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G = C23.388C24  order 128 = 27

105th central stem extension by C23 of C24

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

Aliases: C24.10C23, C23.388C24, C22.1412- 1+4, C22.1892+ 1+4, C22⋊C4.72D4, C42⋊8C4⋊25C2, C23⋊Q8⋊15C2, C23.428(C2×D4), C2.61(D4⋊5D4), C2.40(D4⋊6D4), (C23×C4).96C22, C23.308(C4○D4), C23.11D4⋊28C2, C23.34D4⋊32C2, (C2×C42).516C22, (C22×C4).522C23, C22.268(C22×D4), C24.C22⋊63C2, C23.23D4.26C2, C22.14(C4.4D4), (C22×D4).145C22, (C22×Q8).114C22, C23.67C23⋊48C2, C23.83C23⋊18C2, C2.27(C22.45C24), C2.C42.141C22, C2.20(C22.33C24), C2.46(C23.36C23), C2.35(C22.46C24), (C2×C4).59(C2×D4), (C4×C22⋊C4)⋊72C2, (C2×C22⋊Q8)⋊18C2, C2.14(C2×C4.4D4), (C2×C4).375(C4○D4), (C2×C4⋊C4).258C22, C22.265(C2×C4○D4), (C2×C2.C42)⋊33C2, (C2×C22⋊C4).154C22, (C2×C22.D4).15C2, SmallGroup(128,1220)

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

Derived series
C1 — C23 — C23.388C24
Chief series
C1 — C2 — C22 — C23 — C24 — C23×C4 — C2×C2.C42 — C23.388C24
Lower central
C1 — C23 — C23.388C24
Upper central
C1 — C23 — C23.388C24
Jennings
C1 — C23 — C23.388C24

Generators and relations for C23.388C24
 G = < a,b,c,d,e,f,g | a2=b2=c2=g2=1, d2=c, e2=b, f2=a, 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: 532 in 273 conjugacy classes, 104 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, C22×C4, C22×C4, C2×D4, C2×Q8, C24, C2.C42, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C22⋊Q8, C22.D4, C23×C4, C22×D4, C22×Q8, C2×C2.C42, C4×C22⋊C4, C23.34D4, C42⋊8C4, C23.23D4, C24.C22, C23.67C23, C23⋊Q8, C23.11D4, C23.83C23, C2×C22⋊Q8, C2×C22.D4, C23.388C24
Quotients: C1, C2, C22, D4, C23, C2×D4, C4○D4, C24, C4.4D4, C22×D4, C2×C4○D4, 2+ 1+4, 2- 1+4, C2×C4.4D4, C23.36C23, C22.33C24, D4⋊5D4, D4⋊6D4, C22.45C24, C22.46C24, C23.388C24

Smallest permutation representation of C23.388C24
►On 64 points
Generators in S64
(1 19)(2 20)(3 17)(4 18)(5 46)(6 47)(7 48)(8 45)(9 58)(10 59)(11 60)(12 57)(13 54)(14 55)(15 56)(16 53)(21 35)(22 36)(23 33)(24 34)(25 39)(26 40)(27 37)(28 38)(29 43)(30 44)(31 41)(32 42)(49 64)(50 61)(51 62)(52 63)

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

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

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

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

dim111111111111122244
type+++++++++++++++-
imageC1C2C2C2C2C2C2C2C2C2C2C2C2D4C4○D4C4○D42+ 1+42- 1+4
kernelC23.388C24C2×C2.C42C4×C22⋊C4C23.34D4C42⋊8C4C23.23D4C24.C22C23.67C23C23⋊Q8C23.11D4C23.83C23C2×C22⋊Q8C2×C22.D4C22⋊C4C2×C4C23C22C22
# reps111112211121148811

Matrix representation of C23.388C24 ►in GL6(𝔽5)

100000
010000
001000
000100
000040
000004
,
400000
040000
001000
000100
000040
000004
,
100000
010000
004000
000400
000010
000001
,
320000
120000
002000
000200
000010
000024
,
300000
030000
004400
000100
000032
000002
,
140000
040000
002200
001300
000023
000003
,
400000
040000
004000
000400
000041
000001

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

C23.388C24 in GAP, Magma, Sage, TeX 

C_2^3._{388}C_2^4
 
% in TeX
 
G:=Group("C2^3.388C2^4");
 
// GroupNames label
 
G:=SmallGroup(128,1220);
 
// by ID
 
G=gap.SmallGroup(128,1220);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,2,448,253,232,758,723,100,675]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=g^2=1,d^2=c,e^2=b,f^2=a,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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