Copied to
clipboard

G = C23.449C24order 128 = 27

166th central stem extension by C23 of C24

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

Aliases: C23.449C24, C24.581C23, C22.2342+ 1+4, C22⋊C417Q8, (C22×C4)⋊11Q8, C22⋊C4.75D4, C22.8(C4⋊Q8), C23.98(C2×Q8), C23.430(C2×D4), C2.71(D45D4), C2.30(D43Q8), C2.4(C232Q8), C22.99(C22×Q8), (C2×C42).554C22, (C23×C4).397C22, (C22×C4).838C23, C22.300(C22×D4), C23.8Q8.29C2, C23.7Q8.52C2, (C22×Q8).132C22, C23.65C2385C2, C23.78C2315C2, C23.81C2337C2, C23.67C2361C2, C2.C42.186C22, C2.25(C23.37C23), C2.12(C2×C4⋊Q8), (C2×C4).75(C2×D4), (C2×C4).50(C2×Q8), (C4×C22⋊C4).61C2, (C2×C22⋊Q8).34C2, (C2×C4).385(C4○D4), (C2×C4⋊C4).303C22, C22.326(C2×C4○D4), (C2×C22⋊C4).504C22, (C2×C2.C42).28C2, SmallGroup(128,1281)

Series: Derived Chief Lower central Upper central Jennings

C1C23 — C23.449C24
C1C2C22C23C24C23×C4C2×C2.C42 — C23.449C24
C1C23 — C23.449C24
C1C23 — C23.449C24
C1C23 — C23.449C24

Generators and relations for C23.449C24
 G = < a,b,c,d,e,f,g | a2=b2=c2=g2=1, d2=f2=b, e2=ca=ac, 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: 500 in 272 conjugacy classes, 120 normal (22 characteristic)
C1, C2 [×3], C2 [×4], C2 [×4], C4 [×22], C22 [×3], C22 [×8], C22 [×12], C2×C4 [×16], C2×C4 [×46], Q8 [×8], C23, C23 [×6], C23 [×4], C42 [×2], C22⋊C4 [×8], C22⋊C4 [×4], C4⋊C4 [×20], C22×C4 [×2], C22×C4 [×16], C22×C4 [×12], C2×Q8 [×10], C24, C2.C42 [×4], C2.C42 [×8], C2×C42 [×2], C2×C22⋊C4 [×2], C2×C22⋊C4 [×2], C2×C4⋊C4 [×12], C22⋊Q8 [×8], C23×C4, C23×C4 [×2], C22×Q8 [×2], C2×C2.C42, C4×C22⋊C4, C23.7Q8, C23.8Q8 [×2], C23.65C23 [×2], C23.67C23 [×2], C23.78C23 [×2], C23.81C23 [×2], C2×C22⋊Q8 [×2], C23.449C24
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], Q8 [×8], C23 [×15], C2×D4 [×6], C2×Q8 [×12], C4○D4 [×4], C24, C4⋊Q8 [×4], C22×D4, C22×Q8 [×2], C2×C4○D4 [×2], 2+ 1+4 [×2], C2×C4⋊Q8, C23.37C23, C232Q8, D45D4 [×2], D43Q8 [×2], C23.449C24

Smallest permutation representation of C23.449C24
On 64 points
Generators in S64
(1 39)(2 40)(3 37)(4 38)(5 17)(6 18)(7 19)(8 20)(9 41)(10 42)(11 43)(12 44)(13 45)(14 46)(15 47)(16 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 62)(34 63)(35 64)(36 61)
(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 51)(2 52)(3 49)(4 50)(5 36)(6 33)(7 34)(8 35)(9 53)(10 54)(11 55)(12 56)(13 57)(14 58)(15 59)(16 60)(17 61)(18 62)(19 63)(20 64)(21 37)(22 38)(23 39)(24 40)(25 41)(26 42)(27 43)(28 44)(29 45)(30 46)(31 47)(32 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 31 23 15)(2 60 24 48)(3 29 21 13)(4 58 22 46)(5 41 61 53)(6 10 62 26)(7 43 63 55)(8 12 64 28)(9 36 25 17)(11 34 27 19)(14 38 30 50)(16 40 32 52)(18 42 33 54)(20 44 35 56)(37 57 49 45)(39 59 51 47)
(1 9 3 11)(2 12 4 10)(5 29 7 31)(6 32 8 30)(13 63 15 61)(14 62 16 64)(17 57 19 59)(18 60 20 58)(21 27 23 25)(22 26 24 28)(33 48 35 46)(34 47 36 45)(37 43 39 41)(38 42 40 44)(49 55 51 53)(50 54 52 56)
(1 51)(2 24)(3 49)(4 22)(5 36)(6 62)(7 34)(8 64)(9 53)(10 26)(11 55)(12 28)(13 57)(14 30)(15 59)(16 32)(17 61)(18 33)(19 63)(20 35)(21 37)(23 39)(25 41)(27 43)(29 45)(31 47)(38 50)(40 52)(42 54)(44 56)(46 58)(48 60)

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

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

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

38 conjugacy classes

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

38 irreducible representations

dim111111111122224
type+++++++++++--+
imageC1C2C2C2C2C2C2C2C2C2D4Q8Q8C4○D42+ 1+4
kernelC23.449C24C2×C2.C42C4×C22⋊C4C23.7Q8C23.8Q8C23.65C23C23.67C23C23.78C23C23.81C23C2×C22⋊Q8C22⋊C4C22⋊C4C22×C4C2×C4C22
# reps111122222244482

Matrix representation of C23.449C24 in GL6(𝔽5)

400000
040000
001000
000100
000040
000004
,
100000
010000
001000
000100
000040
000004
,
100000
010000
004000
000400
000010
000001
,
010000
100000
001000
000100
000001
000040
,
200000
030000
002300
000300
000020
000003
,
400000
040000
001000
002400
000020
000003
,
100000
040000
004000
000400
000010
000004

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

C23.449C24 in GAP, Magma, Sage, TeX

C_2^3._{449}C_2^4
% in TeX

G:=Group("C2^3.449C2^4");
// GroupNames label

G:=SmallGroup(128,1281);
// by ID

G=gap.SmallGroup(128,1281);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,224,253,568,758,723,184,675]);
// Polycyclic

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

׿
×
𝔽