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G = C23.349C24  order 128 = 27

66th central stem extension by C23 of C24

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

Aliases: C23.349C24, C24.272C23, C22.1572+ 1+4, C22.1162- 1+4, C2.19D42, C4⋊C4⋊40D4, (C2×D4)⋊12Q8, C2.13(D4×Q8), C22⋊C4.127D4, C23.426(C2×D4), C22⋊3(C22⋊Q8), C2.26(D4⋊6D4), C23.118(C2×Q8), C23.Q8⋊9C2, C23.4Q8⋊7C2, C2.12(D4⋊3Q8), (C23×C4).79C22, C23.8Q8⋊41C2, C23.233(C4○D4), C22.75(C22×Q8), (C2×C42).492C22, (C22×C4).805C23, C22.229(C22×D4), C23.78C23⋊9C2, C23.23D4.18C2, (C22×D4).510C22, (C22×Q8).105C22, C23.63C23⋊36C2, C23.65C23⋊55C2, C23.81C23⋊13C2, C2.26(C22.19C24), C2.C42.106C22, C2.13(C22.33C24), (C2×C4×D4).48C2, (C2×C4).30(C2×Q8), (C22×C4⋊C4)⋊21C2, (C2×C4).329(C2×D4), (C2×C22⋊Q8)⋊11C2, C2.22(C2×C22⋊Q8), (C2×C4).363(C4○D4), (C2×C4⋊C4).231C22, C22.226(C2×C4○D4), (C2×C22⋊C4).128C22, SmallGroup(128,1181)

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

Derived series
C1 — C23 — C23.349C24
Chief series
C1 — C2 — C22 — C23 — C24 — C23×C4 — C22×C4⋊C4 — C23.349C24
Lower central
C1 — C23 — C23.349C24
Upper central
C1 — C23 — C23.349C24
Jennings
C1 — C23 — C23.349C24

Generators and relations for C23.349C24
 G = < a,b,c,d,e,f,g | a2=b2=c2=g2=1, d2=e2=f2=b, ab=ba, ac=ca, ede-1=gdg=ad=da, 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, eg=ge, fg=gf >

Subgroups: 596 in 324 conjugacy classes, 120 normal (82 characteristic)
C1, C2, C2, C4, C22, 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×D4, C2×Q8, C24, C2.C42, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C2×C4⋊C4, C4×D4, C22⋊Q8, C23×C4, C22×D4, C22×Q8, C23.8Q8, C23.23D4, C23.63C23, C23.65C23, C23.78C23, C23.Q8, C23.81C23, C23.4Q8, C22×C4⋊C4, C2×C4×D4, C2×C22⋊Q8, C23.349C24
Quotients: C1, C2, C22, D4, Q8, C23, C2×D4, C2×Q8, C4○D4, C24, C22⋊Q8, C22×D4, C22×Q8, C2×C4○D4, 2+ 1+4, 2- 1+4, C2×C22⋊Q8, C22.19C24, C22.33C24, D42, D4⋊6D4, D4×Q8, D4⋊3Q8, C23.349C24

Smallest permutation representation of C23.349C24
►On 64 points
Generators in S64
(1 9)(2 10)(3 11)(4 12)(5 39)(6 40)(7 37)(8 38)(13 34)(14 35)(15 36)(16 33)(17 54)(18 55)(19 56)(20 53)(21 58)(22 59)(23 60)(24 57)(25 50)(26 51)(27 52)(28 49)(29 46)(30 47)(31 48)(32 45)(41 64)(42 61)(43 62)(44 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 53)(2 54)(3 55)(4 56)(5 41)(6 42)(7 43)(8 44)(9 20)(10 17)(11 18)(12 19)(13 27)(14 28)(15 25)(16 26)(21 45)(22 46)(23 47)(24 48)(29 59)(30 60)(31 57)(32 58)(33 51)(34 52)(35 49)(36 50)(37 62)(38 63)(39 64)(40 61)

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

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

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

38 conjugacy classes

class 1 2A···2G2H2I2J2K2L2M4A4B4C4D4E···4T4U4V4W4X
order12···222222244444···44444
size11···122224422224···48888

38 irreducible representations

dim1111111111112222244
type++++++++++++++-+-
imageC1C2C2C2C2C2C2C2C2C2C2C2D4D4Q8C4○D4C4○D42+ 1+42- 1+4
kernelC23.349C24C23.8Q8C23.23D4C23.63C23C23.65C23C23.78C23C23.Q8C23.81C23C23.4Q8C22×C4⋊C4C2×C4×D4C2×C22⋊Q8C22⋊C4C4⋊C4C2×D4C2×C4C23C22C22
# reps1321111111124444411

Matrix representation of C23.349C24 ►in GL6(𝔽5)

100000
010000
004000
000400
000010
000001
,
400000
040000
004000
000400
000040
000004
,
400000
040000
001000
000100
000010
000001
,
300000
020000
002000
000300
000030
000012
,
200000
030000
000300
003000
000020
000002
,
010000
400000
000300
003000
000032
000002
,
100000
010000
000100
001000
000010
000001

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

C23.349C24 in GAP, Magma, Sage, TeX 

C_2^3._{349}C_2^4
 
% in TeX
 
G:=Group("C2^3.349C2^4");
 
// GroupNames label
 
G:=SmallGroup(128,1181);
 
// by ID
 
G=gap.SmallGroup(128,1181);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,2,224,253,232,758,723,184,675]);
 
// 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=b,a*b=b*a,a*c=c*a,e*d*e^-1=g*d*g=a*d=d*a,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,e*g=g*e,f*g=g*f>;
 
// generators/relations
 

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