C3:S3xD11 - GroupNames

G = C₃⋊S₃×D₁₁ order 396 = 2²·3²·11

Direct product of C₃⋊S₃ and D₁₁

direct product, metabelian, supersoluble, monomial, A-group

Aliases: C₃⋊S₃×D₁₁, C₃₃⋊₁D₆, C₃²⋊₄D₂₂, (C₃×D₁₁)⋊S₃, C₃⋊D₃₃⋊₁C₂, C₃⋊₂(S₃×D₁₁), (C₃×C₃₃)⋊₂C₂², (C₃²×D₁₁)⋊₂C₂, C₁₁⋊₁(C₂×C₃⋊S₃), (C₁₁×C₃⋊S₃)⋊₁C₂, SmallGroup(396,20)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₃×C₃₃ — C₃⋊S₃×D₁₁

Chief series

C₁ — C₁₁ — C₃₃ — C₃×C₃₃ — C₃²×D₁₁ — C₃⋊S₃×D₁₁

Lower central

C₃×C₃₃ — C₃⋊S₃×D₁₁

Upper central

C₁

Generators and relations for C₃⋊S₃×D₁₁
G = < a,b,c,d,e | a³=b³=c²=d¹¹=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: 656 in 60 conjugacy classes, 22 normal (10 characteristic)
C₁, C₂, C₃, C₂², S₃, C₆, C₃², C₁₁, D₆, C₃⋊S₃, C₃⋊S₃, C₃×C₆, D₁₁, D₁₁, C₂₂, C₃₃, C₂×C₃⋊S₃, D₂₂, S₃×C₁₁, C₃×D₁₁, D₃₃, C₃×C₃₃, S₃×D₁₁, C₃²×D₁₁, C₁₁×C₃⋊S₃, C₃⋊D₃₃, C₃⋊S₃×D₁₁
Quotients: C₁, C₂, C₂², S₃, D₆, C₃⋊S₃, D₁₁, C₂×C₃⋊S₃, D₂₂, S₃×D₁₁, C₃⋊S₃×D₁₁

Smallest permutation representation of C₃⋊S₃×D₁₁
►On 99 points

Generators in S₉₉

(1 32 21)(2 33 22)(3 23 12)(4 24 13)(5 25 14)(6 26 15)(7 27 16)(8 28 17)(9 29 18)(10 30 19)(11 31 20)(34 56 45)(35 57 46)(36 58 47)(37 59 48)(38 60 49)(39 61 50)(40 62 51)(41 63 52)(42 64 53)(43 65 54)(44 66 55)(67 89 78)(68 90 79)(69 91 80)(70 92 81)(71 93 82)(72 94 83)(73 95 84)(74 96 85)(75 97 86)(76 98 87)(77 99 88)

(1 43 76)(2 44 77)(3 34 67)(4 35 68)(5 36 69)(6 37 70)(7 38 71)(8 39 72)(9 40 73)(10 41 74)(11 42 75)(12 45 78)(13 46 79)(14 47 80)(15 48 81)(16 49 82)(17 50 83)(18 51 84)(19 52 85)(20 53 86)(21 54 87)(22 55 88)(23 56 89)(24 57 90)(25 58 91)(26 59 92)(27 60 93)(28 61 94)(29 62 95)(30 63 96)(31 64 97)(32 65 98)(33 66 99)

(12 23)(13 24)(14 25)(15 26)(16 27)(17 28)(18 29)(19 30)(20 31)(21 32)(22 33)(34 67)(35 68)(36 69)(37 70)(38 71)(39 72)(40 73)(41 74)(42 75)(43 76)(44 77)(45 89)(46 90)(47 91)(48 92)(49 93)(50 94)(51 95)(52 96)(53 97)(54 98)(55 99)(56 78)(57 79)(58 80)(59 81)(60 82)(61 83)(62 84)(63 85)(64 86)(65 87)(66 88)

(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 73 74 75 76 77)(78 79 80 81 82 83 84 85 86 87 88)(89 90 91 92 93 94 95 96 97 98 99)

(1 11)(2 10)(3 9)(4 8)(5 7)(12 18)(13 17)(14 16)(19 22)(20 21)(23 29)(24 28)(25 27)(30 33)(31 32)(34 40)(35 39)(36 38)(41 44)(42 43)(45 51)(46 50)(47 49)(52 55)(53 54)(56 62)(57 61)(58 60)(63 66)(64 65)(67 73)(68 72)(69 71)(74 77)(75 76)(78 84)(79 83)(80 82)(85 88)(86 87)(89 95)(90 94)(91 93)(96 99)(97 98)

G:=sub<Sym(99)| (1,32,21)(2,33,22)(3,23,12)(4,24,13)(5,25,14)(6,26,15)(7,27,16)(8,28,17)(9,29,18)(10,30,19)(11,31,20)(34,56,45)(35,57,46)(36,58,47)(37,59,48)(38,60,49)(39,61,50)(40,62,51)(41,63,52)(42,64,53)(43,65,54)(44,66,55)(67,89,78)(68,90,79)(69,91,80)(70,92,81)(71,93,82)(72,94,83)(73,95,84)(74,96,85)(75,97,86)(76,98,87)(77,99,88), (1,43,76)(2,44,77)(3,34,67)(4,35,68)(5,36,69)(6,37,70)(7,38,71)(8,39,72)(9,40,73)(10,41,74)(11,42,75)(12,45,78)(13,46,79)(14,47,80)(15,48,81)(16,49,82)(17,50,83)(18,51,84)(19,52,85)(20,53,86)(21,54,87)(22,55,88)(23,56,89)(24,57,90)(25,58,91)(26,59,92)(27,60,93)(28,61,94)(29,62,95)(30,63,96)(31,64,97)(32,65,98)(33,66,99), (12,23)(13,24)(14,25)(15,26)(16,27)(17,28)(18,29)(19,30)(20,31)(21,32)(22,33)(34,67)(35,68)(36,69)(37,70)(38,71)(39,72)(40,73)(41,74)(42,75)(43,76)(44,77)(45,89)(46,90)(47,91)(48,92)(49,93)(50,94)(51,95)(52,96)(53,97)(54,98)(55,99)(56,78)(57,79)(58,80)(59,81)(60,82)(61,83)(62,84)(63,85)(64,86)(65,87)(66,88), (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)(67,68,69,70,71,72,73,74,75,76,77)(78,79,80,81,82,83,84,85,86,87,88)(89,90,91,92,93,94,95,96,97,98,99), (1,11)(2,10)(3,9)(4,8)(5,7)(12,18)(13,17)(14,16)(19,22)(20,21)(23,29)(24,28)(25,27)(30,33)(31,32)(34,40)(35,39)(36,38)(41,44)(42,43)(45,51)(46,50)(47,49)(52,55)(53,54)(56,62)(57,61)(58,60)(63,66)(64,65)(67,73)(68,72)(69,71)(74,77)(75,76)(78,84)(79,83)(80,82)(85,88)(86,87)(89,95)(90,94)(91,93)(96,99)(97,98)>;

G:=Group( (1,32,21)(2,33,22)(3,23,12)(4,24,13)(5,25,14)(6,26,15)(7,27,16)(8,28,17)(9,29,18)(10,30,19)(11,31,20)(34,56,45)(35,57,46)(36,58,47)(37,59,48)(38,60,49)(39,61,50)(40,62,51)(41,63,52)(42,64,53)(43,65,54)(44,66,55)(67,89,78)(68,90,79)(69,91,80)(70,92,81)(71,93,82)(72,94,83)(73,95,84)(74,96,85)(75,97,86)(76,98,87)(77,99,88), (1,43,76)(2,44,77)(3,34,67)(4,35,68)(5,36,69)(6,37,70)(7,38,71)(8,39,72)(9,40,73)(10,41,74)(11,42,75)(12,45,78)(13,46,79)(14,47,80)(15,48,81)(16,49,82)(17,50,83)(18,51,84)(19,52,85)(20,53,86)(21,54,87)(22,55,88)(23,56,89)(24,57,90)(25,58,91)(26,59,92)(27,60,93)(28,61,94)(29,62,95)(30,63,96)(31,64,97)(32,65,98)(33,66,99), (12,23)(13,24)(14,25)(15,26)(16,27)(17,28)(18,29)(19,30)(20,31)(21,32)(22,33)(34,67)(35,68)(36,69)(37,70)(38,71)(39,72)(40,73)(41,74)(42,75)(43,76)(44,77)(45,89)(46,90)(47,91)(48,92)(49,93)(50,94)(51,95)(52,96)(53,97)(54,98)(55,99)(56,78)(57,79)(58,80)(59,81)(60,82)(61,83)(62,84)(63,85)(64,86)(65,87)(66,88), (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)(67,68,69,70,71,72,73,74,75,76,77)(78,79,80,81,82,83,84,85,86,87,88)(89,90,91,92,93,94,95,96,97,98,99), (1,11)(2,10)(3,9)(4,8)(5,7)(12,18)(13,17)(14,16)(19,22)(20,21)(23,29)(24,28)(25,27)(30,33)(31,32)(34,40)(35,39)(36,38)(41,44)(42,43)(45,51)(46,50)(47,49)(52,55)(53,54)(56,62)(57,61)(58,60)(63,66)(64,65)(67,73)(68,72)(69,71)(74,77)(75,76)(78,84)(79,83)(80,82)(85,88)(86,87)(89,95)(90,94)(91,93)(96,99)(97,98) );

G=PermutationGroup([[(1,32,21),(2,33,22),(3,23,12),(4,24,13),(5,25,14),(6,26,15),(7,27,16),(8,28,17),(9,29,18),(10,30,19),(11,31,20),(34,56,45),(35,57,46),(36,58,47),(37,59,48),(38,60,49),(39,61,50),(40,62,51),(41,63,52),(42,64,53),(43,65,54),(44,66,55),(67,89,78),(68,90,79),(69,91,80),(70,92,81),(71,93,82),(72,94,83),(73,95,84),(74,96,85),(75,97,86),(76,98,87),(77,99,88)], [(1,43,76),(2,44,77),(3,34,67),(4,35,68),(5,36,69),(6,37,70),(7,38,71),(8,39,72),(9,40,73),(10,41,74),(11,42,75),(12,45,78),(13,46,79),(14,47,80),(15,48,81),(16,49,82),(17,50,83),(18,51,84),(19,52,85),(20,53,86),(21,54,87),(22,55,88),(23,56,89),(24,57,90),(25,58,91),(26,59,92),(27,60,93),(28,61,94),(29,62,95),(30,63,96),(31,64,97),(32,65,98),(33,66,99)], [(12,23),(13,24),(14,25),(15,26),(16,27),(17,28),(18,29),(19,30),(20,31),(21,32),(22,33),(34,67),(35,68),(36,69),(37,70),(38,71),(39,72),(40,73),(41,74),(42,75),(43,76),(44,77),(45,89),(46,90),(47,91),(48,92),(49,93),(50,94),(51,95),(52,96),(53,97),(54,98),(55,99),(56,78),(57,79),(58,80),(59,81),(60,82),(61,83),(62,84),(63,85),(64,86),(65,87),(66,88)], [(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),(67,68,69,70,71,72,73,74,75,76,77),(78,79,80,81,82,83,84,85,86,87,88),(89,90,91,92,93,94,95,96,97,98,99)], [(1,11),(2,10),(3,9),(4,8),(5,7),(12,18),(13,17),(14,16),(19,22),(20,21),(23,29),(24,28),(25,27),(30,33),(31,32),(34,40),(35,39),(36,38),(41,44),(42,43),(45,51),(46,50),(47,49),(52,55),(53,54),(56,62),(57,61),(58,60),(63,66),(64,65),(67,73),(68,72),(69,71),(74,77),(75,76),(78,84),(79,83),(80,82),(85,88),(86,87),(89,95),(90,94),(91,93),(96,99),(97,98)]])

42 conjugacy classes

class	1	2A	2B	2C	3A	3B	3C	3D	6A	6B	6C	6D	11A	···	11E	22A	···	22E	33A	···	33T
order	1	2	2	2	3	3	3	3	6	6	6	6	11	···	11	22	···	22	33	···	33
size	1	9	11	99	2	2	2	2	22	22	22	22	2	···	2	18	···	18	4	···	4

42 irreducible representations

dim	1	1	1	1	2	2	2	2	4
type	+	+	+	+	+	+	+	+	+
image	C₁	C₂	C₂	C₂	S₃	D₆	D₁₁	D₂₂	S₃×D₁₁
kernel	C₃⋊S₃×D₁₁	C₃²×D₁₁	C₁₁×C₃⋊S₃	C₃⋊D₃₃	C₃×D₁₁	C₃₃	C₃⋊S₃	C₃²	C₃
# reps	1	1	1	1	4	4	5	5	20

Matrix representation of C₃⋊S₃×D₁₁ ►in GL₆(𝔽₆₇)

1	0	0	0	0	0
0	1	0	0	0	0
0	0	1	27	0	0
0	0	52	65	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	1	27	0	0
0	0	52	65	0	0
0	0	0	0	1	17
0	0	0	0	55	65

1	0	0	0	0	0
0	1	0	0	0	0
0	0	1	27	0	0
0	0	0	66	0	0
0	0	0	0	66	50
0	0	0	0	0	1

31	1	0	0	0	0
15	7	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

13	31	0	0	0	0
27	54	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

G:=sub<GL(6,GF(67))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,52,0,0,0,0,27,65,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,52,0,0,0,0,27,65,0,0,0,0,0,0,1,55,0,0,0,0,17,65],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,27,66,0,0,0,0,0,0,66,0,0,0,0,0,50,1],[31,15,0,0,0,0,1,7,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[13,27,0,0,0,0,31,54,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;

C₃⋊S₃×D₁₁ in GAP, Magma, Sage, TeX

C_3\rtimes S_3\times D_{11}

% in TeX

G:=Group("C3:S3xD11");

// GroupNames label

G:=SmallGroup(396,20);

// by ID

G=gap.SmallGroup(396,20);

# by ID

G:=PCGroup([5,-2,-2,-3,-3,-11,67,248,9004]);

// Polycyclic

G:=Group<a,b,c,d,e|a^3=b^3=c^2=d^11=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

G = C3⋊S3×D11 order 396 = 22·32·11

Direct product of C3⋊S3 and D11

G = C₃⋊S₃×D₁₁ order 396 = 2²·3²·11

Direct product of C₃⋊S₃ and D₁₁