
// Polycyclic

G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=d^2=1,e^2=b*a=a*b,f^2=g^2=a,a*c=c*a,e*d*e^-1=a*d=d*a,g*e*g^-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,f*e*f^-1=c*e=e*c,c*f=f*c,c*g=g*c,g*d*g^-1=a*b*d,f*g=g*f>;
// generators/relations

׿
×
𝔽