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G = C3⋊S3×D12  order 432 = 24·33

Direct product of C3⋊S3 and D12

direct product, metabelian, supersoluble, monomial

Aliases: C3⋊S3×D12, C12⋊1S32, (S3×C6)⋊3D6, C3⋊3(S3×D12), (C3×D12)⋊8S3, (C3×C12)⋊12D6, C33⋊14(C2×D4), C3⋊Dic3⋊17D6, C32⋊10(S3×D4), C33⋊7D4⋊3C2, C32⋊13(C2×D12), C33⋊12D4⋊8C2, (C32×D12)⋊12C2, (C32×C12)⋊4C22, (C32×C6).50C23, C4⋊2(S3×C3⋊S3), C3⋊1(D4×C3⋊S3), (C4×C3⋊S3)⋊8S3, C12⋊4(C2×C3⋊S3), C6.60(C2×S32), (C3×C3⋊S3)⋊8D4, D6⋊1(C2×C3⋊S3), (C12×C3⋊S3)⋊7C2, (S3×C3×C6)⋊11C22, (C2×C3⋊S3).53D6, C6.13(C22×C3⋊S3), (C6×C3⋊S3).55C22, (C3×C6).145(C22×S3), (C3×C3⋊Dic3)⋊13C22, (C2×C33⋊C2)⋊5C22, (C2×S3×C3⋊S3)⋊6C2, C2.16(C2×S3×C3⋊S3), SmallGroup(432,672)

Series: Derived ►Chief ►Lower central ►Upper central

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

Generators and relations for C3⋊S3×D12
 G = < a,b,c,d,e | a3=b3=c2=d12=e2=1, ab=ba, cac=a-1, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede=d-1 >

Subgroups: 2904 in 388 conjugacy classes, 72 normal (22 characteristic)
C1, C2, C2, C3, C3, C3, C4, C4, C22, S3, C6, C6, C6, C2×C4, D4, C23, C32, C32, C32, Dic3, C12, C12, C12, D6, D6, C2×C6, C2×D4, C3×S3, C3⋊S3, C3⋊S3, C3×C6, C3×C6, C3×C6, C4×S3, D12, D12, C3⋊D4, C2×C12, C3×D4, C22×S3, C33, C3×Dic3, C3⋊Dic3, C3×C12, C3×C12, C3×C12, S32, S3×C6, S3×C6, C2×C3⋊S3, C2×C3⋊S3, C62, C2×D12, S3×D4, S3×C32, C3×C3⋊S3, C33⋊C2, C32×C6, C3⋊D12, S3×C12, C3×D12, C4×C3⋊S3, C12⋊S3, C32⋊7D4, D4×C32, C2×S32, C22×C3⋊S3, C3×C3⋊Dic3, C32×C12, S3×C3⋊S3, S3×C3×C6, C6×C3⋊S3, C2×C33⋊C2, S3×D12, D4×C3⋊S3, C33⋊7D4, C32×D12, C12×C3⋊S3, C33⋊12D4, C2×S3×C3⋊S3, C3⋊S3×D12
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C3⋊S3, D12, C22×S3, S32, C2×C3⋊S3, C2×D12, S3×D4, C2×S32, C22×C3⋊S3, S3×C3⋊S3, S3×D12, D4×C3⋊S3, C2×S3×C3⋊S3, C3⋊S3×D12

Smallest permutation representation of C3⋊S3×D12
►On 72 points
Generators in S72
(1 19 42)(2 20 43)(3 21 44)(4 22 45)(5 23 46)(6 24 47)(7 13 48)(8 14 37)(9 15 38)(10 16 39)(11 17 40)(12 18 41)(25 62 51)(26 63 52)(27 64 53)(28 65 54)(29 66 55)(30 67 56)(31 68 57)(32 69 58)(33 70 59)(34 71 60)(35 72 49)(36 61 50)

(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 45 41)(38 46 42)(39 47 43)(40 48 44)(49 53 57)(50 54 58)(51 55 59)(52 56 60)(61 65 69)(62 66 70)(63 67 71)(64 68 72)

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

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

54 conjugacy classes

class 1 2A2B2C2D2E2F2G3A···3E3F3G3H3I4A4B6A···6E6F6G6H6I6J···6Q6R6S12A12B12C···12N12O12P
order122222223···33333446···666666···666121212···121212
size11669954542···244442182···2444412···121818224···41818

54 irreducible representations

dim111111222222224444
type++++++++++++++++++
imageC1C2C2C2C2C2S3S3D4D6D6D6D6D12S32S3×D4C2×S32S3×D12
kernelC3⋊S3×D12C33⋊7D4C32×D12C12×C3⋊S3C33⋊12D4C2×S3×C3⋊S3C3×D12C4×C3⋊S3C3×C3⋊S3C3⋊Dic3C3×C12S3×C6C2×C3⋊S3C3⋊S3C12C32C6C3
# reps121112412158144448

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

10000000
01000000
000120000
001120000
00001000
00000100
00000010
00000001
,
10000000
01000000
00100000
00010000
000012100
000012000
00000010
00000001
,
10000000
01000000
00010000
00100000
000001200
000012000
00000010
00000001
,
123000000
81000000
00100000
00010000
000012000
000001200
000000121
000000120
,
120000000
81000000
00100000
00010000
000012000
000001200
000000120
000000121

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

C3⋊S3×D12 in GAP, Magma, Sage, TeX 

C_3\rtimes S_3\times D_{12}
 
% in TeX
 
G:=Group("C3:S3xD12");
 
// GroupNames label
 
G:=SmallGroup(432,672);
 
// by ID
 
G=gap.SmallGroup(432,672);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,254,58,571,2028,14118]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^3=b^3=c^2=d^12=e^2=1,a*b=b*a,c*a*c=a^-1,a*d=d*a,a*e=e*a,c*b*c=b^-1,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e=d^-1>;
 
// generators/relations
 

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