Copied to
clipboard

## G = C2×C6.D4order 96 = 25·3

### Direct product of C2 and C6.D4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C6 — C2×C6.D4
 Chief series C1 — C3 — C6 — C2×C6 — C2×Dic3 — C22×Dic3 — C2×C6.D4
 Lower central C3 — C6 — C2×C6.D4
 Upper central C1 — C23 — C24

Generators and relations for C2×C6.D4
G = < a,b,c,d | a2=b6=c4=1, d2=b3, ab=ba, ac=ca, ad=da, cbc-1=dbd-1=b-1, dcd-1=b3c-1 >

Subgroups: 242 in 132 conjugacy classes, 65 normal (11 characteristic)
C1, C2, C2 [×6], C2 [×4], C3, C4 [×4], C22, C22 [×10], C22 [×12], C6, C6 [×6], C6 [×4], C2×C4 [×8], C23, C23 [×6], C23 [×4], Dic3 [×4], C2×C6, C2×C6 [×10], C2×C6 [×12], C22⋊C4 [×4], C22×C4 [×2], C24, C2×Dic3 [×4], C2×Dic3 [×4], C22×C6, C22×C6 [×6], C22×C6 [×4], C2×C22⋊C4, C6.D4 [×4], C22×Dic3 [×2], C23×C6, C2×C6.D4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3, C2×C4 [×6], D4 [×4], C23, Dic3 [×4], D6 [×3], C22⋊C4 [×4], C22×C4, C2×D4 [×2], C2×Dic3 [×6], C3⋊D4 [×4], C22×S3, C2×C22⋊C4, C6.D4 [×4], C22×Dic3, C2×C3⋊D4 [×2], C2×C6.D4

Smallest permutation representation of C2×C6.D4
On 48 points
Generators in S48
(1 27)(2 28)(3 29)(4 30)(5 25)(6 26)(7 41)(8 42)(9 37)(10 38)(11 39)(12 40)(13 19)(14 20)(15 21)(16 22)(17 23)(18 24)(31 46)(32 47)(33 48)(34 43)(35 44)(36 45)
(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 31 14 38)(2 36 15 37)(3 35 16 42)(4 34 17 41)(5 33 18 40)(6 32 13 39)(7 30 43 23)(8 29 44 22)(9 28 45 21)(10 27 46 20)(11 26 47 19)(12 25 48 24)
(1 41 4 38)(2 40 5 37)(3 39 6 42)(7 30 10 27)(8 29 11 26)(9 28 12 25)(13 35 16 32)(14 34 17 31)(15 33 18 36)(19 44 22 47)(20 43 23 46)(21 48 24 45)

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

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

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

36 conjugacy classes

 class 1 2A ··· 2G 2H 2I 2J 2K 3 4A ··· 4H 6A ··· 6O order 1 2 ··· 2 2 2 2 2 3 4 ··· 4 6 ··· 6 size 1 1 ··· 1 2 2 2 2 2 6 ··· 6 2 ··· 2

36 irreducible representations

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

Matrix representation of C2×C6.D4 in GL4(𝔽13) generated by

 12 0 0 0 0 12 0 0 0 0 1 0 0 0 0 1
,
 12 0 0 0 0 1 0 0 0 0 3 0 0 0 0 9
,
 8 0 0 0 0 1 0 0 0 0 0 1 0 0 12 0
,
 5 0 0 0 0 12 0 0 0 0 0 1 0 0 1 0
G:=sub<GL(4,GF(13))| [12,0,0,0,0,12,0,0,0,0,1,0,0,0,0,1],[12,0,0,0,0,1,0,0,0,0,3,0,0,0,0,9],[8,0,0,0,0,1,0,0,0,0,0,12,0,0,1,0],[5,0,0,0,0,12,0,0,0,0,0,1,0,0,1,0] >;

C2×C6.D4 in GAP, Magma, Sage, TeX

C_2\times C_6.D_4
% in TeX

G:=Group("C2xC6.D4");
// GroupNames label

G:=SmallGroup(96,159);
// by ID

G=gap.SmallGroup(96,159);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-3,48,362,2309]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^6=c^4=1,d^2=b^3,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d^-1=b^-1,d*c*d^-1=b^3*c^-1>;
// generators/relations

׿
×
𝔽