Copied to
clipboard

## G = C25.4S3order 192 = 26·3

### 1st non-split extension by C25 of S3 acting via S3/C3=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C6 — C25.4S3
 Chief series C1 — C3 — C6 — C2×C6 — C22×C6 — C22×Dic3 — C2×C6.D4 — C25.4S3
 Lower central C3 — C2×C6 — C25.4S3
 Upper central C1 — C23 — C25

Generators and relations for C25.4S3
G = < a,b,c,d,e,f,g | a2=b2=c2=d2=e2=f3=1, g2=c, ab=ba, ac=ca, ad=da, gag-1=ae=ea, af=fa, bc=cb, gbg-1=bd=db, be=eb, bf=fb, cd=dc, ce=ec, cf=fc, cg=gc, de=ed, df=fd, dg=gd, ef=fe, eg=ge, gfg-1=f-1 >

Subgroups: 1048 in 506 conjugacy classes, 119 normal (9 characteristic)
C1, C2 [×7], C2 [×12], C3, C4 [×4], C22, C22 [×18], C22 [×68], C6 [×7], C6 [×12], C2×C4 [×12], C23, C23 [×18], C23 [×68], Dic3 [×4], C2×C6, C2×C6 [×18], C2×C6 [×68], C22⋊C4 [×12], C22×C4 [×4], C24 [×7], C24 [×12], C2×Dic3 [×12], C22×C6, C22×C6 [×18], C22×C6 [×68], C2×C22⋊C4 [×6], C25, C6.D4 [×12], C22×Dic3 [×4], C23×C6 [×7], C23×C6 [×12], C243C4, C2×C6.D4 [×6], C24×C6, C25.4S3
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3, C2×C4 [×6], D4 [×12], C23, Dic3 [×4], D6 [×3], C22⋊C4 [×12], C22×C4, C2×D4 [×6], C2×Dic3 [×6], C3⋊D4 [×12], C22×S3, C2×C22⋊C4 [×3], C22≀C2 [×4], C6.D4 [×12], C22×Dic3, C2×C3⋊D4 [×6], C243C4, C2×C6.D4 [×3], C244S3 [×4], C25.4S3

Smallest permutation representation of C25.4S3
On 48 points
Generators in S48
```(1 35)(2 10)(3 33)(4 12)(5 42)(6 47)(7 44)(8 45)(9 28)(11 26)(13 22)(14 31)(15 24)(16 29)(17 21)(18 30)(19 23)(20 32)(25 36)(27 34)(37 46)(38 43)(39 48)(40 41)
(1 26)(2 36)(3 28)(4 34)(5 46)(6 40)(7 48)(8 38)(9 33)(10 25)(11 35)(12 27)(13 30)(14 17)(15 32)(16 19)(18 22)(20 24)(21 31)(23 29)(37 42)(39 44)(41 47)(43 45)
(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)
(1 11)(2 12)(3 9)(4 10)(5 44)(6 41)(7 42)(8 43)(13 24)(14 21)(15 22)(16 23)(17 31)(18 32)(19 29)(20 30)(25 34)(26 35)(27 36)(28 33)(37 48)(38 45)(39 46)(40 47)
(1 28)(2 25)(3 26)(4 27)(5 37)(6 38)(7 39)(8 40)(9 35)(10 36)(11 33)(12 34)(13 18)(14 19)(15 20)(16 17)(21 29)(22 30)(23 31)(24 32)(41 45)(42 46)(43 47)(44 48)
(1 23 43)(2 44 24)(3 21 41)(4 42 22)(5 13 12)(6 9 14)(7 15 10)(8 11 16)(17 40 33)(18 34 37)(19 38 35)(20 36 39)(25 48 32)(26 29 45)(27 46 30)(28 31 47)
(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)```

`G:=sub<Sym(48)| (1,35)(2,10)(3,33)(4,12)(5,42)(6,47)(7,44)(8,45)(9,28)(11,26)(13,22)(14,31)(15,24)(16,29)(17,21)(18,30)(19,23)(20,32)(25,36)(27,34)(37,46)(38,43)(39,48)(40,41), (1,26)(2,36)(3,28)(4,34)(5,46)(6,40)(7,48)(8,38)(9,33)(10,25)(11,35)(12,27)(13,30)(14,17)(15,32)(16,19)(18,22)(20,24)(21,31)(23,29)(37,42)(39,44)(41,47)(43,45), (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), (1,11)(2,12)(3,9)(4,10)(5,44)(6,41)(7,42)(8,43)(13,24)(14,21)(15,22)(16,23)(17,31)(18,32)(19,29)(20,30)(25,34)(26,35)(27,36)(28,33)(37,48)(38,45)(39,46)(40,47), (1,28)(2,25)(3,26)(4,27)(5,37)(6,38)(7,39)(8,40)(9,35)(10,36)(11,33)(12,34)(13,18)(14,19)(15,20)(16,17)(21,29)(22,30)(23,31)(24,32)(41,45)(42,46)(43,47)(44,48), (1,23,43)(2,44,24)(3,21,41)(4,42,22)(5,13,12)(6,9,14)(7,15,10)(8,11,16)(17,40,33)(18,34,37)(19,38,35)(20,36,39)(25,48,32)(26,29,45)(27,46,30)(28,31,47), (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)>;`

`G:=Group( (1,35)(2,10)(3,33)(4,12)(5,42)(6,47)(7,44)(8,45)(9,28)(11,26)(13,22)(14,31)(15,24)(16,29)(17,21)(18,30)(19,23)(20,32)(25,36)(27,34)(37,46)(38,43)(39,48)(40,41), (1,26)(2,36)(3,28)(4,34)(5,46)(6,40)(7,48)(8,38)(9,33)(10,25)(11,35)(12,27)(13,30)(14,17)(15,32)(16,19)(18,22)(20,24)(21,31)(23,29)(37,42)(39,44)(41,47)(43,45), (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), (1,11)(2,12)(3,9)(4,10)(5,44)(6,41)(7,42)(8,43)(13,24)(14,21)(15,22)(16,23)(17,31)(18,32)(19,29)(20,30)(25,34)(26,35)(27,36)(28,33)(37,48)(38,45)(39,46)(40,47), (1,28)(2,25)(3,26)(4,27)(5,37)(6,38)(7,39)(8,40)(9,35)(10,36)(11,33)(12,34)(13,18)(14,19)(15,20)(16,17)(21,29)(22,30)(23,31)(24,32)(41,45)(42,46)(43,47)(44,48), (1,23,43)(2,44,24)(3,21,41)(4,42,22)(5,13,12)(6,9,14)(7,15,10)(8,11,16)(17,40,33)(18,34,37)(19,38,35)(20,36,39)(25,48,32)(26,29,45)(27,46,30)(28,31,47), (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) );`

`G=PermutationGroup([(1,35),(2,10),(3,33),(4,12),(5,42),(6,47),(7,44),(8,45),(9,28),(11,26),(13,22),(14,31),(15,24),(16,29),(17,21),(18,30),(19,23),(20,32),(25,36),(27,34),(37,46),(38,43),(39,48),(40,41)], [(1,26),(2,36),(3,28),(4,34),(5,46),(6,40),(7,48),(8,38),(9,33),(10,25),(11,35),(12,27),(13,30),(14,17),(15,32),(16,19),(18,22),(20,24),(21,31),(23,29),(37,42),(39,44),(41,47),(43,45)], [(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)], [(1,11),(2,12),(3,9),(4,10),(5,44),(6,41),(7,42),(8,43),(13,24),(14,21),(15,22),(16,23),(17,31),(18,32),(19,29),(20,30),(25,34),(26,35),(27,36),(28,33),(37,48),(38,45),(39,46),(40,47)], [(1,28),(2,25),(3,26),(4,27),(5,37),(6,38),(7,39),(8,40),(9,35),(10,36),(11,33),(12,34),(13,18),(14,19),(15,20),(16,17),(21,29),(22,30),(23,31),(24,32),(41,45),(42,46),(43,47),(44,48)], [(1,23,43),(2,44,24),(3,21,41),(4,42,22),(5,13,12),(6,9,14),(7,15,10),(8,11,16),(17,40,33),(18,34,37),(19,38,35),(20,36,39),(25,48,32),(26,29,45),(27,46,30),(28,31,47)], [(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)])`

60 conjugacy classes

 class 1 2A ··· 2G 2H ··· 2S 3 4A ··· 4H 6A ··· 6AE order 1 2 ··· 2 2 ··· 2 3 4 ··· 4 6 ··· 6 size 1 1 ··· 1 2 ··· 2 2 12 ··· 12 2 ··· 2

60 irreducible representations

 dim 1 1 1 1 2 2 2 2 2 type + + + + + - + image C1 C2 C2 C4 S3 D4 Dic3 D6 C3⋊D4 kernel C25.4S3 C2×C6.D4 C24×C6 C23×C6 C25 C22×C6 C24 C24 C23 # reps 1 6 1 8 1 12 4 3 24

Matrix representation of C25.4S3 in GL5(𝔽13)

 1 0 0 0 0 0 12 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 12
,
 12 0 0 0 0 0 1 0 0 0 0 0 12 0 0 0 0 0 1 0 0 0 0 0 1
,
 12 0 0 0 0 0 12 0 0 0 0 0 12 0 0 0 0 0 12 0 0 0 0 0 12
,
 1 0 0 0 0 0 12 0 0 0 0 0 12 0 0 0 0 0 1 0 0 0 0 0 1
,
 1 0 0 0 0 0 12 0 0 0 0 0 12 0 0 0 0 0 12 0 0 0 0 0 12
,
 1 0 0 0 0 0 9 0 0 0 0 0 3 0 0 0 0 0 9 0 0 0 0 0 3
,
 8 0 0 0 0 0 0 1 0 0 0 12 0 0 0 0 0 0 0 1 0 0 0 12 0

`G:=sub<GL(5,GF(13))| [1,0,0,0,0,0,12,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,12],[12,0,0,0,0,0,1,0,0,0,0,0,12,0,0,0,0,0,1,0,0,0,0,0,1],[12,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,12],[1,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,12],[1,0,0,0,0,0,9,0,0,0,0,0,3,0,0,0,0,0,9,0,0,0,0,0,3],[8,0,0,0,0,0,0,12,0,0,0,1,0,0,0,0,0,0,0,12,0,0,0,1,0] >;`

C25.4S3 in GAP, Magma, Sage, TeX

`C_2^5._4S_3`
`% in TeX`

`G:=Group("C2^5.4S3");`
`// GroupNames label`

`G:=SmallGroup(192,806);`
`// by ID`

`G=gap.SmallGroup(192,806);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,56,477,422,6278]);`
`// Polycyclic`

`G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=d^2=e^2=f^3=1,g^2=c,a*b=b*a,a*c=c*a,a*d=d*a,g*a*g^-1=a*e=e*a,a*f=f*a,b*c=c*b,g*b*g^-1=b*d=d*b,b*e=e*b,b*f=f*b,c*d=d*c,c*e=e*c,c*f=f*c,c*g=g*c,d*e=e*d,d*f=f*d,d*g=g*d,e*f=f*e,e*g=g*e,g*f*g^-1=f^-1>;`
`// generators/relations`

׿
×
𝔽