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G = C23.443C24  order 128 = 27

160th central stem extension by C23 of C24

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

Aliases: C24.28C23, C23.443C24, C22.2312+ 1+4, (C2×D4)⋊36D4, (C22×C4)⋊30D4, C23.49(C2×D4), C23⋊2D4⋊20C2, C2.69(D4⋊5D4), C23.331(C4○D4), C23.11D4⋊41C2, C23.10D4⋊40C2, C23.23D4⋊54C2, C22.12(C4⋊D4), C2.13(C23⋊3D4), (C2×C42).548C22, (C22×C4).836C23, (C23×C4).112C22, C22.294(C22×D4), C24.C22⋊78C2, (C22×D4).164C22, C23.81C23⋊36C2, C2.C42.549C22, C2.30(C22.26C24), C2.53(C22.47C24), (C2×C4×D4)⋊46C2, (C2×C4⋊D4)⋊19C2, C2.35(C2×C4⋊D4), (C2×C4).1195(C2×D4), (C2×C4).384(C4○D4), (C2×C4⋊C4).867C22, C22.320(C2×C4○D4), (C2×C2.C42)⋊36C2, (C2×C22⋊C4).53C22, SmallGroup(128,1275)

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

Derived series
C1 — C23 — C23.443C24
Chief series
C1 — C2 — C22 — C23 — C24 — C23×C4 — C2×C2.C42 — C23.443C24
Lower central
C1 — C23 — C23.443C24
Upper central
C1 — C23 — C23.443C24
Jennings
C1 — C23 — C23.443C24

Generators and relations for C23.443C24
 G = < a,b,c,d,e,f,g | a2=b2=c2=d2=g2=1, e2=ca=ac, f2=b, ab=ba, 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: 772 in 362 conjugacy classes, 112 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, C2×D4, C24, C24, C2.C42, C2.C42, C2×C42, C2×C22⋊C4, C2×C22⋊C4, C2×C4⋊C4, C2×C4⋊C4, C4×D4, C4⋊D4, C23×C4, C23×C4, C22×D4, C22×D4, C2×C2.C42, C23.23D4, C24.C22, C23⋊2D4, C23.10D4, C23.11D4, C23.81C23, C2×C4×D4, C2×C4⋊D4, C2×C4⋊D4, C23.443C24
Quotients: C1, C2, C22, D4, C23, C2×D4, C4○D4, C24, C4⋊D4, C22×D4, C2×C4○D4, 2+ 1+4, C2×C4⋊D4, C22.26C24, C23⋊3D4, D4⋊5D4, C22.47C24, C23.443C24

Smallest permutation representation of C23.443C24
►On 64 points
Generators in S64
(1 10)(2 11)(3 12)(4 9)(5 37)(6 38)(7 39)(8 40)(13 52)(14 49)(15 50)(16 51)(17 46)(18 47)(19 48)(20 45)(21 43)(22 44)(23 41)(24 42)(25 54)(26 55)(27 56)(28 53)(29 60)(30 57)(31 58)(32 59)(33 64)(34 61)(35 62)(36 63)

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

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

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

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

38 conjugacy classes

class 1 2A···2G2H2I2J2K2L2M2N2O4A4B4C4D4E···4T4U4V
order12···22222222244444···444
size11···12222448822224···488

38 irreducible representations

dim111111111122224
type+++++++++++++
imageC1C2C2C2C2C2C2C2C2C2D4D4C4○D4C4○D42+ 1+4
kernelC23.443C24C2×C2.C42C23.23D4C24.C22C23⋊2D4C23.10D4C23.11D4C23.81C23C2×C4×D4C2×C4⋊D4C22×C4C2×D4C2×C4C23C22
# reps114211111344842

Matrix representation of C23.443C24 ►in GL6(𝔽5)

400000
040000
001000
000100
000010
000001
,
100000
010000
004000
000400
000040
000004
,
100000
010000
001000
000100
000040
000004
,
400000
410000
004300
000100
000001
000010
,
340000
020000
004000
000400
000002
000020
,
400000
040000
004300
001100
000001
000040
,
130000
040000
001000
000100
000040
000004

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

C23.443C24 in GAP, Magma, Sage, TeX 

C_2^3._{443}C_2^4
 
% in TeX
 
G:=Group("C2^3.443C2^4");
 
// GroupNames label
 
G:=SmallGroup(128,1275);
 
// by ID
 
G=gap.SmallGroup(128,1275);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,2,253,568,758,723,184,675]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=d^2=g^2=1,e^2=c*a=a*c,f^2=b,a*b=b*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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