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## G = Dic3×C3⋊S3order 216 = 23·33

### Direct product of Dic3 and C3⋊S3

Series: Derived Chief Lower central Upper central

 Derived series C1 — C33 — Dic3×C3⋊S3
 Chief series C1 — C3 — C32 — C33 — C32×C6 — C32×Dic3 — Dic3×C3⋊S3
 Lower central C33 — Dic3×C3⋊S3
 Upper central C1 — C2

Generators and relations for Dic3×C3⋊S3
G = < a,b,c,d,e | a6=c3=d3=e2=1, b2=a3, bab-1=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ece=c-1, ede=d-1 >

Subgroups: 484 in 120 conjugacy classes, 38 normal (14 characteristic)
C1, C2, C2 [×2], C3, C3 [×4], C3 [×4], C4 [×2], C22, S3 [×8], C6, C6 [×4], C6 [×6], C2×C4, C32, C32 [×4], C32 [×4], Dic3, Dic3 [×9], C12 [×4], D6 [×4], C2×C6, C3×S3 [×8], C3⋊S3 [×2], C3×C6, C3×C6 [×4], C3×C6 [×4], C4×S3 [×4], C2×Dic3, C33, C3×Dic3 [×4], C3⋊Dic3 [×9], C3×C12, S3×C6 [×4], C2×C3⋊S3, C3×C3⋊S3 [×2], C32×C6, S3×Dic3 [×4], C4×C3⋊S3, C32×Dic3, C335C4, C6×C3⋊S3, Dic3×C3⋊S3
Quotients: C1, C2 [×3], C4 [×2], C22, S3 [×5], C2×C4, Dic3 [×2], D6 [×5], C3⋊S3, C4×S3 [×4], C2×Dic3, S32 [×4], C2×C3⋊S3, S3×Dic3 [×4], C4×C3⋊S3, S3×C3⋊S3, Dic3×C3⋊S3

Smallest permutation representation of Dic3×C3⋊S3
On 72 points
Generators in S72
(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 52 4 49)(2 51 5 54)(3 50 6 53)(7 42 10 39)(8 41 11 38)(9 40 12 37)(13 48 16 45)(14 47 17 44)(15 46 18 43)(19 59 22 56)(20 58 23 55)(21 57 24 60)(25 65 28 62)(26 64 29 61)(27 63 30 66)(31 68 34 71)(32 67 35 70)(33 72 36 69)
(1 19 17)(2 20 18)(3 21 13)(4 22 14)(5 23 15)(6 24 16)(7 67 65)(8 68 66)(9 69 61)(10 70 62)(11 71 63)(12 72 64)(25 39 32)(26 40 33)(27 41 34)(28 42 35)(29 37 36)(30 38 31)(43 51 58)(44 52 59)(45 53 60)(46 54 55)(47 49 56)(48 50 57)
(1 5 3)(2 6 4)(7 11 9)(8 12 10)(13 17 15)(14 18 16)(19 23 21)(20 24 22)(25 27 29)(26 28 30)(31 33 35)(32 34 36)(37 39 41)(38 40 42)(43 45 47)(44 46 48)(49 51 53)(50 52 54)(55 57 59)(56 58 60)(61 65 63)(62 66 64)(67 71 69)(68 72 70)
(1 35)(2 36)(3 31)(4 32)(5 33)(6 34)(7 56)(8 57)(9 58)(10 59)(11 60)(12 55)(13 30)(14 25)(15 26)(16 27)(17 28)(18 29)(19 42)(20 37)(21 38)(22 39)(23 40)(24 41)(43 61)(44 62)(45 63)(46 64)(47 65)(48 66)(49 67)(50 68)(51 69)(52 70)(53 71)(54 72)

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

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

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

Dic3×C3⋊S3 is a maximal subgroup of
C33⋊(C4⋊C4)  S32×Dic3  Dic36S32  D6⋊S32  C335(C2×Q8)  D6.S32  D6.6S32  C12.39S32  C12.57S32  C4×S3×C3⋊S3  C62.91D6  C62.93D6
Dic3×C3⋊S3 is a maximal quotient of
C338M4(2)  C62.78D6  C62.82D6

36 conjugacy classes

 class 1 2A 2B 2C 3A ··· 3E 3F 3G 3H 3I 4A 4B 4C 4D 6A ··· 6E 6F 6G 6H 6I 6J 6K 12A ··· 12H order 1 2 2 2 3 ··· 3 3 3 3 3 4 4 4 4 6 ··· 6 6 6 6 6 6 6 12 ··· 12 size 1 1 9 9 2 ··· 2 4 4 4 4 3 3 27 27 2 ··· 2 4 4 4 4 18 18 6 ··· 6

36 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 4 4 type + + + + + + - + + - image C1 C2 C2 C2 C4 S3 S3 Dic3 D6 C4×S3 S32 S3×Dic3 kernel Dic3×C3⋊S3 C32×Dic3 C33⋊5C4 C6×C3⋊S3 C3×C3⋊S3 C3×Dic3 C2×C3⋊S3 C3⋊S3 C3×C6 C32 C6 C3 # reps 1 1 1 1 4 4 1 2 5 8 4 4

Matrix representation of Dic3×C3⋊S3 in GL6(𝔽13)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 12 1
,
 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 0 8 0 0 0 0 8 0
,
 3 0 0 0 0 0 4 9 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 12 1 0 0 0 0 12 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 12 5 0 0 0 0 0 1 0 0 0 0 0 0 0 12 0 0 0 0 12 0 0 0 0 0 0 0 12 0 0 0 0 0 0 12

G:=sub<GL(6,GF(13))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,12,0,0,0,0,1,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,8,0,0,0,0,8,0],[3,4,0,0,0,0,0,9,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,12,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[12,0,0,0,0,0,5,1,0,0,0,0,0,0,0,12,0,0,0,0,12,0,0,0,0,0,0,0,12,0,0,0,0,0,0,12] >;

Dic3×C3⋊S3 in GAP, Magma, Sage, TeX

{\rm Dic}_3\times C_3\rtimes S_3
% in TeX

G:=Group("Dic3xC3:S3");
// GroupNames label

G:=SmallGroup(216,125);
// by ID

G=gap.SmallGroup(216,125);
# by ID

G:=PCGroup([6,-2,-2,-2,-3,-3,-3,24,201,730,5189]);
// Polycyclic

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

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