Copied to
clipboard

G = C24.374C23  order 128 = 27

214th non-split extension by C24 of C23 acting via C23/C2=C22

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

Aliases: C24.374C23, C23.538C24, C22.2312- 1+4, C22.3142+ 1+4, C23⋊Q8⋊31C2, (C22×C4).406D4, C23.199(C2×D4), C23.69(C4○D4), C23.8Q8⋊88C2, C23.34D4⋊44C2, C23.11D4⋊62C2, C2.29(C23⋊3D4), (C2×C42).615C22, (C23×C4).434C22, (C22×C4).148C23, C22.363(C22×D4), C23.10D4.33C2, (C22×D4).545C22, (C22×Q8).157C22, C23.81C23⋊65C2, C23.78C23⋊30C2, C2.89(C22.19C24), C23.63C23⋊114C2, C2.C42.263C22, C2.51(C22.36C24), C2.42(C22.33C24), C2.44(C23.38C23), (C2×C4×D4).69C2, (C2×C4).397(C2×D4), (C2×C22⋊Q8)⋊29C2, (C2×C4).172(C4○D4), (C2×C4⋊C4).364C22, C22.410(C2×C4○D4), (C2×C22⋊C4).226C22, SmallGroup(128,1370)

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

Derived series
C1 — C23 — C24.374C23
Chief series
C1 — C2 — C22 — C23 — C22×C4 — C2×C4⋊C4 — C23.8Q8 — C24.374C23
Lower central
C1 — C23 — C24.374C23
Upper central
C1 — C23 — C24.374C23
Jennings
C1 — C23 — C24.374C23

Generators and relations for C24.374C23
 G = < a,b,c,d,e,f,g | a2=b2=c2=d2=e2=1, f2=b, g2=cb=bc, eae=ab=ba, faf-1=ac=ca, ad=da, ag=ga, bd=db, be=eb, gfg-1=bf=fb, bg=gb, cd=dc, ce=ec, cf=fc, cg=gc, fef-1=de=ed, df=fd, dg=gd, eg=ge >

Subgroups: 516 in 260 conjugacy classes, 96 normal (34 characteristic)
C1, C2, C2, C2, C4, C22, C22, C22, C2×C4, C2×C4, D4, Q8, C23, C23, C23, C42, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C22×C4, C2×D4, C2×Q8, C24, C2.C42, C2.C42, C2×C42, C2×C22⋊C4, C2×C22⋊C4, C2×C4⋊C4, C2×C4⋊C4, C4×D4, C22⋊Q8, C23×C4, C22×D4, C22×Q8, C23.34D4, C23.8Q8, C23.63C23, C23⋊Q8, C23.10D4, C23.78C23, C23.11D4, C23.11D4, C23.81C23, C2×C4×D4, C2×C22⋊Q8, C24.374C23
Quotients: C1, C2, C22, D4, C23, C2×D4, C4○D4, C24, C22×D4, C2×C4○D4, 2+ 1+4, 2- 1+4, C22.19C24, C23⋊3D4, C23.38C23, C22.33C24, C22.36C24, C24.374C23

Smallest permutation representation of C24.374C23
►On 64 points
Generators in S64
(1 59)(2 46)(3 57)(4 48)(5 52)(6 37)(7 50)(8 39)(9 45)(10 60)(11 47)(12 58)(13 54)(14 41)(15 56)(16 43)(17 34)(18 31)(19 36)(20 29)(21 38)(22 51)(23 40)(24 49)(25 44)(26 55)(27 42)(28 53)(30 62)(32 64)(33 61)(35 63)

(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 9)(2 10)(3 11)(4 12)(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 33)(30 34)(31 35)(32 36)(37 49)(38 50)(39 51)(40 52)(41 55)(42 56)(43 53)(44 54)(45 59)(46 60)(47 57)(48 58)

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

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

32 conjugacy classes

class 1 2A···2G2H2I2J2K4A4B4C4D4E···4L4M···4T
order12···2222244444···44···4
size11···1444422224···48···8

32 irreducible representations

dim1111111111122244
type+++++++++++++-
imageC1C2C2C2C2C2C2C2C2C2C2D4C4○D4C4○D42+ 1+42- 1+4
kernelC24.374C23C23.34D4C23.8Q8C23.63C23C23⋊Q8C23.10D4C23.78C23C23.11D4C23.81C23C2×C4×D4C2×C22⋊Q8C22×C4C2×C4C23C22C22
# reps1122121311144422

Matrix representation of C24.374C23 ►in GL8(𝔽5)

10000000
01000000
00100000
00140000
00000011
00000004
00001100
00000400
,
10000000
01000000
00100000
00010000
00004000
00000400
00000040
00000004
,
10000000
01000000
00400000
00040000
00004000
00000400
00000040
00000004
,
40000000
04000000
00100000
00010000
00004000
00000400
00000040
00000004
,
10000000
04000000
00400000
00040000
00001000
00000100
00000040
00000004
,
01000000
10000000
00210000
00230000
00000010
00000001
00004000
00000400
,
40000000
04000000
00200000
00020000
00003300
00004200
00000022
00000013

G:=sub<GL(8,GF(5))| [1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,4,0,0,0,0,1,0,0,0,0,0,0,0,1,4,0,0],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4],[4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4],[1,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4],[0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,2,2,0,0,0,0,0,0,1,3,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0],[4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,3,4,0,0,0,0,0,0,3,2,0,0,0,0,0,0,0,0,2,1,0,0,0,0,0,0,2,3] >;
 

C24.374C23 in GAP, Magma, Sage, TeX 

C_2^4._{374}C_2^3
 
% in TeX
 
G:=Group("C2^4.374C2^3");
 
// GroupNames label
 
G:=SmallGroup(128,1370);
 
// by ID
 
G=gap.SmallGroup(128,1370);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,2,253,758,723,100,185,192]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=d^2=e^2=1,f^2=b,g^2=c*b=b*c,e*a*e=a*b=b*a,f*a*f^-1=a*c=c*a,a*d=d*a,a*g=g*a,b*d=d*b,b*e=e*b,g*f*g^-1=b*f=f*b,b*g=g*b,c*d=d*c,c*e=e*c,c*f=f*c,c*g=g*c,f*e*f^-1=d*e=e*d,d*f=f*d,d*g=g*d,e*g=g*e>;
 
// generators/relations
 

׿
×
⋊
ℤ
𝔽
○
≀
ℚ
◁