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G = C33⋊6D4  order 216 = 23·33

3rd semidirect product of C33 and D4 acting via D4/C2=C22

metabelian, supersoluble, monomial

Aliases: C33⋊6D4, C6.12S32, (S3×C6)⋊3S3, D6⋊1(C3⋊S3), (C3×C6).31D6, C33⋊5C4⋊3C2, C3⋊2(D6⋊S3), C32⋊6(C3⋊D4), C3⋊2(C32⋊7D4), (C32×C6).9C22, (S3×C3×C6)⋊3C2, (C6×C3⋊S3)⋊2C2, (C2×C3⋊S3)⋊4S3, C2.4(S3×C3⋊S3), C6.4(C2×C3⋊S3), SmallGroup(216,127)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C32×C6 — C33⋊6D4
Chief series
C1 — C3 — C32 — C33 — C32×C6 — S3×C3×C6 — C33⋊6D4
Lower central
C33 — C32×C6 — C33⋊6D4
Upper central
C1 — C2

Generators and relations for C33⋊6D4
 G = < a,b,c,d,e | a3=b3=c3=d4=e2=1, ab=ba, ac=ca, dad-1=a-1, ae=ea, bc=cb, dbd-1=b-1, be=eb, dcd-1=ece=c-1, ede=d-1 >

Subgroups: 532 in 120 conjugacy classes, 34 normal (14 characteristic)
C1, C2, C2, C3, C3, C3, C4, C22, S3, C6, C6, C6, D4, C32, C32, C32, Dic3, D6, D6, C2×C6, C3×S3, C3⋊S3, C3×C6, C3×C6, C3×C6, C3⋊D4, C33, C3⋊Dic3, S3×C6, S3×C6, C2×C3⋊S3, C62, S3×C32, C3×C3⋊S3, C32×C6, D6⋊S3, C32⋊7D4, C33⋊5C4, S3×C3×C6, C6×C3⋊S3, C33⋊6D4
Quotients: C1, C2, C22, S3, D4, D6, C3⋊S3, C3⋊D4, S32, C2×C3⋊S3, D6⋊S3, C32⋊7D4, S3×C3⋊S3, C33⋊6D4

Smallest permutation representation of C33⋊6D4
►On 72 points
Generators in S72
(1 9 63)(2 64 10)(3 11 61)(4 62 12)(5 59 16)(6 13 60)(7 57 14)(8 15 58)(17 71 30)(18 31 72)(19 69 32)(20 29 70)(21 38 55)(22 56 39)(23 40 53)(24 54 37)(25 48 67)(26 68 45)(27 46 65)(28 66 47)(33 51 42)(34 43 52)(35 49 44)(36 41 50)

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

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

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

C33⋊6D4 is a maximal subgroup of
 S3×D6⋊S3  D6⋊S32  D6.S32  D6.4S32  (C3×D12)⋊S3  C12.73S32  C12.57S32  C12⋊S32  C62.91D6  S3×C32⋊7D4  C3⋊S3×C3⋊D4
C33⋊6D4 is a maximal quotient of
 C33⋊6D8  C33⋊12SD16  C33⋊13SD16  C33⋊6Q16  C62.77D6  C62.78D6  C62.81D6

33 conjugacy classes

class 1 2A2B2C3A···3E3F3G3H3I 4 6A···6E6F6G6H6I6J···6Q6R6S
order12223···3333346···666666···666
size116182···24444542···244446···61818

33 irreducible representations

dim11112222244
type+++++++++-
imageC1C2C2C2S3S3D4D6C3⋊D4S32D6⋊S3
kernelC33⋊6D4C33⋊5C4S3×C3×C6C6×C3⋊S3S3×C6C2×C3⋊S3C33C3×C6C32C6C3
# reps111141151044

Matrix representation of C33⋊6D4 ►in GL6(𝔽13)

12120000
100000
001000
000100
0000012
0000112
,
010000
12120000
001000
000100
000010
000001
,
100000
010000
0001200
0011200
000010
000001
,
1190000
1120000
000100
001000
000001
000010
,
1190000
420000
0001200
0012000
000010
000001

G:=sub<GL(6,GF(13))| [12,1,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,12,12],[0,12,0,0,0,0,1,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,12,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[11,11,0,0,0,0,9,2,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[11,4,0,0,0,0,9,2,0,0,0,0,0,0,0,12,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;
 

C33⋊6D4 in GAP, Magma, Sage, TeX 

C_3^3\rtimes_6D_4
 
% in TeX
 
G:=Group("C3^3:6D4");
 
// GroupNames label
 
G:=SmallGroup(216,127);
 
// by ID
 
G=gap.SmallGroup(216,127);
 
# by ID
 
G:=PCGroup([6,-2,-2,-2,-3,-3,-3,73,201,730,5189]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^3=b^3=c^3=d^4=e^2=1,a*b=b*a,a*c=c*a,d*a*d^-1=a^-1,a*e=e*a,b*c=c*b,d*b*d^-1=b^-1,b*e=e*b,d*c*d^-1=e*c*e=c^-1,e*d*e=d^-1>;
 
// generators/relations
 

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