Copied to
clipboard

G = C24.573C23  order 128 = 27

54th non-split extension by C24 of C23 acting via C23/C22=C2

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

Aliases: C24.573C23, C23.381C24, C22.1372- 1+4, C22.1842+ 1+4, C4⋊C4⋊47D4, C2.60(D4⋊5D4), C2.32(Q8⋊5D4), (C2×C42).39C22, (C22×C4).67C23, C23.8Q8⋊60C2, C23.Q8⋊22C2, C23.7Q8⋊52C2, C22⋊3(C42⋊2C2), C23.237(C4○D4), C23.11D4⋊23C2, (C23×C4).368C22, C22.261(C22×D4), C24.C22⋊60C2, C23.23D4.24C2, (C22×D4).144C22, C23.63C23⋊58C2, C23.83C23⋊15C2, C2.53(C22.19C24), C2.C42.136C22, C2.19(C22.33C24), C2.29(C22.47C24), C2.31(C22.46C24), (C4×C22⋊C4)⋊14C2, (C22×C4⋊C4)⋊22C2, (C2×C4).905(C2×D4), (C2×C42⋊2C2)⋊6C2, (C2×C4).373(C4○D4), (C2×C4⋊C4).855C22, C2.11(C2×C42⋊2C2), C22.258(C2×C4○D4), (C2×C22⋊C4).149C22, SmallGroup(128,1213)

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

Derived series
C1 — C23 — C24.573C23
Chief series
C1 — C2 — C22 — C23 — C22×C4 — C2×C42 — C4×C22⋊C4 — C24.573C23
Lower central
C1 — C23 — C24.573C23
Upper central
C1 — C23 — C24.573C23
Jennings
C1 — C23 — C24.573C23

Generators and relations for C24.573C23
 G = < a,b,c,d,e,f,g | a2=b2=c2=1, d2=f2=a, e2=g2=ba=ab, ac=ca, ede-1=gdg-1=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 (82 characteristic)
C1, C2, C2, C4, C22, C22, C22, C2×C4, C2×C4, D4, C23, C23, C23, C42, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C24, C2.C42, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C2×C4⋊C4, C42⋊2C2, C23×C4, C22×D4, C4×C22⋊C4, C23.7Q8, C23.8Q8, C23.23D4, C23.63C23, C24.C22, C23.Q8, C23.11D4, C23.83C23, C22×C4⋊C4, C2×C42⋊2C2, C24.573C23
Quotients: C1, C2, C22, D4, C23, C2×D4, C4○D4, C24, C42⋊2C2, C22×D4, C2×C4○D4, 2+ 1+4, 2- 1+4, C22.19C24, C2×C42⋊2C2, C22.33C24, D4⋊5D4, Q8⋊5D4, C22.46C24, C22.47C24, C24.573C23

Smallest permutation representation of C24.573C23
►On 64 points
Generators in S64
(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 43)(2 44)(3 41)(4 42)(5 39)(6 40)(7 37)(8 38)(9 50)(10 51)(11 52)(12 49)(13 22)(14 23)(15 24)(16 21)(17 45)(18 46)(19 47)(20 48)(25 54)(26 55)(27 56)(28 53)(29 60)(30 57)(31 58)(32 59)(33 62)(34 63)(35 64)(36 61)

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

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

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

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

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
kernelC24.573C23C4×C22⋊C4C23.7Q8C23.8Q8C23.23D4C23.63C23C24.C22C23.Q8C23.11D4C23.83C23C22×C4⋊C4C2×C42⋊2C2C4⋊C4C2×C4C23C22C22
# reps111132111211441211

Matrix representation of C24.573C23 ►in GL6(𝔽5)

400000
040000
001000
000100
000010
000001
,
100000
010000
001000
000100
000040
000004
,
100000
010000
004000
000400
000010
000001
,
020000
200000
001000
000100
000044
000001
,
010000
400000
004300
000100
000030
000003
,
300000
030000
004000
001100
000033
000042
,
010000
400000
004000
000400
000030
000003

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

C24.573C23 in GAP, Magma, Sage, TeX 

C_2^4._{573}C_2^3
 
% in TeX
 
G:=Group("C2^4.573C2^3");
 
// GroupNames label
 
G:=SmallGroup(128,1213);
 
// by ID
 
G=gap.SmallGroup(128,1213);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,2,112,253,344,758,723,100,675,192]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=1,d^2=f^2=a,e^2=g^2=b*a=a*b,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^-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
 

׿
×
⋊
ℤ
𝔽
○
≀
ℚ
◁