C3:S3xD13 - GroupNames

G = C₃⋊S₃×D₁₃ order 468 = 2²·3²·13

Direct product of C₃⋊S₃ and D₁₃

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

Aliases: C₃⋊S₃×D₁₃, C₃₉⋊₁D₆, C₃²⋊₄D₂₆, (C₃×D₁₃)⋊S₃, C₃⋊D₃₉⋊₁C₂, C₃⋊₂(S₃×D₁₃), (C₃×C₃₉)⋊₂C₂², (C₃²×D₁₃)⋊₂C₂, C₁₃⋊₁(C₂×C₃⋊S₃), (C₁₃×C₃⋊S₃)⋊₁C₂, SmallGroup(468,43)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₃×C₃₉ — C₃⋊S₃×D₁₃

Chief series

C₁ — C₁₃ — C₃₉ — C₃×C₃₉ — C₃²×D₁₃ — C₃⋊S₃×D₁₃

Lower central

C₃×C₃₉ — C₃⋊S₃×D₁₃

Upper central

C₁

Generators and relations for C₃⋊S₃×D₁₃
G = < a,b,c,d,e | a³=b³=c²=d¹³=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: 756 in 60 conjugacy classes, 22 normal (10 characteristic)
C₁, C₂, C₃, C₂², S₃, C₆, C₃², D₆, C₁₃, C₃⋊S₃, C₃⋊S₃, C₃×C₆, D₁₃, D₁₃, C₂₆, C₂×C₃⋊S₃, C₃₉, D₂₆, S₃×C₁₃, C₃×D₁₃, D₃₉, C₃×C₃₉, S₃×D₁₃, C₃²×D₁₃, C₁₃×C₃⋊S₃, C₃⋊D₃₉, C₃⋊S₃×D₁₃
Quotients: C₁, C₂, C₂², S₃, D₆, C₃⋊S₃, D₁₃, C₂×C₃⋊S₃, D₂₆, S₃×D₁₃, C₃⋊S₃×D₁₃

Smallest permutation representation of C₃⋊S₃×D₁₃
►On 117 points

Generators in S₁₁₇

(1 35 15)(2 36 16)(3 37 17)(4 38 18)(5 39 19)(6 27 20)(7 28 21)(8 29 22)(9 30 23)(10 31 24)(11 32 25)(12 33 26)(13 34 14)(40 67 58)(41 68 59)(42 69 60)(43 70 61)(44 71 62)(45 72 63)(46 73 64)(47 74 65)(48 75 53)(49 76 54)(50 77 55)(51 78 56)(52 66 57)(79 111 102)(80 112 103)(81 113 104)(82 114 92)(83 115 93)(84 116 94)(85 117 95)(86 105 96)(87 106 97)(88 107 98)(89 108 99)(90 109 100)(91 110 101)

(1 40 83)(2 41 84)(3 42 85)(4 43 86)(5 44 87)(6 45 88)(7 46 89)(8 47 90)(9 48 91)(10 49 79)(11 50 80)(12 51 81)(13 52 82)(14 57 92)(15 58 93)(16 59 94)(17 60 95)(18 61 96)(19 62 97)(20 63 98)(21 64 99)(22 65 100)(23 53 101)(24 54 102)(25 55 103)(26 56 104)(27 72 107)(28 73 108)(29 74 109)(30 75 110)(31 76 111)(32 77 112)(33 78 113)(34 66 114)(35 67 115)(36 68 116)(37 69 117)(38 70 105)(39 71 106)

(14 34)(15 35)(16 36)(17 37)(18 38)(19 39)(20 27)(21 28)(22 29)(23 30)(24 31)(25 32)(26 33)(40 83)(41 84)(42 85)(43 86)(44 87)(45 88)(46 89)(47 90)(48 91)(49 79)(50 80)(51 81)(52 82)(53 110)(54 111)(55 112)(56 113)(57 114)(58 115)(59 116)(60 117)(61 105)(62 106)(63 107)(64 108)(65 109)(66 92)(67 93)(68 94)(69 95)(70 96)(71 97)(72 98)(73 99)(74 100)(75 101)(76 102)(77 103)(78 104)

(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 99 100 101 102 103 104)(105 106 107 108 109 110 111 112 113 114 115 116 117)

(1 13)(2 12)(3 11)(4 10)(5 9)(6 8)(14 15)(16 26)(17 25)(18 24)(19 23)(20 22)(27 29)(30 39)(31 38)(32 37)(33 36)(34 35)(40 52)(41 51)(42 50)(43 49)(44 48)(45 47)(53 62)(54 61)(55 60)(56 59)(57 58)(63 65)(66 67)(68 78)(69 77)(70 76)(71 75)(72 74)(79 86)(80 85)(81 84)(82 83)(87 91)(88 90)(92 93)(94 104)(95 103)(96 102)(97 101)(98 100)(105 111)(106 110)(107 109)(112 117)(113 116)(114 115)

G:=sub<Sym(117)| (1,35,15)(2,36,16)(3,37,17)(4,38,18)(5,39,19)(6,27,20)(7,28,21)(8,29,22)(9,30,23)(10,31,24)(11,32,25)(12,33,26)(13,34,14)(40,67,58)(41,68,59)(42,69,60)(43,70,61)(44,71,62)(45,72,63)(46,73,64)(47,74,65)(48,75,53)(49,76,54)(50,77,55)(51,78,56)(52,66,57)(79,111,102)(80,112,103)(81,113,104)(82,114,92)(83,115,93)(84,116,94)(85,117,95)(86,105,96)(87,106,97)(88,107,98)(89,108,99)(90,109,100)(91,110,101), (1,40,83)(2,41,84)(3,42,85)(4,43,86)(5,44,87)(6,45,88)(7,46,89)(8,47,90)(9,48,91)(10,49,79)(11,50,80)(12,51,81)(13,52,82)(14,57,92)(15,58,93)(16,59,94)(17,60,95)(18,61,96)(19,62,97)(20,63,98)(21,64,99)(22,65,100)(23,53,101)(24,54,102)(25,55,103)(26,56,104)(27,72,107)(28,73,108)(29,74,109)(30,75,110)(31,76,111)(32,77,112)(33,78,113)(34,66,114)(35,67,115)(36,68,116)(37,69,117)(38,70,105)(39,71,106), (14,34)(15,35)(16,36)(17,37)(18,38)(19,39)(20,27)(21,28)(22,29)(23,30)(24,31)(25,32)(26,33)(40,83)(41,84)(42,85)(43,86)(44,87)(45,88)(46,89)(47,90)(48,91)(49,79)(50,80)(51,81)(52,82)(53,110)(54,111)(55,112)(56,113)(57,114)(58,115)(59,116)(60,117)(61,105)(62,106)(63,107)(64,108)(65,109)(66,92)(67,93)(68,94)(69,95)(70,96)(71,97)(72,98)(73,99)(74,100)(75,101)(76,102)(77,103)(78,104), (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)(92,93,94,95,96,97,98,99,100,101,102,103,104)(105,106,107,108,109,110,111,112,113,114,115,116,117), (1,13)(2,12)(3,11)(4,10)(5,9)(6,8)(14,15)(16,26)(17,25)(18,24)(19,23)(20,22)(27,29)(30,39)(31,38)(32,37)(33,36)(34,35)(40,52)(41,51)(42,50)(43,49)(44,48)(45,47)(53,62)(54,61)(55,60)(56,59)(57,58)(63,65)(66,67)(68,78)(69,77)(70,76)(71,75)(72,74)(79,86)(80,85)(81,84)(82,83)(87,91)(88,90)(92,93)(94,104)(95,103)(96,102)(97,101)(98,100)(105,111)(106,110)(107,109)(112,117)(113,116)(114,115)>;

G:=Group( (1,35,15)(2,36,16)(3,37,17)(4,38,18)(5,39,19)(6,27,20)(7,28,21)(8,29,22)(9,30,23)(10,31,24)(11,32,25)(12,33,26)(13,34,14)(40,67,58)(41,68,59)(42,69,60)(43,70,61)(44,71,62)(45,72,63)(46,73,64)(47,74,65)(48,75,53)(49,76,54)(50,77,55)(51,78,56)(52,66,57)(79,111,102)(80,112,103)(81,113,104)(82,114,92)(83,115,93)(84,116,94)(85,117,95)(86,105,96)(87,106,97)(88,107,98)(89,108,99)(90,109,100)(91,110,101), (1,40,83)(2,41,84)(3,42,85)(4,43,86)(5,44,87)(6,45,88)(7,46,89)(8,47,90)(9,48,91)(10,49,79)(11,50,80)(12,51,81)(13,52,82)(14,57,92)(15,58,93)(16,59,94)(17,60,95)(18,61,96)(19,62,97)(20,63,98)(21,64,99)(22,65,100)(23,53,101)(24,54,102)(25,55,103)(26,56,104)(27,72,107)(28,73,108)(29,74,109)(30,75,110)(31,76,111)(32,77,112)(33,78,113)(34,66,114)(35,67,115)(36,68,116)(37,69,117)(38,70,105)(39,71,106), (14,34)(15,35)(16,36)(17,37)(18,38)(19,39)(20,27)(21,28)(22,29)(23,30)(24,31)(25,32)(26,33)(40,83)(41,84)(42,85)(43,86)(44,87)(45,88)(46,89)(47,90)(48,91)(49,79)(50,80)(51,81)(52,82)(53,110)(54,111)(55,112)(56,113)(57,114)(58,115)(59,116)(60,117)(61,105)(62,106)(63,107)(64,108)(65,109)(66,92)(67,93)(68,94)(69,95)(70,96)(71,97)(72,98)(73,99)(74,100)(75,101)(76,102)(77,103)(78,104), (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)(92,93,94,95,96,97,98,99,100,101,102,103,104)(105,106,107,108,109,110,111,112,113,114,115,116,117), (1,13)(2,12)(3,11)(4,10)(5,9)(6,8)(14,15)(16,26)(17,25)(18,24)(19,23)(20,22)(27,29)(30,39)(31,38)(32,37)(33,36)(34,35)(40,52)(41,51)(42,50)(43,49)(44,48)(45,47)(53,62)(54,61)(55,60)(56,59)(57,58)(63,65)(66,67)(68,78)(69,77)(70,76)(71,75)(72,74)(79,86)(80,85)(81,84)(82,83)(87,91)(88,90)(92,93)(94,104)(95,103)(96,102)(97,101)(98,100)(105,111)(106,110)(107,109)(112,117)(113,116)(114,115) );

G=PermutationGroup([[(1,35,15),(2,36,16),(3,37,17),(4,38,18),(5,39,19),(6,27,20),(7,28,21),(8,29,22),(9,30,23),(10,31,24),(11,32,25),(12,33,26),(13,34,14),(40,67,58),(41,68,59),(42,69,60),(43,70,61),(44,71,62),(45,72,63),(46,73,64),(47,74,65),(48,75,53),(49,76,54),(50,77,55),(51,78,56),(52,66,57),(79,111,102),(80,112,103),(81,113,104),(82,114,92),(83,115,93),(84,116,94),(85,117,95),(86,105,96),(87,106,97),(88,107,98),(89,108,99),(90,109,100),(91,110,101)], [(1,40,83),(2,41,84),(3,42,85),(4,43,86),(5,44,87),(6,45,88),(7,46,89),(8,47,90),(9,48,91),(10,49,79),(11,50,80),(12,51,81),(13,52,82),(14,57,92),(15,58,93),(16,59,94),(17,60,95),(18,61,96),(19,62,97),(20,63,98),(21,64,99),(22,65,100),(23,53,101),(24,54,102),(25,55,103),(26,56,104),(27,72,107),(28,73,108),(29,74,109),(30,75,110),(31,76,111),(32,77,112),(33,78,113),(34,66,114),(35,67,115),(36,68,116),(37,69,117),(38,70,105),(39,71,106)], [(14,34),(15,35),(16,36),(17,37),(18,38),(19,39),(20,27),(21,28),(22,29),(23,30),(24,31),(25,32),(26,33),(40,83),(41,84),(42,85),(43,86),(44,87),(45,88),(46,89),(47,90),(48,91),(49,79),(50,80),(51,81),(52,82),(53,110),(54,111),(55,112),(56,113),(57,114),(58,115),(59,116),(60,117),(61,105),(62,106),(63,107),(64,108),(65,109),(66,92),(67,93),(68,94),(69,95),(70,96),(71,97),(72,98),(73,99),(74,100),(75,101),(76,102),(77,103),(78,104)], [(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),(92,93,94,95,96,97,98,99,100,101,102,103,104),(105,106,107,108,109,110,111,112,113,114,115,116,117)], [(1,13),(2,12),(3,11),(4,10),(5,9),(6,8),(14,15),(16,26),(17,25),(18,24),(19,23),(20,22),(27,29),(30,39),(31,38),(32,37),(33,36),(34,35),(40,52),(41,51),(42,50),(43,49),(44,48),(45,47),(53,62),(54,61),(55,60),(56,59),(57,58),(63,65),(66,67),(68,78),(69,77),(70,76),(71,75),(72,74),(79,86),(80,85),(81,84),(82,83),(87,91),(88,90),(92,93),(94,104),(95,103),(96,102),(97,101),(98,100),(105,111),(106,110),(107,109),(112,117),(113,116),(114,115)]])

48 conjugacy classes

class	1	2A	2B	2C	3A	3B	3C	3D	6A	6B	6C	6D	13A	···	13F	26A	···	26F	39A	···	39X
order	1	2	2	2	3	3	3	3	6	6	6	6	13	···	13	26	···	26	39	···	39
size	1	9	13	117	2	2	2	2	26	26	26	26	2	···	2	18	···	18	4	···	4

48 irreducible representations

dim	1	1	1	1	2	2	2	2	4
type	+	+	+	+	+	+	+	+	+
image	C₁	C₂	C₂	C₂	S₃	D₆	D₁₃	D₂₆	S₃×D₁₃
kernel	C₃⋊S₃×D₁₃	C₃²×D₁₃	C₁₃×C₃⋊S₃	C₃⋊D₃₉	C₃×D₁₃	C₃₉	C₃⋊S₃	C₃²	C₃
# reps	1	1	1	1	4	4	6	6	24

Matrix representation of C₃⋊S₃×D₁₃ ►in GL₆(𝔽₇₉)

1	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	0	1
0	0	0	0	78	78

1	0	0	0	0	0
0	1	0	0	0	0
0	0	77	32	0	0
0	0	32	1	0	0
0	0	0	0	0	1
0	0	0	0	78	78

1	0	0	0	0	0
0	1	0	0	0	0
0	0	1	0	0	0
0	0	47	78	0	0
0	0	0	0	1	0
0	0	0	0	78	78

0	1	0	0	0	0
78	18	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

0	1	0	0	0	0
1	0	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(79))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,78,0,0,0,0,1,78],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,77,32,0,0,0,0,32,1,0,0,0,0,0,0,0,78,0,0,0,0,1,78],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,47,0,0,0,0,0,78,0,0,0,0,0,0,1,78,0,0,0,0,0,78],[0,78,0,0,0,0,1,18,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;

C₃⋊S₃×D₁₃ in GAP, Magma, Sage, TeX

C_3\rtimes S_3\times D_{13}

% in TeX

G:=Group("C3:S3xD13");

// GroupNames label

G:=SmallGroup(468,43);

// by ID

G=gap.SmallGroup(468,43);

# by ID

G:=PCGroup([5,-2,-2,-3,-3,-13,67,248,10804]);

// Polycyclic

G:=Group<a,b,c,d,e|a^3=b^3=c^2=d^13=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×D13 order 468 = 22·32·13

Direct product of C3⋊S3 and D13

G = C₃⋊S₃×D₁₃ order 468 = 2²·3²·13

Direct product of C₃⋊S₃ and D₁₃