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G = C23.493C24order 128 = 27

210th central stem extension by C23 of C24

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

Aliases: C24.34C23, C23.493C24, C22.2752+ 1+4, (C2×C42).77C22, (C22×C4).115C23, C24.C2294C2, C23.10D4.26C2, (C22×D4).180C22, C23.83C2350C2, C23.81C2349C2, C2.33(C22.32C24), C24.3C22.52C2, C2.C42.227C22, C2.26(C22.53C24), C2.95(C23.36C23), C2.70(C22.47C24), C2.30(C22.49C24), (C4×C4⋊C4)⋊106C2, (C2×C4).859(C4○D4), (C2×C4⋊C4).882C22, C22.369(C2×C4○D4), (C2×C22⋊C4).57C22, SmallGroup(128,1325)

Series: Derived Chief Lower central Upper central Jennings

C1C23 — C23.493C24
C1C2C22C23C22×C4C2×C42C4×C4⋊C4 — C23.493C24
C1C23 — C23.493C24
C1C23 — C23.493C24
C1C23 — C23.493C24

Generators and relations for C23.493C24
 G = < a,b,c,d,e,f,g | a2=b2=c2=1, d2=e2=ca=ac, f2=c, g2=b, ab=ba, 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: 436 in 220 conjugacy classes, 92 normal (22 characteristic)
C1, C2 [×3], C2 [×4], C2 [×2], C4 [×18], C22 [×3], C22 [×4], C22 [×14], C2×C4 [×10], C2×C4 [×34], D4 [×4], C23, C23 [×14], C42 [×7], C22⋊C4 [×20], C4⋊C4 [×11], C22×C4 [×3], C22×C4 [×10], C2×D4 [×4], C24 [×2], C2.C42 [×2], C2.C42 [×8], C2×C42, C2×C42 [×4], C2×C22⋊C4 [×12], C2×C4⋊C4 [×3], C2×C4⋊C4 [×4], C22×D4, C4×C4⋊C4 [×2], C24.C22 [×8], C24.3C22, C23.10D4 [×2], C23.81C23, C23.83C23, C23.493C24
Quotients: C1, C2 [×15], C22 [×35], C23 [×15], C4○D4 [×10], C24, C2×C4○D4 [×5], 2+ 1+4 [×2], C23.36C23 [×2], C22.32C24, C22.47C24 [×2], C22.49C24, C22.53C24, C23.493C24

Smallest permutation representation of C23.493C24
On 64 points
Generators in S64
(1 9)(2 10)(3 11)(4 12)(5 28)(6 25)(7 26)(8 27)(13 19)(14 20)(15 17)(16 18)(21 47)(22 48)(23 45)(24 46)(29 63)(30 64)(31 61)(32 62)(33 54)(34 55)(35 56)(36 53)(37 57)(38 58)(39 59)(40 60)(41 51)(42 52)(43 49)(44 50)
(1 42)(2 43)(3 44)(4 41)(5 55)(6 56)(7 53)(8 54)(9 52)(10 49)(11 50)(12 51)(13 23)(14 24)(15 21)(16 22)(17 47)(18 48)(19 45)(20 46)(25 35)(26 36)(27 33)(28 34)(29 39)(30 40)(31 37)(32 38)(57 61)(58 62)(59 63)(60 64)
(1 11)(2 12)(3 9)(4 10)(5 26)(6 27)(7 28)(8 25)(13 17)(14 18)(15 19)(16 20)(21 45)(22 46)(23 47)(24 48)(29 61)(30 62)(31 63)(32 64)(33 56)(34 53)(35 54)(36 55)(37 59)(38 60)(39 57)(40 58)(41 49)(42 50)(43 51)(44 52)
(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 55 3 53)(2 35 4 33)(5 44 7 42)(6 51 8 49)(9 34 11 36)(10 56 12 54)(13 37 15 39)(14 58 16 60)(17 59 19 57)(18 40 20 38)(21 29 23 31)(22 64 24 62)(25 41 27 43)(26 52 28 50)(30 46 32 48)(45 61 47 63)
(1 14 11 18)(2 21 12 45)(3 16 9 20)(4 23 10 47)(5 30 26 62)(6 37 27 59)(7 32 28 64)(8 39 25 57)(13 49 17 41)(15 51 19 43)(22 52 46 44)(24 50 48 42)(29 35 61 54)(31 33 63 56)(34 60 53 38)(36 58 55 40)
(1 15 42 21)(2 18 43 48)(3 13 44 23)(4 20 41 46)(5 29 55 39)(6 64 56 60)(7 31 53 37)(8 62 54 58)(9 17 52 47)(10 16 49 22)(11 19 50 45)(12 14 51 24)(25 30 35 40)(26 61 36 57)(27 32 33 38)(28 63 34 59)

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

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

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

38 conjugacy classes

class 1 2A···2G2H2I4A···4H4I···4Z4AA4AB
order12···2224···44···444
size11···1882···24···488

38 irreducible representations

dim111111124
type++++++++
imageC1C2C2C2C2C2C2C4○D42+ 1+4
kernelC23.493C24C4×C4⋊C4C24.C22C24.3C22C23.10D4C23.81C23C23.83C23C2×C4C22
# reps1281211202

Matrix representation of C23.493C24 in GL6(𝔽5)

400000
040000
001000
000100
000010
000001
,
400000
040000
001000
000100
000040
000004
,
100000
010000
004000
000400
000010
000001
,
200000
030000
003000
000300
000002
000030
,
040000
100000
004200
004100
000040
000004
,
020000
300000
003000
003200
000001
000010
,
010000
400000
004000
000400
000020
000002

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

C23.493C24 in GAP, Magma, Sage, TeX

C_2^3._{493}C_2^4
% in TeX

G:=Group("C2^3.493C2^4");
// GroupNames label

G:=SmallGroup(128,1325);
// by ID

G=gap.SmallGroup(128,1325);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,560,253,568,758,723,352,675,136]);
// Polycyclic

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

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