Copied to
clipboard

## G = C3×C23⋊2Q8order 192 = 26·3

### Direct product of C3 and C23⋊2Q8

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — C3×C23⋊2Q8
 Chief series C1 — C2 — C22 — C2×C6 — C2×C12 — C3×C22⋊C4 — C3×C22⋊Q8 — C3×C23⋊2Q8
 Lower central C1 — C22 — C3×C23⋊2Q8
 Upper central C1 — C2×C6 — C3×C23⋊2Q8

Generators and relations for C3×C232Q8
G = < a,b,c,d,e,f | a3=b2=c2=d2=e4=1, f2=e2, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, ebe-1=fbf-1=bd=db, fcf-1=cd=dc, ce=ec, de=ed, df=fd, fef-1=e-1 >

Subgroups: 386 in 242 conjugacy classes, 162 normal (10 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, C22, C6, C6, C6, C2×C4, C2×C4, Q8, C23, C23, C12, C2×C6, C2×C6, C2×C6, C22⋊C4, C4⋊C4, C22×C4, C2×Q8, C24, C2×C12, C2×C12, C3×Q8, C22×C6, C22×C6, C2×C22⋊C4, C22⋊Q8, C3×C22⋊C4, C3×C4⋊C4, C22×C12, C6×Q8, C23×C6, C232Q8, C6×C22⋊C4, C3×C22⋊Q8, C3×C232Q8
Quotients: C1, C2, C3, C22, C6, Q8, C23, C2×C6, C2×Q8, C24, C3×Q8, C22×C6, C22×Q8, 2+ 1+4, C6×Q8, C23×C6, C232Q8, Q8×C2×C6, C3×2+ 1+4, C3×C232Q8

Smallest permutation representation of C3×C232Q8
On 48 points
Generators in S48
(1 19 7)(2 20 8)(3 17 5)(4 18 6)(9 28 21)(10 25 22)(11 26 23)(12 27 24)(13 47 34)(14 48 35)(15 45 36)(16 46 33)(29 39 42)(30 40 43)(31 37 44)(32 38 41)
(1 3)(2 10)(4 12)(5 7)(6 24)(8 22)(9 11)(13 40)(14 16)(15 38)(17 19)(18 27)(20 25)(21 23)(26 28)(29 31)(30 34)(32 36)(33 35)(37 39)(41 45)(42 44)(43 47)(46 48)
(1 3)(2 4)(5 7)(6 8)(9 11)(10 12)(13 40)(14 37)(15 38)(16 39)(17 19)(18 20)(21 23)(22 24)(25 27)(26 28)(29 33)(30 34)(31 35)(32 36)(41 45)(42 46)(43 47)(44 48)
(1 11)(2 12)(3 9)(4 10)(5 21)(6 22)(7 23)(8 24)(13 38)(14 39)(15 40)(16 37)(17 28)(18 25)(19 26)(20 27)(29 35)(30 36)(31 33)(32 34)(41 47)(42 48)(43 45)(44 46)
(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)
(1 32 3 30)(2 31 4 29)(5 43 7 41)(6 42 8 44)(9 36 11 34)(10 35 12 33)(13 28 15 26)(14 27 16 25)(17 40 19 38)(18 39 20 37)(21 45 23 47)(22 48 24 46)

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

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

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

66 conjugacy classes

 class 1 2A 2B 2C 2D ··· 2I 3A 3B 4A ··· 4L 6A ··· 6F 6G ··· 6R 12A ··· 12X 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 4 ··· 4 1 ··· 1 2 ··· 2 4 ··· 4

66 irreducible representations

 dim 1 1 1 1 1 1 2 2 4 4 type + + + - + image C1 C2 C2 C3 C6 C6 Q8 C3×Q8 2+ 1+4 C3×2+ 1+4 kernel C3×C23⋊2Q8 C6×C22⋊C4 C3×C22⋊Q8 C23⋊2Q8 C2×C22⋊C4 C22⋊Q8 C22×C6 C23 C6 C2 # reps 1 3 12 2 6 24 4 8 2 4

Matrix representation of C3×C232Q8 in GL6(𝔽13)

 9 0 0 0 0 0 0 9 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 3
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 12 12 0 0 0 0 1 0 12 0 0 0 0 0 0 1
,
 12 0 0 0 0 0 0 12 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 1 0 12 0 0 0 12 0 0 12
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 0 12
,
 12 11 0 0 0 0 1 1 0 0 0 0 0 0 12 11 0 0 0 0 0 1 0 0 0 0 0 12 0 1 0 0 0 1 1 0
,
 6 5 0 0 0 0 3 7 0 0 0 0 0 0 1 0 11 0 0 0 0 0 1 1 0 0 0 0 12 0 0 0 0 1 1 0

G:=sub<GL(6,GF(13))| [9,0,0,0,0,0,0,9,0,0,0,0,0,0,3,0,0,0,0,0,0,3,0,0,0,0,0,0,3,0,0,0,0,0,0,3],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,12,1,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,1],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,1,12,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[12,1,0,0,0,0,11,1,0,0,0,0,0,0,12,0,0,0,0,0,11,1,12,1,0,0,0,0,0,1,0,0,0,0,1,0],[6,3,0,0,0,0,5,7,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,11,1,12,1,0,0,0,1,0,0] >;

C3×C232Q8 in GAP, Magma, Sage, TeX

C_3\times C_2^3\rtimes_2Q_8
% in TeX

G:=Group("C3xC2^3:2Q8");
// GroupNames label

G:=SmallGroup(192,1432);
// by ID

G=gap.SmallGroup(192,1432);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,-2,-2,672,701,2102,555,520,1571]);
// Polycyclic

G:=Group<a,b,c,d,e,f|a^3=b^2=c^2=d^2=e^4=1,f^2=e^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,b*c=c*b,e*b*e^-1=f*b*f^-1=b*d=d*b,f*c*f^-1=c*d=d*c,c*e=e*c,d*e=e*d,d*f=f*d,f*e*f^-1=e^-1>;
// generators/relations

׿
×
𝔽