Copied to
clipboard

## G = C3×C22.11C24order 192 = 26·3

### Direct product of C3 and C22.11C24

direct product, metabelian, nilpotent (class 2), monomial, 2-elementary

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — C3×C22.11C24
 Chief series C1 — C2 — C22 — C2×C6 — C2×C12 — C3×C22⋊C4 — D4×C12 — C3×C22.11C24
 Lower central C1 — C2 — C3×C22.11C24
 Upper central C1 — C2×C6 — C3×C22.11C24

Generators and relations for C3×C22.11C24
G = < a,b,c,d,e,f,g | a3=b2=c2=e2=f2=g2=1, d2=c, ab=ba, ac=ca, ad=da, ae=ea, af=fa, ag=ga, bc=cb, ede=gdg=bd=db, fef=be=eb, bf=fb, bg=gb, cd=dc, ce=ec, cf=fc, cg=gc, df=fd, eg=ge, fg=gf >

Subgroups: 514 in 338 conjugacy classes, 242 normal (14 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, C22, C6, C6, C6, C2×C4, C2×C4, D4, C23, C23, C23, C12, C12, C2×C6, C2×C6, C2×C6, C42, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C24, C2×C12, C2×C12, C3×D4, C22×C6, C22×C6, C22×C6, C2×C22⋊C4, C42⋊C2, C4×D4, C22×D4, C4×C12, C3×C22⋊C4, C3×C4⋊C4, C22×C12, C22×C12, C6×D4, C23×C6, C22.11C24, C6×C22⋊C4, C3×C42⋊C2, D4×C12, D4×C2×C6, C3×C22.11C24
Quotients: C1, C2, C3, C4, C22, C6, C2×C4, C23, C12, C2×C6, C22×C4, C24, C2×C12, C22×C6, C23×C4, 2+ 1+4, C22×C12, C23×C6, C22.11C24, C23×C12, C3×2+ 1+4, C3×C22.11C24

Smallest permutation representation of C3×C22.11C24
On 48 points
Generators in S48
(1 35 11)(2 36 12)(3 33 9)(4 34 10)(5 30 14)(6 31 15)(7 32 16)(8 29 13)(17 37 41)(18 38 42)(19 39 43)(20 40 44)(21 27 45)(22 28 46)(23 25 47)(24 26 48)
(1 39)(2 40)(3 37)(4 38)(5 28)(6 25)(7 26)(8 27)(9 17)(10 18)(11 19)(12 20)(13 21)(14 22)(15 23)(16 24)(29 45)(30 46)(31 47)(32 48)(33 41)(34 42)(35 43)(36 44)
(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 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)
(2 40)(4 38)(6 25)(8 27)(10 18)(12 20)(13 21)(15 23)(29 45)(31 47)(34 42)(36 44)
(1 15)(2 16)(3 13)(4 14)(5 34)(6 35)(7 36)(8 33)(9 29)(10 30)(11 31)(12 32)(17 45)(18 46)(19 47)(20 48)(21 37)(22 38)(23 39)(24 40)(25 43)(26 44)(27 41)(28 42)
(2 40)(4 38)(5 28)(7 26)(10 18)(12 20)(14 22)(16 24)(30 46)(32 48)(34 42)(36 44)

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

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

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

102 conjugacy classes

 class 1 2A 2B 2C 2D ··· 2M 3A 3B 4A ··· 4T 6A ··· 6F 6G ··· 6Z 12A ··· 12AN order 1 2 2 2 2 ··· 2 3 3 4 ··· 4 6 ··· 6 6 ··· 6 12 ··· 12 size 1 1 1 1 2 ··· 2 1 1 2 ··· 2 1 ··· 1 2 ··· 2 2 ··· 2

102 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 4 4 type + + + + + + image C1 C2 C2 C2 C2 C3 C4 C6 C6 C6 C6 C12 2+ 1+4 C3×2+ 1+4 kernel C3×C22.11C24 C6×C22⋊C4 C3×C42⋊C2 D4×C12 D4×C2×C6 C22.11C24 C6×D4 C2×C22⋊C4 C42⋊C2 C4×D4 C22×D4 C2×D4 C6 C2 # reps 1 4 2 8 1 2 16 8 4 16 2 32 2 4

Matrix representation of C3×C22.11C24 in GL5(𝔽13)

 3 0 0 0 0 0 9 0 0 0 0 0 9 0 0 0 0 0 9 0 0 0 0 0 9
,
 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
,
 12 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1
,
 5 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 1 0 0 0 0 0 1 0 0
,
 12 0 0 0 0 0 1 0 0 0 0 0 12 0 0 0 0 0 12 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 1 0
,
 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 12 0 0 0 0 0 12

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

C3×C22.11C24 in GAP, Magma, Sage, TeX

C_3\times C_2^2._{11}C_2^4
% in TeX

G:=Group("C3xC2^2.11C2^4");
// GroupNames label

G:=SmallGroup(192,1407);
// by ID

G=gap.SmallGroup(192,1407);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,-2,-2,672,701,555,1571]);
// Polycyclic

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

׿
×
𝔽