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## G = D12⋊(C3⋊S3)  order 432 = 24·33

### 3rd semidirect product of D12 and C3⋊S3 acting via C3⋊S3/C32=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32×C6 — D12⋊(C3⋊S3)
 Chief series C1 — C3 — C32 — C33 — C32×C6 — S3×C3×C6 — S3×C3⋊Dic3 — D12⋊(C3⋊S3)
 Lower central C33 — C32×C6 — D12⋊(C3⋊S3)
 Upper central C1 — C2 — C4

Generators and relations for D12⋊(C3⋊S3)
G = < a,b,c,d,e | a12=b2=c3=d3=e2=1, bab=a-1, ac=ca, ad=da, eae=a5, bc=cb, bd=db, ebe=a10b, cd=dc, ece=c-1, ede=d-1 >

Subgroups: 1744 in 304 conjugacy classes, 68 normal (20 characteristic)
C1, C2, C2, C3, C3, C3, C4, C4, C22, S3, C6, C6, C6, C2×C4, D4, Q8, C32, C32, C32, Dic3, C12, C12, C12, D6, D6, C2×C6, C4○D4, C3×S3, C3⋊S3, C3×C6, C3×C6, C3×C6, Dic6, C4×S3, D12, D12, C2×Dic3, C3⋊D4, C3×D4, C3×Q8, C33, C3×Dic3, C3⋊Dic3, C3⋊Dic3, C3×C12, C3×C12, C3×C12, S3×C6, C2×C3⋊S3, C62, D42S3, Q83S3, S3×C32, C33⋊C2, C32×C6, S3×Dic3, C3⋊D12, C3×Dic6, C3×D12, C324Q8, C4×C3⋊S3, C2×C3⋊Dic3, C327D4, D4×C32, C3×C3⋊Dic3, C335C4, C32×C12, S3×C3×C6, C2×C33⋊C2, D12⋊S3, C12.D6, S3×C3⋊Dic3, C337D4, C32×D12, C3×C324Q8, C4×C33⋊C2, D12⋊(C3⋊S3)
Quotients: C1, C2, C22, S3, C23, D6, C4○D4, C3⋊S3, C22×S3, S32, C2×C3⋊S3, D42S3, Q83S3, C2×S32, C22×C3⋊S3, S3×C3⋊S3, D12⋊S3, C12.D6, C2×S3×C3⋊S3, D12⋊(C3⋊S3)

Smallest permutation representation of D12⋊(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 17)(2 16)(3 15)(4 14)(5 13)(6 24)(7 23)(8 22)(9 21)(10 20)(11 19)(12 18)(25 58)(26 57)(27 56)(28 55)(29 54)(30 53)(31 52)(32 51)(33 50)(34 49)(35 60)(36 59)(37 66)(38 65)(39 64)(40 63)(41 62)(42 61)(43 72)(44 71)(45 70)(46 69)(47 68)(48 67)
(1 40 49)(2 41 50)(3 42 51)(4 43 52)(5 44 53)(6 45 54)(7 46 55)(8 47 56)(9 48 57)(10 37 58)(11 38 59)(12 39 60)(13 71 30)(14 72 31)(15 61 32)(16 62 33)(17 63 34)(18 64 35)(19 65 36)(20 66 25)(21 67 26)(22 68 27)(23 69 28)(24 70 29)
(1 48 53)(2 37 54)(3 38 55)(4 39 56)(5 40 57)(6 41 58)(7 42 59)(8 43 60)(9 44 49)(10 45 50)(11 46 51)(12 47 52)(13 63 26)(14 64 27)(15 65 28)(16 66 29)(17 67 30)(18 68 31)(19 69 32)(20 70 33)(21 71 34)(22 72 35)(23 61 36)(24 62 25)
(2 6)(3 11)(5 9)(8 12)(13 23)(14 16)(15 21)(17 19)(18 24)(20 22)(25 68)(26 61)(27 66)(28 71)(29 64)(30 69)(31 62)(32 67)(33 72)(34 65)(35 70)(36 63)(37 58)(38 51)(39 56)(40 49)(41 54)(42 59)(43 52)(44 57)(45 50)(46 55)(47 60)(48 53)

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

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

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

51 conjugacy classes

 class 1 2A 2B 2C 2D 3A ··· 3E 3F 3G 3H 3I 4A 4B 4C 4D 4E 6A ··· 6E 6F 6G 6H 6I 6J ··· 6Q 12A ··· 12M 12N 12O order 1 2 2 2 2 3 ··· 3 3 3 3 3 4 4 4 4 4 6 ··· 6 6 6 6 6 6 ··· 6 12 ··· 12 12 12 size 1 1 6 6 54 2 ··· 2 4 4 4 4 2 18 18 27 27 2 ··· 2 4 4 4 4 12 ··· 12 4 ··· 4 36 36

51 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 4 4 4 4 4 type + + + + + + + + + + + + - + + image C1 C2 C2 C2 C2 C2 S3 S3 D6 D6 D6 C4○D4 S32 D4⋊2S3 Q8⋊3S3 C2×S32 D12⋊S3 kernel D12⋊(C3⋊S3) S3×C3⋊Dic3 C33⋊7D4 C32×D12 C3×C32⋊4Q8 C4×C33⋊C2 C3×D12 C32⋊4Q8 C3⋊Dic3 C3×C12 S3×C6 C33 C12 C32 C32 C6 C3 # reps 1 2 2 1 1 1 4 1 2 5 8 2 4 4 1 4 8

Matrix representation of D12⋊(C3⋊S3) in GL8(𝔽13)

 8 0 0 0 0 0 0 0 2 5 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 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 12 12
,
 10 11 0 0 0 0 0 0 4 3 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 12 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0
,
 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 10 0 0 0 0 0 0 1 11 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 1 0 0 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 10 0 0 0 0 0 0 1 11 0 0 0 0 0 0 0 0 12 1 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 0 10 12 0 0 0 0 0 0 0 0 12 3 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 12 12

G:=sub<GL(8,GF(13))| [8,2,0,0,0,0,0,0,0,5,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,1,12],[10,4,0,0,0,0,0,0,11,3,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,10,11,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,10,11,0,0,0,0,0,0,0,0,12,12,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[1,10,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,3,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,12,0,0,0,0,0,0,0,12] >;

D12⋊(C3⋊S3) in GAP, Magma, Sage, TeX

D_{12}\rtimes (C_3\rtimes S_3)
% in TeX

G:=Group("D12:(C3:S3)");
// GroupNames label

G:=SmallGroup(432,662);
// by ID

G=gap.SmallGroup(432,662);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,64,254,135,58,571,2028,14118]);
// Polycyclic

G:=Group<a,b,c,d,e|a^12=b^2=c^3=d^3=e^2=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,e*a*e=a^5,b*c=c*b,b*d=d*b,e*b*e=a^10*b,c*d=d*c,e*c*e=c^-1,e*d*e=d^-1>;
// generators/relations

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