Copied to
clipboard

G = C23.574C24  order 128 = 27

291st central stem extension by C23 of C24

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

Aliases: C24.59C23, C23.574C24, C22.3482+ 1+4, C22.2592- 1+4, C2.38D42, C4⋊C4.115D4, C22⋊C4.8D4, C23⋊Q8⋊38C2, C23.203(C2×D4), C2.56(D4⋊6D4), C2.42(Q8⋊5D4), C23.4Q8⋊41C2, C23.Q8⋊48C2, C23.8Q8⋊94C2, C23.11D4⋊74C2, (C23×C4).444C22, (C2×C42).634C22, (C22×C4).863C23, C22.383(C22×D4), C23.10D4.38C2, (C22×D4).215C22, (C22×Q8).173C22, C23.78C23⋊37C2, C24.C22⋊118C2, C23.65C23⋊114C2, C23.63C23⋊125C2, C2.C42.285C22, C2.7(C22.57C24), C2.68(C22.36C24), C2.52(C23.38C23), (C2×C4⋊Q8)⋊20C2, (C2×C4).84(C2×D4), (C2×C22⋊Q8)⋊33C2, (C2×C4).189(C4○D4), (C2×C4⋊C4).392C22, (C2×C4.4D4).30C2, C22.440(C2×C4○D4), (C2×C22⋊C4).245C22, (C2×C22.D4).22C2, SmallGroup(128,1406)

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

Derived series
C1 — C23 — C23.574C24
Chief series
C1 — C2 — C22 — C23 — C24 — C23×C4 — C2×C22.D4 — C23.574C24
Lower central
C1 — C23 — C23.574C24
Upper central
C1 — C23 — C23.574C24
Jennings
C1 — C23 — C23.574C24

Generators and relations for C23.574C24
 G = < a,b,c,d,e,f,g | a2=b2=c2=g2=1, d2=e2=f2=a, ab=ba, ac=ca, ede-1=ad=da, geg=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=abd, fg=gf >

Subgroups: 548 in 281 conjugacy classes, 104 normal (82 characteristic)
C1, C2, C2, C4, C22, C22, C2×C4, C2×C4, D4, Q8, C23, C23, C23, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×Q8, C24, C2.C42, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C22⋊Q8, C22.D4, C4.4D4, C4⋊Q8, C23×C4, C22×D4, C22×Q8, C23.8Q8, C23.63C23, C24.C22, C23.65C23, C23⋊Q8, C23.10D4, C23.78C23, C23.Q8, C23.11D4, C23.4Q8, C2×C22⋊Q8, C2×C22.D4, C2×C4.4D4, C2×C4⋊Q8, C23.574C24
Quotients: C1, C2, C22, D4, C23, C2×D4, C4○D4, C24, C22×D4, C2×C4○D4, 2+ 1+4, 2- 1+4, C23.38C23, C22.36C24, D42, D4⋊6D4, Q8⋊5D4, C22.57C24, C23.574C24

Smallest permutation representation of C23.574C24
►On 64 points
Generators in S64
(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 41)(2 42)(3 43)(4 44)(5 20)(6 17)(7 18)(8 19)(9 57)(10 58)(11 59)(12 60)(13 25)(14 26)(15 27)(16 28)(21 34)(22 35)(23 36)(24 33)(29 45)(30 46)(31 47)(32 48)(37 56)(38 53)(39 54)(40 55)(49 61)(50 62)(51 63)(52 64)

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

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

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

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

32 conjugacy classes

class 1 2A···2G2H2I2J4A···4P4Q···4U
order12···22224···44···4
size11···14484···48···8

32 irreducible representations

dim11111111111111122244
type++++++++++++++++++-
imageC1C2C2C2C2C2C2C2C2C2C2C2C2C2C2D4D4C4○D42+ 1+42- 1+4
kernelC23.574C24C23.8Q8C23.63C23C24.C22C23.65C23C23⋊Q8C23.10D4C23.78C23C23.Q8C23.11D4C23.4Q8C2×C22⋊Q8C2×C22.D4C2×C4.4D4C2×C4⋊Q8C22⋊C4C4⋊C4C2×C4C22C22
# reps11121111111111144413

Matrix representation of C23.574C24 ►in GL6(𝔽5)

100000
010000
001000
000100
000040
000004
,
400000
040000
001000
000100
000010
000001
,
100000
010000
004000
000400
000010
000001
,
410000
010000
001000
000100
000020
000003
,
400000
040000
004000
000100
000004
000010
,
400000
310000
000100
001000
000030
000003
,
400000
310000
004000
000400
000001
000010

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

C23.574C24 in GAP, Magma, Sage, TeX 

C_2^3._{574}C_2^4
 
% in TeX
 
G:=Group("C2^3.574C2^4");
 
// GroupNames label
 
G:=SmallGroup(128,1406);
 
// by ID
 
G=gap.SmallGroup(128,1406);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,2,112,253,120,758,723,100,1571,346]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=g^2=1,d^2=e^2=f^2=a,a*b=b*a,a*c=c*a,e*d*e^-1=a*d=d*a,g*e*g=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=a*b*d,f*g=g*f>;
 
// generators/relations
 

׿
×
⋊
ℤ
𝔽
○
≀
ℚ
◁