Copied to
clipboard

G = C24.579C23order 128 = 27

60th non-split extension by C24 of C23 acting via C23/C22=C2

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

Aliases: C24.579C23, C23.403C24, C22.1522- 1+4, C428C431C2, (C22×C4).390D4, C23.617(C2×D4), C23.311(C4○D4), (C2×C42).523C22, (C22×C4).524C23, (C23×C4).100C22, C22.279(C22×D4), C23.34D4.16C2, C4.53(C22.D4), C22.29(C4.4D4), (C22×Q8).120C22, C23.83C2326C2, C23.67C2353C2, C2.C42.154C22, C2.43(C22.46C24), C2.19(C23.38C23), (C2×C4).832(C2×D4), (C22×C4⋊C4).38C2, (C4×C22⋊C4).53C2, C2.19(C2×C4.4D4), (C2×C22⋊Q8).33C2, (C2×C4).815(C4○D4), (C2×C4⋊C4).270C22, C22.280(C2×C4○D4), C2.38(C2×C22.D4), (C2×C22⋊C4).465C22, SmallGroup(128,1235)

Series: Derived Chief Lower central Upper central Jennings

C1C23 — C24.579C23
C1C2C22C23C22×C4C2×C42C4×C22⋊C4 — C24.579C23
C1C23 — C24.579C23
C1C23 — C24.579C23
C1C23 — C24.579C23

Generators and relations for C24.579C23
 G = < a,b,c,d,e,f,g | a2=b2=c2=d2=1, e2=d, f2=cb=bc, g2=b, ab=ba, ac=ca, faf-1=ad=da, ae=ea, ag=ga, 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 256 conjugacy classes, 108 normal (16 characteristic)
C1, C2 [×3], C2 [×4], C2 [×4], C4 [×4], C4 [×14], C22, C22 [×10], C22 [×12], C2×C4 [×8], C2×C4 [×54], Q8 [×4], C23, C23 [×6], C23 [×4], C42 [×4], C22⋊C4 [×8], C4⋊C4 [×14], C22×C4 [×2], C22×C4 [×16], C22×C4 [×16], C2×Q8 [×6], C24, C2.C42 [×18], C2×C42 [×2], C2×C22⋊C4 [×4], C2×C4⋊C4, C2×C4⋊C4 [×6], C2×C4⋊C4 [×4], C22⋊Q8 [×4], C23×C4, C23×C4 [×2], C22×Q8, C4×C22⋊C4, C23.34D4 [×4], C428C4 [×2], C23.67C23 [×2], C23.83C23 [×4], C22×C4⋊C4, C2×C22⋊Q8, C24.579C23
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], C2×D4 [×6], C4○D4 [×8], C24, C22.D4 [×4], C4.4D4 [×4], C22×D4, C2×C4○D4 [×4], 2- 1+4 [×2], C2×C22.D4, C2×C4.4D4, C23.38C23, C22.46C24 [×4], C24.579C23

Smallest permutation representation of C24.579C23
On 64 points
Generators in S64
(1 51)(2 52)(3 49)(4 50)(5 64)(6 61)(7 62)(8 63)(9 23)(10 24)(11 21)(12 22)(13 27)(14 28)(15 25)(16 26)(17 29)(18 30)(19 31)(20 32)(33 37)(34 38)(35 39)(36 40)(41 55)(42 56)(43 53)(44 54)(45 57)(46 58)(47 59)(48 60)
(1 9)(2 10)(3 11)(4 12)(5 38)(6 39)(7 40)(8 37)(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 53)(26 54)(27 55)(28 56)(29 57)(30 58)(31 59)(32 60)(33 63)(34 64)(35 61)(36 62)
(1 51)(2 52)(3 49)(4 50)(5 62)(6 63)(7 64)(8 61)(9 23)(10 24)(11 21)(12 22)(13 27)(14 28)(15 25)(16 26)(17 31)(18 32)(19 29)(20 30)(33 39)(34 40)(35 37)(36 38)(41 55)(42 56)(43 53)(44 54)(45 59)(46 60)(47 57)(48 58)
(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 31 23 45)(2 18 24 60)(3 29 21 47)(4 20 22 58)(5 54 36 16)(6 41 33 27)(7 56 34 14)(8 43 35 25)(9 59 51 17)(10 46 52 32)(11 57 49 19)(12 48 50 30)(13 63 55 39)(15 61 53 37)(26 62 44 38)(28 64 42 40)
(1 55 9 27)(2 28 10 56)(3 53 11 25)(4 26 12 54)(5 60 38 32)(6 29 39 57)(7 58 40 30)(8 31 37 59)(13 51 41 23)(14 24 42 52)(15 49 43 21)(16 22 44 50)(17 35 45 61)(18 62 46 36)(19 33 47 63)(20 64 48 34)

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

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

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

38 conjugacy classes

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

38 irreducible representations

dim111111112224
type+++++++++-
imageC1C2C2C2C2C2C2C2D4C4○D4C4○D42- 1+4
kernelC24.579C23C4×C22⋊C4C23.34D4C428C4C23.67C23C23.83C23C22×C4⋊C4C2×C22⋊Q8C22×C4C2×C4C23C22
# reps114224114882

Matrix representation of C24.579C23 in GL6(𝔽5)

100000
040000
004000
000100
000040
000004
,
100000
010000
001000
000100
000040
000004
,
100000
010000
004000
000400
000040
000004
,
400000
040000
004000
000400
000010
000001
,
200000
020000
003000
000200
000012
000004
,
010000
100000
000100
004000
000024
000033
,
400000
010000
001000
000400
000043
000011

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

C24.579C23 in GAP, Magma, Sage, TeX

C_2^4._{579}C_2^3
% in TeX

G:=Group("C2^4.579C2^3");
// GroupNames label

G:=SmallGroup(128,1235);
// by ID

G=gap.SmallGroup(128,1235);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,448,253,120,758,723,268,675,80]);
// Polycyclic

G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=d^2=1,e^2=d,f^2=c*b=b*c,g^2=b,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*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

׿
×
𝔽