Copied to
clipboard

## G = C3×Q8○D12order 288 = 25·32

### Direct product of C3 and Q8○D12

Series: Derived Chief Lower central Upper central

 Derived series C1 — C6 — C3×Q8○D12
 Chief series C1 — C3 — C6 — C3×C6 — S3×C6 — S3×C12 — C3×S3×Q8 — C3×Q8○D12
 Lower central C3 — C6 — C3×Q8○D12
 Upper central C1 — C6 — C3×C4○D4

Generators and relations for C3×Q8○D12
G = < a,b,c,d,e | a3=b4=e2=1, c2=d6=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=b-1, bd=db, be=eb, cd=dc, ce=ec, ede=b2d5 >

Subgroups: 578 in 312 conjugacy classes, 170 normal (24 characteristic)
C1, C2, C2, C3, C3, C4, C4, C4, C22, C22, S3, C6, C6, C2×C4, C2×C4, D4, D4, Q8, Q8, C32, Dic3, C12, C12, C12, D6, C2×C6, C2×C6, C2×Q8, C4○D4, C4○D4, C3×S3, C3×C6, C3×C6, Dic6, C4×S3, D12, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C3×D4, C3×D4, C3×Q8, C3×Q8, 2- 1+4, C3×Dic3, C3×C12, C3×C12, S3×C6, C62, C2×Dic6, C4○D12, D42S3, S3×Q8, C6×Q8, C3×C4○D4, C3×C4○D4, C3×Dic6, S3×C12, C3×D12, C6×Dic3, C3×C3⋊D4, C6×C12, D4×C32, Q8×C32, Q8○D12, C3×2- 1+4, C6×Dic6, C3×C4○D12, C3×D42S3, C3×S3×Q8, C32×C4○D4, C3×Q8○D12
Quotients: C1, C2, C3, C22, S3, C6, C23, D6, C2×C6, C24, C3×S3, C22×S3, C22×C6, 2- 1+4, S3×C6, S3×C23, C23×C6, S3×C2×C6, Q8○D12, C3×2- 1+4, S3×C22×C6, C3×Q8○D12

Smallest permutation representation of C3×Q8○D12
On 48 points
Generators in S48
(1 9 5)(2 10 6)(3 11 7)(4 12 8)(13 21 17)(14 22 18)(15 23 19)(16 24 20)(25 29 33)(26 30 34)(27 31 35)(28 32 36)(37 41 45)(38 42 46)(39 43 47)(40 44 48)
(1 21 7 15)(2 22 8 16)(3 23 9 17)(4 24 10 18)(5 13 11 19)(6 14 12 20)(25 37 31 43)(26 38 32 44)(27 39 33 45)(28 40 34 46)(29 41 35 47)(30 42 36 48)
(1 18 7 24)(2 19 8 13)(3 20 9 14)(4 21 10 15)(5 22 11 16)(6 23 12 17)(25 40 31 46)(26 41 32 47)(27 42 33 48)(28 43 34 37)(29 44 35 38)(30 45 36 39)
(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 27)(2 26)(3 25)(4 36)(5 35)(6 34)(7 33)(8 32)(9 31)(10 30)(11 29)(12 28)(13 47)(14 46)(15 45)(16 44)(17 43)(18 42)(19 41)(20 40)(21 39)(22 38)(23 37)(24 48)

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

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

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

81 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 3A 3B 3C 3D 3E 4A 4B 4C 4D 4E ··· 4J 6A 6B 6C ··· 6K 6L ··· 6T 6U 6V 6W 6X 12A ··· 12N 12O ··· 12W 12X ··· 12AI order 1 2 2 2 2 2 2 3 3 3 3 3 4 4 4 4 4 ··· 4 6 6 6 ··· 6 6 ··· 6 6 6 6 6 12 ··· 12 12 ··· 12 12 ··· 12 size 1 1 2 2 2 6 6 1 1 2 2 2 2 2 2 2 6 ··· 6 1 1 2 ··· 2 4 ··· 4 6 6 6 6 2 ··· 2 4 ··· 4 6 ··· 6

81 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + + + - - image C1 C2 C2 C2 C2 C2 C3 C6 C6 C6 C6 C6 S3 D6 D6 D6 C3×S3 S3×C6 S3×C6 S3×C6 2- 1+4 Q8○D12 C3×2- 1+4 C3×Q8○D12 kernel C3×Q8○D12 C6×Dic6 C3×C4○D12 C3×D4⋊2S3 C3×S3×Q8 C32×C4○D4 Q8○D12 C2×Dic6 C4○D12 D4⋊2S3 S3×Q8 C3×C4○D4 C3×C4○D4 C2×C12 C3×D4 C3×Q8 C4○D4 C2×C4 D4 Q8 C32 C3 C3 C1 # reps 1 3 3 6 2 1 2 6 6 12 4 2 1 3 3 1 2 6 6 2 1 2 2 4

Matrix representation of C3×Q8○D12 in GL4(𝔽13) generated by

 3 0 0 0 0 3 0 0 0 0 3 0 0 0 0 3
,
 0 1 0 0 12 0 0 0 0 0 0 1 0 0 12 0
,
 0 8 0 0 8 0 0 0 0 0 0 8 0 0 8 0
,
 6 0 0 0 0 6 0 0 0 0 11 0 0 0 0 11
,
 0 0 2 0 0 0 0 2 7 0 0 0 0 7 0 0
G:=sub<GL(4,GF(13))| [3,0,0,0,0,3,0,0,0,0,3,0,0,0,0,3],[0,12,0,0,1,0,0,0,0,0,0,12,0,0,1,0],[0,8,0,0,8,0,0,0,0,0,0,8,0,0,8,0],[6,0,0,0,0,6,0,0,0,0,11,0,0,0,0,11],[0,0,7,0,0,0,0,7,2,0,0,0,0,2,0,0] >;

C3×Q8○D12 in GAP, Magma, Sage, TeX

C_3\times Q_8\circ D_{12}
% in TeX

G:=Group("C3xQ8oD12");
// GroupNames label

G:=SmallGroup(288,1000);
// by ID

G=gap.SmallGroup(288,1000);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,-2,-3,344,555,268,1571,192,9414]);
// Polycyclic

G:=Group<a,b,c,d,e|a^3=b^4=e^2=1,c^2=d^6=b^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c^-1=b^-1,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e=b^2*d^5>;
// generators/relations

׿
×
𝔽