Copied to
clipboard

G = C24.549C23  order 128 = 27

30th non-split extension by C24 of C23 acting via C23/C22=C2

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

Aliases: C24.549C23, C23.221C24, C22.412- 1+4, C22.592+ 1+4, D4⋊7(C22⋊C4), (C2×D4).340D4, C2.2(D4⋊5D4), C2.1(D4⋊6D4), C23.412(C2×D4), D4○2(C2.C42), (C23×C4).52C22, C23.7Q8⋊20C2, C23.34D4⋊14C2, C23.23D4⋊11C2, C22.112(C23×C4), (C2×C42).424C22, C23.127(C22×C4), C22.100(C22×D4), (C22×C4).1242C23, (C22×D4).608C22, (C22×Q8).401C22, C23.67C23⋊18C2, C2.C42.55C22, C2.22(C23.33C23), (C2×C4×D4)⋊10C2, (C2×C4○D4)⋊15C4, (C2×D4)⋊42(C2×C4), (C2×Q8)⋊34(C2×C4), C2.23(C4×C4○D4), (C2×C4)⋊16(C4○D4), (C4×C22⋊C4)⋊37C2, (C22×C4)⋊29(C2×C4), C4.25(C2×C22⋊C4), (C2×C4).1066(C2×D4), (C22×C4○D4).8C2, C22.1(C2×C22⋊C4), (C2×C4⋊C4).815C22, (C2×C4).489(C22×C4), C22.106(C2×C4○D4), C2.17(C22×C22⋊C4), (C2×C2.C42)⋊18C2, (C2×D4)○2(C2.C42), (C2×C22⋊C4).433C22, SmallGroup(128,1071)

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

Derived series
C1 — C22 — C24.549C23
Chief series
C1 — C2 — C22 — C23 — C24 — C23×C4 — C22×C4○D4 — C24.549C23
Lower central
C1 — C22 — C24.549C23
Upper central
C1 — C23 — C24.549C23
Jennings
C1 — C23 — C24.549C23

Generators and relations for C24.549C23
 G = < a,b,c,d,e,f,g | a2=b2=c2=d2=f2=1, e2=d, g2=b, faf=ab=ba, ac=ca, ad=da, ae=ea, ag=ga, bc=cb, bd=db, fef=be=eb, bf=fb, bg=gb, cd=dc, geg-1=ce=ec, cf=fc, cg=gc, de=ed, df=fd, dg=gd, fg=gf >

Subgroups: 812 in 464 conjugacy classes, 184 normal (20 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C2×C4, C2×C4, D4, D4, Q8, C23, C23, C23, C42, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C2×Q8, C4○D4, C24, C24, C2.C42, C2.C42, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C4×D4, C23×C4, C23×C4, C22×D4, C22×D4, C22×Q8, C2×C4○D4, C2×C4○D4, C2×C2.C42, C4×C22⋊C4, C23.7Q8, C23.34D4, C23.23D4, C23.67C23, C2×C4×D4, C22×C4○D4, C24.549C23
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22⋊C4, C22×C4, C2×D4, C4○D4, C24, C2×C22⋊C4, C23×C4, C22×D4, C2×C4○D4, 2+ 1+4, 2- 1+4, C22×C22⋊C4, C4×C4○D4, C23.33C23, D4⋊5D4, D4⋊6D4, C24.549C23

Smallest permutation representation of C24.549C23
►On 64 points
Generators in S64
(1 43)(2 44)(3 41)(4 42)(5 18)(6 19)(7 20)(8 17)(9 13)(10 14)(11 15)(12 16)(21 25)(22 26)(23 27)(24 28)(29 35)(30 36)(31 33)(32 34)(37 45)(38 46)(39 47)(40 48)(49 53)(50 54)(51 55)(52 56)(57 61)(58 62)(59 63)(60 64)

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

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

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

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

50 conjugacy classes

class 1 2A···2G2H···2O2P2Q4A···4P4Q···4AF
order12···22···2224···44···4
size11···12···2442···24···4

50 irreducible representations

dim11111111112244
type+++++++++++-
imageC1C2C2C2C2C2C2C2C2C4D4C4○D42+ 1+42- 1+4
kernelC24.549C23C2×C2.C42C4×C22⋊C4C23.7Q8C23.34D4C23.23D4C23.67C23C2×C4×D4C22×C4○D4C2×C4○D4C2×D4C2×C4C22C22
# reps121124221168811

Matrix representation of C24.549C23 ►in GL6(𝔽5)

400000
040000
001000
000100
000044
000001
,
100000
010000
001000
000100
000040
000004
,
400000
040000
004000
000400
000010
000001
,
400000
040000
001000
000100
000040
000004
,
010000
400000
000100
001000
000033
000002
,
100000
010000
001000
000100
000033
000042
,
400000
010000
001000
000400
000030
000003

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

C24.549C23 in GAP, Magma, Sage, TeX 

C_2^4._{549}C_2^3
 
% in TeX
 
G:=Group("C2^4.549C2^3");
 
// GroupNames label
 
G:=SmallGroup(128,1071);
 
// by ID
 
G=gap.SmallGroup(128,1071);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,2,448,253,758,100,346]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=d^2=f^2=1,e^2=d,g^2=b,f*a*f=a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*g=g*a,b*c=c*b,b*d=d*b,f*e*f=b*e=e*b,b*f=f*b,b*g=g*b,c*d=d*c,g*e*g^-1=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
◁