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G = C23.398C24  order 128 = 27

115th central stem extension by C23 of C24

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

Aliases: C24.12C23, C23.398C24, C22.1492- 1+4, C22.1972+ 1+4, (C22×C4).385D4, C23.368(C2×D4), (C2×C42).47C22, C23.4Q8⋊15C2, C23.7Q8⋊58C2, C23.310(C4○D4), C23.11D4⋊30C2, (C23×C4).382C22, (C22×C4).523C23, C22.274(C22×D4), C24.C22⋊66C2, C23.23D4.27C2, (C22×D4).148C22, C23.63C23⋊65C2, C23.83C23⋊25C2, C23.81C23⋊25C2, C2.29(C22.45C24), C2.C42.150C22, C2.23(C22.26C24), C22.16(C22.D4), C2.11(C22.31C24), C2.41(C22.46C24), (C2×C4).1192(C2×D4), (C2×C42⋊C2)⋊30C2, (C2×C4).376(C4○D4), (C2×C4⋊C4).267C22, C22.275(C2×C4○D4), (C2×C2.C42)⋊34C2, (C2×C22⋊C4).46C22, C2.33(C2×C22.D4), (C2×C22.D4).16C2, SmallGroup(128,1230)

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

Derived series
C1 — C23 — C23.398C24
Chief series
C1 — C2 — C22 — C23 — C24 — C23×C4 — C2×C2.C42 — C23.398C24
Lower central
C1 — C23 — C23.398C24
Upper central
C1 — C23 — C23.398C24
Jennings
C1 — C23 — C23.398C24

Generators and relations for C23.398C24
 G = < a,b,c,d,e,f,g | a2=b2=c2=g2=1, d2=c, e2=a, 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: 516 in 273 conjugacy classes, 104 normal (34 characteristic)
C1, C2, C2, C2, C4, C22, C22, C22, C2×C4, C2×C4, D4, C23, C23, C23, C42, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C22×C4, C2×D4, C24, C2.C42, C2.C42, C2×C42, C2×C22⋊C4, C2×C22⋊C4, C2×C4⋊C4, C2×C4⋊C4, C42⋊C2, C22.D4, C23×C4, C23×C4, C22×D4, C2×C2.C42, C23.7Q8, C23.23D4, C23.63C23, C24.C22, C23.11D4, C23.81C23, C23.4Q8, C23.83C23, C2×C42⋊C2, C2×C22.D4, C23.398C24
Quotients: C1, C2, C22, D4, C23, C2×D4, C4○D4, C24, C22.D4, C22×D4, C2×C4○D4, 2+ 1+4, 2- 1+4, C2×C22.D4, C22.26C24, C22.31C24, C22.45C24, C22.46C24, C23.398C24

Smallest permutation representation of C23.398C24
►On 64 points
Generators in S64
(1 21)(2 22)(3 23)(4 24)(5 46)(6 47)(7 48)(8 45)(9 58)(10 59)(11 60)(12 57)(13 54)(14 55)(15 56)(16 53)(17 33)(18 34)(19 35)(20 36)(25 41)(26 42)(27 43)(28 44)(29 37)(30 38)(31 39)(32 40)(49 64)(50 61)(51 62)(52 63)

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

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

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

38 conjugacy classes

class 1 2A···2G2H2I2J2K2L4A4B4C4D4E···4V4W4X4Y
order12···22222244444···4444
size11···12222822224···4888

38 irreducible representations

dim11111111111122244
type++++++++++++++-
imageC1C2C2C2C2C2C2C2C2C2C2C2D4C4○D4C4○D42+ 1+42- 1+4
kernelC23.398C24C2×C2.C42C23.7Q8C23.23D4C23.63C23C24.C22C23.11D4C23.81C23C23.4Q8C23.83C23C2×C42⋊C2C2×C22.D4C22×C4C2×C4C23C22C22
# reps11222211111148811

Matrix representation of C23.398C24 ►in GL6(𝔽5)

400000
040000
004000
000400
000010
000001
,
400000
040000
001000
000100
000010
000001
,
100000
010000
001000
000100
000040
000004
,
400000
010000
000200
003000
000020
000002
,
030000
300000
000200
002000
000002
000030
,
030000
300000
001000
000100
000001
000010
,
010000
100000
000100
001000
000010
000001

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

C23.398C24 in GAP, Magma, Sage, TeX 

C_2^3._{398}C_2^4
 
% in TeX
 
G:=Group("C2^3.398C2^4");
 
// GroupNames label
 
G:=SmallGroup(128,1230);
 
// by ID
 
G=gap.SmallGroup(128,1230);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,2,448,253,120,758,723,184,675]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=g^2=1,d^2=c,e^2=a,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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