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G = C24.542C23  order 128 = 27

23rd non-split extension by C24 of C23 acting via C23/C22=C2

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

Aliases: C24.542C23, C23.193C24, C22.322+ 1+4, C22.192- 1+4, C23.364(C2×D4), (C22×C4).360D4, C23.23D4⋊4C2, C22.84(C23×C4), C23.80(C22×C4), (C23×C4).40C22, C23.7Q8⋊11C2, C22.87(C22×D4), (C22×C4).458C23, (C2×C42).406C22, C24.3C22⋊12C2, C2.2(C22.29C24), (C22×D4).474C22, (C22×Q8).396C22, C23.67C23⋊12C2, C2.C42.32C22, C2.2(C23.38C23), C2.1(C22.31C24), C2.5(C23.33C23), (C2×C4○D4)⋊14C4, (C2×D4)⋊40(C2×C4), (C22×C4⋊C4)⋊7C2, (C2×Q8)⋊33(C2×C4), (C2×C4)⋊7(C22⋊C4), (C22×C4)⋊20(C2×C4), C4.65(C2×C22⋊C4), (C2×C42⋊C2)⋊9C2, (C2×C4).1392(C2×D4), (C22×C4○D4).7C2, (C2×C4⋊C4).805C22, (C2×C4).216(C22×C4), C22.16(C2×C22⋊C4), C2.15(C22×C22⋊C4), (C2×C22⋊C4).22C22, SmallGroup(128,1043)

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

Derived series
C1 — C22 — C24.542C23
Chief series
C1 — C2 — C22 — C23 — C22×C4 — C23×C4 — C22×C4○D4 — C24.542C23
Lower central
C1 — C22 — C24.542C23
Upper central
C1 — C23 — C24.542C23
Jennings
C1 — C23 — C24.542C23

Generators and relations for C24.542C23
 G = < a,b,c,d,e,f,g | a2=b2=c2=d2=f2=1, e2=d, g2=b, eae-1=faf=ab=ba, ac=ca, ad=da, ag=ga, bc=cb, bd=db, geg-1=be=eb, bf=fb, bg=gb, cd=dc, fef=ce=ec, cf=fc, cg=gc, de=ed, df=fd, dg=gd, fg=gf >

Subgroups: 780 in 440 conjugacy classes, 180 normal (18 characteristic)
C1, C2, C2, C2, C4, 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×D4, C2×Q8, C2×Q8, C4○D4, C24, C24, C2.C42, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C2×C4⋊C4, C42⋊C2, C23×C4, C23×C4, C22×D4, C22×D4, C22×Q8, C2×C4○D4, C2×C4○D4, C23.7Q8, C23.23D4, C24.3C22, C23.67C23, C22×C4⋊C4, C2×C42⋊C2, C22×C4○D4, C24.542C23
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22⋊C4, C22×C4, C2×D4, C24, C2×C22⋊C4, C23×C4, C22×D4, 2+ 1+4, 2- 1+4, C22×C22⋊C4, C23.33C23, C22.29C24, C23.38C23, C22.31C24, C24.542C23

Smallest permutation representation of C24.542C23
►On 64 points
Generators in S64
(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 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 47)(2 60)(3 45)(4 58)(5 26)(6 15)(7 28)(8 13)(9 17)(10 30)(11 19)(12 32)(14 36)(16 34)(18 22)(20 24)(21 29)(23 31)(25 35)(27 33)(37 41)(38 54)(39 43)(40 56)(42 62)(44 64)(46 50)(48 52)(49 57)(51 59)(53 61)(55 63)

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

44 conjugacy classes

class 1 2A···2G2H2I2J2K2L2M2N2O4A···4H4I···4AB
order12···2222222224···44···4
size11···1222244442···24···4

44 irreducible representations

dim111111111244
type++++++++++-
imageC1C2C2C2C2C2C2C2C4D42+ 1+42- 1+4
kernelC24.542C23C23.7Q8C23.23D4C24.3C22C23.67C23C22×C4⋊C4C2×C42⋊C2C22×C4○D4C2×C4○D4C22×C4C22C22
# reps1442211116822

Matrix representation of C24.542C23 ►in GL8(𝔽5)

40000000
04000000
00400000
00040000
00000200
00003000
00000003
00000020
,
10000000
01000000
00100000
00010000
00004000
00000400
00000040
00000004
,
40000000
04000000
00400000
00040000
00004000
00000400
00000040
00000004
,
40000000
04000000
00100000
00010000
00004000
00000400
00000040
00000004
,
01000000
40000000
00010000
00100000
00000010
00000001
00004000
00000400
,
40000000
01000000
00100000
00040000
00000100
00001000
00000004
00000040
,
10000000
01000000
00400000
00040000
00002000
00000200
00000030
00000003

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

C24.542C23 in GAP, Magma, Sage, TeX 

C_2^4._{542}C_2^3
 
% in TeX
 
G:=Group("C2^4.542C2^3");
 
// GroupNames label
 
G:=SmallGroup(128,1043);
 
// by ID
 
G=gap.SmallGroup(128,1043);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,2,448,253,758,219,675,80]);
 
// 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,e*a*e^-1=f*a*f=a*b=b*a,a*c=c*a,a*d=d*a,a*g=g*a,b*c=c*b,b*d=d*b,g*e*g^-1=b*e=e*b,b*f=f*b,b*g=g*b,c*d=d*c,f*e*f=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
 

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