Copied to
clipboard

G = C23.316C24  order 128 = 27

33rd central stem extension by C23 of C24

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

Aliases: C23.316C24, C24.251C23, C22.932- 1+4, C22.1302+ 1+4, C2.9D42, C4⋊C4⋊35D4, (C2×D4)⋊30D4, C4⋊4(C4⋊D4), (C22×C4)⋊24D4, C23.36(C2×D4), C22⋊2(C4⋊D4), C2.13(D4⋊6D4), C23.Q8⋊3C2, C2.11(Q8⋊5D4), (C22×C4).51C23, C23.7Q8⋊36C2, C23.326(C4○D4), C23.10D4⋊11C2, C23.23D4⋊28C2, (C2×C42).465C22, (C23×C4).334C22, C22.196(C22×D4), C24.3C22⋊29C2, (C22×D4).120C22, C23.65C23⋊36C2, C2.C42.80C22, C2.7(C22.31C24), C2.13(C22.47C24), (C2×C4×D4)⋊22C2, (C2×C4⋊D4)⋊6C2, (C2×C4).46(C2×D4), (C22×C4⋊C4)⋊18C2, C2.20(C2×C4⋊D4), (C2×C4).805(C4○D4), (C2×C4⋊C4).845C22, C22.195(C2×C4○D4), (C2×C22⋊C4).111C22, SmallGroup(128,1148)

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

Derived series
C1 — C23 — C23.316C24
Chief series
C1 — C2 — C22 — C23 — C22×C4 — C23×C4 — C2×C4×D4 — C23.316C24
Lower central
C1 — C23 — C23.316C24
Upper central
C1 — C23 — C23.316C24
Jennings
C1 — C23 — C23.316C24

Generators and relations for C23.316C24
 G = < a,b,c,d,e,f,g | a2=b2=c2=d2=1, e2=c, f2=g2=b, ab=ba, eae-1=ac=ca, faf-1=ad=da, ag=ga, bc=cb, bd=db, fef-1=geg-1=be=eb, bf=fb, bg=gb, cd=dc, ce=ec, cf=fc, cg=gc, de=ed, df=fd, dg=gd, fg=gf >

Subgroups: 804 in 394 conjugacy classes, 124 normal (42 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C2×C4, C2×C4, D4, C23, C23, C23, C42, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C22×C4, C2×D4, C2×D4, C24, C24, C2.C42, C2×C42, C2×C22⋊C4, C2×C22⋊C4, C2×C4⋊C4, C2×C4⋊C4, C2×C4⋊C4, C4×D4, C4⋊D4, C23×C4, C23×C4, C22×D4, C22×D4, C23.7Q8, C23.23D4, C23.65C23, C24.3C22, C23.10D4, C23.Q8, C22×C4⋊C4, C2×C4×D4, C2×C4⋊D4, C2×C4⋊D4, C23.316C24
Quotients: C1, C2, C22, D4, C23, C2×D4, C4○D4, C24, C4⋊D4, C22×D4, C2×C4○D4, 2+ 1+4, 2- 1+4, C2×C4⋊D4, C22.31C24, D42, D4⋊6D4, Q8⋊5D4, C22.47C24, C23.316C24

Smallest permutation representation of C23.316C24
►On 64 points
Generators in S64
(1 14)(2 13)(3 16)(4 15)(5 46)(6 45)(7 48)(8 47)(9 19)(10 18)(11 17)(12 20)(21 30)(22 29)(23 32)(24 31)(25 40)(26 39)(27 38)(28 37)(33 41)(34 44)(35 43)(36 42)(49 59)(50 58)(51 57)(52 60)(53 63)(54 62)(55 61)(56 64)

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

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

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

38 conjugacy classes

class 1 2A···2G2H2I2J2K2L2M2N2O4A4B4C4D4E···4T4U4V
order12···22222222244444···444
size11···12222448822224···488

38 irreducible representations

dim11111111112222244
type++++++++++++++-
imageC1C2C2C2C2C2C2C2C2C2D4D4D4C4○D4C4○D42+ 1+42- 1+4
kernelC23.316C24C23.7Q8C23.23D4C23.65C23C24.3C22C23.10D4C23.Q8C22×C4⋊C4C2×C4×D4C2×C4⋊D4C4⋊C4C22×C4C2×D4C2×C4C23C22C22
# reps11211221144444411

Matrix representation of C23.316C24 ►in GL6(𝔽5)

100000
040000
000100
001000
000003
000020
,
100000
010000
001000
000100
000040
000004
,
100000
010000
004000
000400
000040
000004
,
400000
040000
004000
000400
000010
000001
,
100000
010000
000400
001000
000030
000002
,
010000
100000
000200
003000
000001
000040
,
100000
010000
001000
000100
000001
000040

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

C23.316C24 in GAP, Magma, Sage, TeX 

C_2^3._{316}C_2^4
 
% in TeX
 
G:=Group("C2^3.316C2^4");
 
// GroupNames label
 
G:=SmallGroup(128,1148);
 
// by ID
 
G=gap.SmallGroup(128,1148);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,2,224,253,120,758,723,675,80]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=d^2=1,e^2=c,f^2=g^2=b,a*b=b*a,e*a*e^-1=a*c=c*a,f*a*f^-1=a*d=d*a,a*g=g*a,b*c=c*b,b*d=d*b,f*e*f^-1=g*e*g^-1=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,d*e=e*d,d*f=f*d,d*g=g*d,f*g=g*f>;
 
// generators/relations
 

׿
×
⋊
ℤ
𝔽
○
≀
ℚ
◁