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G = C23.457C24  order 128 = 27

174th central stem extension by C23 of C24

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

Aliases: C24.31C23, C23.457C24, C22.2422+ 1+4, C22⋊C4⋊29D4, C23.54(C2×D4), C23⋊Q8⋊17C2, C2.77(D4⋊5D4), C23⋊2D4.11C2, (C2×C42).63C22, C23.154(C4○D4), C23.11D4⋊44C2, C23.23D4⋊58C2, (C23×C4).115C22, C22.308(C22×D4), (C22×C4).1261C23, C24.C22⋊83C2, C24.3C22⋊59C2, (C22×D4).171C22, (C22×Q8).137C22, C23.63C23⋊86C2, C23.67C23⋊63C2, C2.24(C22.32C24), C2.48(C22.45C24), C2.C42.194C22, C2.37(C22.26C24), C2.18(C22.53C24), C2.79(C23.36C23), (C4×C22⋊C4)⋊87C2, (C2×C4).908(C2×D4), (C2×C4.4D4)⋊18C2, (C2×C4).524(C4○D4), (C2×C4⋊C4).309C22, C22.333(C2×C4○D4), (C2×C22.D4)⋊22C2, (C2×C22⋊C4).55C22, SmallGroup(128,1289)

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

Derived series
C1 — C23 — C23.457C24
Chief series
C1 — C2 — C22 — C23 — C22×C4 — C23×C4 — C4×C22⋊C4 — C23.457C24
Lower central
C1 — C23 — C23.457C24
Upper central
C1 — C23 — C23.457C24
Jennings
C1 — C23 — C23.457C24

Generators and relations for C23.457C24
 G = < a,b,c,d,e,f,g | a2=b2=c2=d2=f2=1, e2=abc, g2=b, ab=ba, ac=ca, ede-1=gdg-1=ad=da, ae=ea, af=fa, ag=ga, bc=cb, fdf=bd=db, be=eb, bf=fb, bg=gb, cd=dc, fef=ce=ec, cf=fc, cg=gc, eg=ge, fg=gf >

Subgroups: 612 in 291 conjugacy classes, 100 normal (82 characteristic)
C1, C2, C2, C4, 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.D4, C4.4D4, C23×C4, C22×D4, C22×Q8, C4×C22⋊C4, C23.23D4, C23.63C23, C24.C22, C24.3C22, C23.67C23, C23⋊2D4, C23⋊Q8, C23.11D4, C2×C22.D4, C2×C4.4D4, C23.457C24
Quotients: C1, C2, C22, D4, C23, C2×D4, C4○D4, C24, C22×D4, C2×C4○D4, 2+ 1+4, C23.36C23, C22.26C24, C22.32C24, D4⋊5D4, C22.45C24, C22.53C24, C23.457C24

Smallest permutation representation of C23.457C24
►On 64 points
Generators in S64
(1 55)(2 56)(3 53)(4 54)(5 37)(6 38)(7 39)(8 40)(9 43)(10 44)(11 41)(12 42)(13 45)(14 46)(15 47)(16 48)(17 49)(18 50)(19 51)(20 52)(21 26)(22 27)(23 28)(24 25)(29 58)(30 59)(31 60)(32 57)(33 64)(34 61)(35 62)(36 63)

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

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

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

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

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

38 conjugacy classes

class 1 2A···2G2H2I2J2K2L4A···4H4I···4V4W4X4Y
order12···2222224···44···4444
size11···1444482···24···4888

38 irreducible representations

dim1111111111112224
type++++++++++++++
imageC1C2C2C2C2C2C2C2C2C2C2C2D4C4○D4C4○D42+ 1+4
kernelC23.457C24C4×C22⋊C4C23.23D4C23.63C23C24.C22C24.3C22C23.67C23C23⋊2D4C23⋊Q8C23.11D4C2×C22.D4C2×C4.4D4C22⋊C4C2×C4C23C22
# reps12311111121141242

Matrix representation of C23.457C24 ►in GL6(𝔽5)

100000
010000
001000
000100
000040
000004
,
100000
010000
004000
000400
000040
000004
,
400000
040000
001000
000100
000010
000001
,
100000
010000
000300
002000
000010
000034
,
040000
100000
003000
000300
000044
000001
,
040000
400000
000100
001000
000044
000001
,
100000
010000
003000
000300
000033
000002

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],[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],[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],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,2,0,0,0,0,3,0,0,0,0,0,0,0,1,3,0,0,0,0,0,4],[0,1,0,0,0,0,4,0,0,0,0,0,0,0,3,0,0,0,0,0,0,3,0,0,0,0,0,0,4,0,0,0,0,0,4,1],[0,4,0,0,0,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,4,0,0,0,0,0,4,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,3,0,0,0,0,0,0,3,0,0,0,0,0,0,3,0,0,0,0,0,3,2] >;
 

C23.457C24 in GAP, Magma, Sage, TeX 

C_2^3._{457}C_2^4
 
% in TeX
 
G:=Group("C2^3.457C2^4");
 
// GroupNames label
 
G:=SmallGroup(128,1289);
 
// by ID
 
G=gap.SmallGroup(128,1289);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,2,253,792,758,723,675,136]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=d^2=f^2=1,e^2=a*b*c,g^2=b,a*b=b*a,a*c=c*a,e*d*e^-1=g*d*g^-1=a*d=d*a,a*e=e*a,a*f=f*a,a*g=g*a,b*c=c*b,f*d*f=b*d=d*b,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,e*g=g*e,f*g=g*f>;
 
// generators/relations
 

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