Copied to
clipboard

## G = C3⋊S3⋊Dic6order 432 = 24·33

### The semidirect product of C3⋊S3 and Dic6 acting via Dic6/C4=S3

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3 — C2×He3 — C3⋊S3⋊Dic6
 Chief series C1 — C3 — C32 — He3 — C2×He3 — C2×C32⋊C6 — C6.S32 — C3⋊S3⋊Dic6
 Lower central He3 — C2×He3 — C3⋊S3⋊Dic6
 Upper central C1 — C2 — C4

Generators and relations for C3⋊S3⋊Dic6
G = < a,b,c,d,e | a3=b3=c2=d12=1, e2=d6, dbd-1=ab=ba, cac=eae-1=a-1, ad=da, cbc=b-1, be=eb, cd=dc, ce=ec, ede-1=d-1 >

Subgroups: 771 in 145 conjugacy classes, 39 normal (25 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, S3, C6, C6, C2×C4, Q8, C32, C32, Dic3, C12, C12, D6, C2×C6, C2×Q8, C3×S3, C3⋊S3, C3×C6, C3×C6, Dic6, C4×S3, C2×Dic3, C2×C12, C3×Q8, He3, C3×Dic3, C3⋊Dic3, C3⋊Dic3, C3×C12, C3×C12, S3×C6, C2×C3⋊S3, C2×Dic6, S3×Q8, C32⋊C6, C2×He3, S3×Dic3, C6.D6, C322Q8, C3×Dic6, S3×C12, C324Q8, C4×C3⋊S3, C32⋊C12, C32⋊C12, He33C4, C4×He3, C2×C32⋊C6, S3×Dic6, Dic3.D6, He32Q8, C6.S32, He33Q8, C4×C32⋊C6, He34Q8, C3⋊S3⋊Dic6
Quotients: C1, C2, C22, S3, Q8, C23, D6, C2×Q8, Dic6, C22×S3, S32, C2×Dic6, S3×Q8, C2×S32, C32⋊D6, S3×Dic6, C2×C32⋊D6, C3⋊S3⋊Dic6

Smallest permutation representation of C3⋊S3⋊Dic6
On 72 points
Generators in S72
(1 34 22)(2 35 23)(3 36 24)(4 25 13)(5 26 14)(6 27 15)(7 28 16)(8 29 17)(9 30 18)(10 31 19)(11 32 20)(12 33 21)(37 57 65)(38 58 66)(39 59 67)(40 60 68)(41 49 69)(42 50 70)(43 51 71)(44 52 72)(45 53 61)(46 54 62)(47 55 63)(48 56 64)
(2 35 23)(3 24 36)(5 26 14)(6 15 27)(8 29 17)(9 18 30)(11 32 20)(12 21 33)(37 57 65)(38 66 58)(40 60 68)(41 69 49)(43 51 71)(44 72 52)(46 54 62)(47 63 55)
(13 25)(14 26)(15 27)(16 28)(17 29)(18 30)(19 31)(20 32)(21 33)(22 34)(23 35)(24 36)(49 69)(50 70)(51 71)(52 72)(53 61)(54 62)(55 63)(56 64)(57 65)(58 66)(59 67)(60 68)
(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)(49 50 51 52 53 54 55 56 57 58 59 60)(61 62 63 64 65 66 67 68 69 70 71 72)
(1 42 7 48)(2 41 8 47)(3 40 9 46)(4 39 10 45)(5 38 11 44)(6 37 12 43)(13 59 19 53)(14 58 20 52)(15 57 21 51)(16 56 22 50)(17 55 23 49)(18 54 24 60)(25 67 31 61)(26 66 32 72)(27 65 33 71)(28 64 34 70)(29 63 35 69)(30 62 36 68)

G:=sub<Sym(72)| (1,34,22)(2,35,23)(3,36,24)(4,25,13)(5,26,14)(6,27,15)(7,28,16)(8,29,17)(9,30,18)(10,31,19)(11,32,20)(12,33,21)(37,57,65)(38,58,66)(39,59,67)(40,60,68)(41,49,69)(42,50,70)(43,51,71)(44,52,72)(45,53,61)(46,54,62)(47,55,63)(48,56,64), (2,35,23)(3,24,36)(5,26,14)(6,15,27)(8,29,17)(9,18,30)(11,32,20)(12,21,33)(37,57,65)(38,66,58)(40,60,68)(41,69,49)(43,51,71)(44,72,52)(46,54,62)(47,63,55), (13,25)(14,26)(15,27)(16,28)(17,29)(18,30)(19,31)(20,32)(21,33)(22,34)(23,35)(24,36)(49,69)(50,70)(51,71)(52,72)(53,61)(54,62)(55,63)(56,64)(57,65)(58,66)(59,67)(60,68), (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)(49,50,51,52,53,54,55,56,57,58,59,60)(61,62,63,64,65,66,67,68,69,70,71,72), (1,42,7,48)(2,41,8,47)(3,40,9,46)(4,39,10,45)(5,38,11,44)(6,37,12,43)(13,59,19,53)(14,58,20,52)(15,57,21,51)(16,56,22,50)(17,55,23,49)(18,54,24,60)(25,67,31,61)(26,66,32,72)(27,65,33,71)(28,64,34,70)(29,63,35,69)(30,62,36,68)>;

G:=Group( (1,34,22)(2,35,23)(3,36,24)(4,25,13)(5,26,14)(6,27,15)(7,28,16)(8,29,17)(9,30,18)(10,31,19)(11,32,20)(12,33,21)(37,57,65)(38,58,66)(39,59,67)(40,60,68)(41,49,69)(42,50,70)(43,51,71)(44,52,72)(45,53,61)(46,54,62)(47,55,63)(48,56,64), (2,35,23)(3,24,36)(5,26,14)(6,15,27)(8,29,17)(9,18,30)(11,32,20)(12,21,33)(37,57,65)(38,66,58)(40,60,68)(41,69,49)(43,51,71)(44,72,52)(46,54,62)(47,63,55), (13,25)(14,26)(15,27)(16,28)(17,29)(18,30)(19,31)(20,32)(21,33)(22,34)(23,35)(24,36)(49,69)(50,70)(51,71)(52,72)(53,61)(54,62)(55,63)(56,64)(57,65)(58,66)(59,67)(60,68), (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)(49,50,51,52,53,54,55,56,57,58,59,60)(61,62,63,64,65,66,67,68,69,70,71,72), (1,42,7,48)(2,41,8,47)(3,40,9,46)(4,39,10,45)(5,38,11,44)(6,37,12,43)(13,59,19,53)(14,58,20,52)(15,57,21,51)(16,56,22,50)(17,55,23,49)(18,54,24,60)(25,67,31,61)(26,66,32,72)(27,65,33,71)(28,64,34,70)(29,63,35,69)(30,62,36,68) );

G=PermutationGroup([[(1,34,22),(2,35,23),(3,36,24),(4,25,13),(5,26,14),(6,27,15),(7,28,16),(8,29,17),(9,30,18),(10,31,19),(11,32,20),(12,33,21),(37,57,65),(38,58,66),(39,59,67),(40,60,68),(41,49,69),(42,50,70),(43,51,71),(44,52,72),(45,53,61),(46,54,62),(47,55,63),(48,56,64)], [(2,35,23),(3,24,36),(5,26,14),(6,15,27),(8,29,17),(9,18,30),(11,32,20),(12,21,33),(37,57,65),(38,66,58),(40,60,68),(41,69,49),(43,51,71),(44,72,52),(46,54,62),(47,63,55)], [(13,25),(14,26),(15,27),(16,28),(17,29),(18,30),(19,31),(20,32),(21,33),(22,34),(23,35),(24,36),(49,69),(50,70),(51,71),(52,72),(53,61),(54,62),(55,63),(56,64),(57,65),(58,66),(59,67),(60,68)], [(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),(49,50,51,52,53,54,55,56,57,58,59,60),(61,62,63,64,65,66,67,68,69,70,71,72)], [(1,42,7,48),(2,41,8,47),(3,40,9,46),(4,39,10,45),(5,38,11,44),(6,37,12,43),(13,59,19,53),(14,58,20,52),(15,57,21,51),(16,56,22,50),(17,55,23,49),(18,54,24,60),(25,67,31,61),(26,66,32,72),(27,65,33,71),(28,64,34,70),(29,63,35,69),(30,62,36,68)]])

32 conjugacy classes

 class 1 2A 2B 2C 3A 3B 3C 3D 4A 4B ··· 4F 6A 6B 6C 6D 6E 6F 12A 12B 12C 12D 12E 12F 12G 12H 12I 12J 12K 12L order 1 2 2 2 3 3 3 3 4 4 ··· 4 6 6 6 6 6 6 12 12 12 12 12 12 12 12 12 12 12 12 size 1 1 9 9 2 6 6 12 2 18 ··· 18 2 6 6 12 18 18 4 6 6 12 12 12 18 18 36 36 36 36

32 irreducible representations

 dim 1 1 1 1 1 1 12 2 2 2 2 2 2 2 4 4 4 4 6 6 type + + + + + + - + + - + + + - + - + - + + image C1 C2 C2 C2 C2 C2 C3⋊S3⋊Dic6 S3 S3 Q8 D6 D6 D6 Dic6 S32 S3×Q8 C2×S32 S3×Dic6 C32⋊D6 C2×C32⋊D6 kernel C3⋊S3⋊Dic6 He3⋊2Q8 C6.S32 He3⋊3Q8 C4×C32⋊C6 He3⋊4Q8 C1 C32⋊4Q8 C4×C3⋊S3 C32⋊C6 C3⋊Dic3 C3×C12 C2×C3⋊S3 C3⋊S3 C12 C32 C6 C3 C4 C2 # reps 1 2 2 1 1 1 1 1 1 2 3 2 1 4 1 1 1 2 2 2

Matrix representation of C3⋊S3⋊Dic6 in GL10(𝔽13)

 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 12 1 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 0 0 0 12 1 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 1 0 1 0 0 1 0 0 0 0 0 12 0 12 12 12
,
 12 1 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 0 0 0 12 1 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 1 12 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 12 12 12 0 12 12
,
 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 12 12 12 12 12 12
,
 1 0 3 0 0 0 0 0 0 0 0 1 0 3 0 0 0 0 0 0 8 0 12 0 0 0 0 0 0 0 0 8 0 12 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 12 1 0 0 0 0 12 12 12 12 11 12 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 1 0
,
 1 0 4 0 0 0 0 0 0 0 0 1 0 4 0 0 0 0 0 0 6 0 12 0 0 0 0 0 0 0 0 6 0 12 0 0 0 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 0 1 1 1 1 2 1 0 0 0 0 0 0 0 0 1 12 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 12 12 0

G:=sub<GL(10,GF(13))| [1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,12,12,0,0,1,0,0,0,0,0,1,0,0,0,0,12,0,0,0,0,0,0,12,12,1,0,0,0,0,0,0,0,1,0,0,12,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,12],[12,12,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,12,12,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,12,0,0,0,0,0,1,0,0,0,12,0,0,0,0,0,0,0,1,0,12,0,0,0,0,0,0,12,12,1,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,12],[0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,12,0,0,0,0,1,0,0,0,0,12,0,0,0,0,0,0,0,1,0,12,0,0,0,0,0,0,1,0,0,12,0,0,0,0,0,0,0,0,1,12,0,0,0,0,0,0,0,0,0,12],[1,0,8,0,0,0,0,0,0,0,0,1,0,8,0,0,0,0,0,0,3,0,12,0,0,0,0,0,0,0,0,3,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,12,0,1,0,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,12,0,0,0,0,0,0,0,1,0,12,0,0,0,0,0,0,0,0,12,11,1,1,0,0,0,0,0,0,1,12,0,0],[1,0,6,0,0,0,0,0,0,0,0,1,0,6,0,0,0,0,0,0,4,0,12,0,0,0,0,0,0,0,0,4,0,12,0,0,0,0,0,0,0,0,0,0,0,12,1,0,0,0,0,0,0,0,12,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,12,0,0,0,0,0,0,2,1,12,12,0,0,0,0,0,0,1,12,0,0] >;

C3⋊S3⋊Dic6 in GAP, Magma, Sage, TeX

C_3\rtimes S_3\rtimes {\rm Dic}_6
% in TeX

G:=Group("C3:S3:Dic6");
// GroupNames label

G:=SmallGroup(432,294);
// by ID

G=gap.SmallGroup(432,294);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,64,135,58,571,4037,537,14118,7069]);
// Polycyclic

G:=Group<a,b,c,d,e|a^3=b^3=c^2=d^12=1,e^2=d^6,d*b*d^-1=a*b=b*a,c*a*c=e*a*e^-1=a^-1,a*d=d*a,c*b*c=b^-1,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=d^-1>;
// generators/relations

׿
×
𝔽