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G = C24.572C23  order 128 = 27

53rd non-split extension by C24 of C23 acting via C23/C22=C2

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

Aliases: C24.572C23, C23.373C24, C22.1312- 1+4, C22.1782+ 1+4, C2.18(D4×Q8), C4⋊C4.333D4, C22⋊C4.6Q8, C2.55(D4⋊5D4), C23.119(C2×Q8), C2.17(D4⋊3Q8), C22⋊3(C42.C2), C23.236(C4○D4), C22.83(C22×Q8), (C23×C4).361C22, (C2×C42).513C22, (C22×C4).517C23, C22.253(C22×D4), C23.7Q8.41C2, C23.8Q8.16C2, C23.63C23⋊51C2, C23.65C23⋊63C2, C23.81C23⋊18C2, C23.83C23⋊12C2, C2.45(C22.19C24), C2.C42.129C22, C2.18(C22.33C24), C2.26(C22.46C24), (C2×C4).36(C2×Q8), (C2×C4).900(C2×D4), (C2×C42.C2)⋊6C2, C2.9(C2×C42.C2), (C4×C22⋊C4).44C2, (C22×C4⋊C4).35C2, (C2×C4).368(C4○D4), (C2×C4⋊C4).251C22, C22.250(C2×C4○D4), (C2×C22⋊C4).457C22, SmallGroup(128,1205)

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

Derived series
C1 — C23 — C24.572C23
Chief series
C1 — C2 — C22 — C23 — C22×C4 — C2×C42 — C4×C22⋊C4 — C24.572C23
Lower central
C1 — C23 — C24.572C23
Upper central
C1 — C23 — C24.572C23
Jennings
C1 — C23 — C24.572C23

Generators and relations for C24.572C23
 G = < a,b,c,d,e,f,g | a2=b2=c2=d2=1, e2=f2=b, g2=c, ab=ba, ac=ca, faf-1=ad=da, ae=ea, ag=ga, bc=cb, bd=db, geg-1=be=eb, bf=fb, bg=gb, cd=dc, fef-1=ce=ec, cf=fc, cg=gc, de=ed, gfg-1=df=fd, dg=gd >

Subgroups: 452 in 260 conjugacy classes, 112 normal (82 characteristic)
C1, C2, C2, C4, C22, C22, C22, C2×C4, C2×C4, C23, C23, C23, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C24, C2.C42, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C2×C4⋊C4, C42.C2, C23×C4, C4×C22⋊C4, C23.7Q8, C23.8Q8, C23.63C23, C23.65C23, C23.81C23, C23.83C23, C22×C4⋊C4, C2×C42.C2, C24.572C23
Quotients: C1, C2, C22, D4, Q8, C23, C2×D4, C2×Q8, C4○D4, C24, C42.C2, C22×D4, C22×Q8, C2×C4○D4, 2+ 1+4, 2- 1+4, C22.19C24, C2×C42.C2, C22.33C24, D4⋊5D4, D4×Q8, C22.46C24, D4⋊3Q8, C24.572C23

Smallest permutation representation of C24.572C23
►On 64 points
Generators in S64
(1 9)(2 10)(3 11)(4 12)(5 24)(6 21)(7 22)(8 23)(13 41)(14 42)(15 43)(16 44)(17 61)(18 62)(19 63)(20 64)(25 55)(26 56)(27 53)(28 54)(29 59)(30 60)(31 57)(32 58)(33 46)(34 47)(35 48)(36 45)(37 50)(38 51)(39 52)(40 49)

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

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

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

38 conjugacy classes

class 1 2A···2G2H2I2J2K4A4B4C4D4E···4V4W4X4Y4Z
order12···2222244444···44444
size11···1222222224···48888

38 irreducible representations

dim1111111111222244
type++++++++++-++-
imageC1C2C2C2C2C2C2C2C2C2Q8D4C4○D4C4○D42+ 1+42- 1+4
kernelC24.572C23C4×C22⋊C4C23.7Q8C23.8Q8C23.63C23C23.65C23C23.81C23C23.83C23C22×C4⋊C4C2×C42.C2C22⋊C4C4⋊C4C2×C4C23C22C22
# reps1114213111444811

Matrix representation of C24.572C23 ►in GL6(𝔽5)

400000
040000
001000
000400
000010
000001
,
100000
010000
001000
000100
000040
000004
,
400000
040000
001000
000100
000010
000001
,
100000
010000
004000
000400
000010
000001
,
010000
100000
001000
000100
000004
000010
,
100000
040000
000100
001000
000030
000003
,
200000
020000
004000
000100
000001
000010

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

C24.572C23 in GAP, Magma, Sage, TeX 

C_2^4._{572}C_2^3
 
% in TeX
 
G:=Group("C2^4.572C2^3");
 
// GroupNames label
 
G:=SmallGroup(128,1205);
 
// by ID
 
G=gap.SmallGroup(128,1205);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,2,112,253,232,758,723,100,675,136]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=d^2=1,e^2=f^2=b,g^2=c,a*b=b*a,a*c=c*a,f*a*f^-1=a*d=d*a,a*e=e*a,a*g=g*a,b*c=c*b,b*d=d*b,g*e*g^-1=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,d*e=e*d,g*f*g^-1=d*f=f*d,d*g=g*d>;
 
// generators/relations
 

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