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G = C3⋊S3⋊Dic6  order 432 = 24·33

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

non-abelian, supersoluble, monomial

Aliases: C3⋊S3⋊Dic6, C32⋊C6⋊Q8, C12.21S32, He3⋊2(C2×Q8), C32⋊1(S3×Q8), He3⋊4Q8⋊3C2, He3⋊2Q8⋊2C2, He3⋊3Q8⋊4C2, (C3×C12).20D6, C3⋊Dic3.5D6, C3.3(S3×Dic6), C32⋊4Q8⋊3S3, C4.5(C32⋊D6), C32⋊1(C2×Dic6), (C2×He3).1C23, C32⋊C12.5C22, (C4×He3).16C22, He3⋊3C4.4C22, C6.75(C2×S32), (C4×C3⋊S3).1S3, C6.S32.C2, (C2×C3⋊S3).5D6, C2.4(C2×C32⋊D6), (C4×C32⋊C6).2C2, (C3×C6).1(C22×S3), (C2×C32⋊C6).4C22, SmallGroup(432,294)

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, C32⋊2Q8, C3×Dic6, S3×C12, C32⋊4Q8, C4×C3⋊S3, C32⋊C12, C32⋊C12, He3⋊3C4, C4×He3, C2×C32⋊C6, S3×Dic6, Dic3.D6, He3⋊2Q8, C6.S32, He3⋊3Q8, C4×C32⋊C6, He3⋊4Q8, 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 2A2B2C3A3B3C3D4A4B···4F6A6B6C6D6E6F12A12B12C12D12E12F12G12H12I12J12K12L
order1222333344···4666666121212121212121212121212
size119926612218···18266121818466121212181836363636

32 irreducible representations

dim111111122222222444466
type++++++-++-+++-+-+-++
imageC1C2C2C2C2C2C3⋊S3⋊Dic6S3S3Q8D6D6D6Dic6S32S3×Q8C2×S32S3×Dic6C32⋊D6C2×C32⋊D6
kernelC3⋊S3⋊Dic6He3⋊2Q8C6.S32He3⋊3Q8C4×C32⋊C6He3⋊4Q8C1C32⋊4Q8C4×C3⋊S3C32⋊C6C3⋊Dic3C3×C12C2×C3⋊S3C3⋊S3C12C32C6C3C4C2
# reps12211111123214111222

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

1000000000
0100000000
0010000000
0001000000
00001210000
00001200000
00000012100
00000012000
0000101001
00000120121212
,
12100000000
12000000000
00121000000
00120000000
0000100000
0000010000
00000001200
00000011200
0000000101
000012121201212
,
0100000000
1000000000
0001000000
0010000000
0000010000
0000100000
0000000100
0000001000
0000000010
0000121212121212
,
1030000000
0103000000
80120000000
08012000000
0000001000
0000000100
00000000121
0000121212121112
0000000010
0000100010
,
1040000000
0104000000
60120000000
06012000000
00000120000
00001200000
0000111121
00000000112
00000000120
000000012120

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
 

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