C3^3:D5 - GroupNames

G = C₃³⋊D₅ order 270 = 2·3³·5

3^rd semidirect product of C₃³ and D₅ acting via D₅/C₅=C₂

metabelian, supersoluble, monomial, A-group

Aliases: C₃³⋊₃D₅, C₃²⋊₄D₁₅, C₃⋊(C₃⋊D₁₅), (C₃×C₁₅)⋊₅S₃, C₅⋊(C₃³⋊C₂), C₁₅⋊₁(C₃⋊S₃), (C₃²×C₁₅)⋊₁C₂, SmallGroup(270,29)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₃²×C₁₅ — C₃³⋊D₅

Chief series

C₁ — C₅ — C₁₅ — C₃×C₁₅ — C₃²×C₁₅ — C₃³⋊D₅

Lower central

C₃²×C₁₅ — C₃³⋊D₅

Upper central

C₁

Generators and relations for C₃³⋊D₅
G = < a,b,c,d,e | a³=b³=c³=d⁵=e²=1, ab=ba, ac=ca, ad=da, eae=a^-1, bc=cb, bd=db, ebe=b^-1, cd=dc, ece=c^-1, ede=d^-1 >

Subgroups: 1160 in 112 conjugacy classes, 57 normal (5 characteristic)
C₁, C₂, C₃, C₅, S₃, C₃², D₅, C₁₅, C₃⋊S₃, C₃³, D₁₅, C₃×C₁₅, C₃³⋊C₂, C₃⋊D₁₅, C₃²×C₁₅, C₃³⋊D₅
Quotients: C₁, C₂, S₃, D₅, C₃⋊S₃, D₁₅, C₃³⋊C₂, C₃⋊D₁₅, C₃³⋊D₅

Smallest permutation representation of C₃³⋊D₅
►On 135 points

Generators in S₁₃₅

(1 94 49)(2 95 50)(3 91 46)(4 92 47)(5 93 48)(6 96 51)(7 97 52)(8 98 53)(9 99 54)(10 100 55)(11 101 56)(12 102 57)(13 103 58)(14 104 59)(15 105 60)(16 106 61)(17 107 62)(18 108 63)(19 109 64)(20 110 65)(21 111 66)(22 112 67)(23 113 68)(24 114 69)(25 115 70)(26 116 71)(27 117 72)(28 118 73)(29 119 74)(30 120 75)(31 121 76)(32 122 77)(33 123 78)(34 124 79)(35 125 80)(36 126 81)(37 127 82)(38 128 83)(39 129 84)(40 130 85)(41 131 86)(42 132 87)(43 133 88)(44 134 89)(45 135 90)

(1 34 19)(2 35 20)(3 31 16)(4 32 17)(5 33 18)(6 36 21)(7 37 22)(8 38 23)(9 39 24)(10 40 25)(11 41 26)(12 42 27)(13 43 28)(14 44 29)(15 45 30)(46 76 61)(47 77 62)(48 78 63)(49 79 64)(50 80 65)(51 81 66)(52 82 67)(53 83 68)(54 84 69)(55 85 70)(56 86 71)(57 87 72)(58 88 73)(59 89 74)(60 90 75)(91 121 106)(92 122 107)(93 123 108)(94 124 109)(95 125 110)(96 126 111)(97 127 112)(98 128 113)(99 129 114)(100 130 115)(101 131 116)(102 132 117)(103 133 118)(104 134 119)(105 135 120)

(1 14 9)(2 15 10)(3 11 6)(4 12 7)(5 13 8)(16 26 21)(17 27 22)(18 28 23)(19 29 24)(20 30 25)(31 41 36)(32 42 37)(33 43 38)(34 44 39)(35 45 40)(46 56 51)(47 57 52)(48 58 53)(49 59 54)(50 60 55)(61 71 66)(62 72 67)(63 73 68)(64 74 69)(65 75 70)(76 86 81)(77 87 82)(78 88 83)(79 89 84)(80 90 85)(91 101 96)(92 102 97)(93 103 98)(94 104 99)(95 105 100)(106 116 111)(107 117 112)(108 118 113)(109 119 114)(110 120 115)(121 131 126)(122 132 127)(123 133 128)(124 134 129)(125 135 130)

(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)(116 117 118 119 120)(121 122 123 124 125)(126 127 128 129 130)(131 132 133 134 135)

(1 5)(2 4)(6 11)(7 15)(8 14)(9 13)(10 12)(16 31)(17 35)(18 34)(19 33)(20 32)(21 41)(22 45)(23 44)(24 43)(25 42)(26 36)(27 40)(28 39)(29 38)(30 37)(46 91)(47 95)(48 94)(49 93)(50 92)(51 101)(52 105)(53 104)(54 103)(55 102)(56 96)(57 100)(58 99)(59 98)(60 97)(61 121)(62 125)(63 124)(64 123)(65 122)(66 131)(67 135)(68 134)(69 133)(70 132)(71 126)(72 130)(73 129)(74 128)(75 127)(76 106)(77 110)(78 109)(79 108)(80 107)(81 116)(82 120)(83 119)(84 118)(85 117)(86 111)(87 115)(88 114)(89 113)(90 112)

G:=sub<Sym(135)| (1,94,49)(2,95,50)(3,91,46)(4,92,47)(5,93,48)(6,96,51)(7,97,52)(8,98,53)(9,99,54)(10,100,55)(11,101,56)(12,102,57)(13,103,58)(14,104,59)(15,105,60)(16,106,61)(17,107,62)(18,108,63)(19,109,64)(20,110,65)(21,111,66)(22,112,67)(23,113,68)(24,114,69)(25,115,70)(26,116,71)(27,117,72)(28,118,73)(29,119,74)(30,120,75)(31,121,76)(32,122,77)(33,123,78)(34,124,79)(35,125,80)(36,126,81)(37,127,82)(38,128,83)(39,129,84)(40,130,85)(41,131,86)(42,132,87)(43,133,88)(44,134,89)(45,135,90), (1,34,19)(2,35,20)(3,31,16)(4,32,17)(5,33,18)(6,36,21)(7,37,22)(8,38,23)(9,39,24)(10,40,25)(11,41,26)(12,42,27)(13,43,28)(14,44,29)(15,45,30)(46,76,61)(47,77,62)(48,78,63)(49,79,64)(50,80,65)(51,81,66)(52,82,67)(53,83,68)(54,84,69)(55,85,70)(56,86,71)(57,87,72)(58,88,73)(59,89,74)(60,90,75)(91,121,106)(92,122,107)(93,123,108)(94,124,109)(95,125,110)(96,126,111)(97,127,112)(98,128,113)(99,129,114)(100,130,115)(101,131,116)(102,132,117)(103,133,118)(104,134,119)(105,135,120), (1,14,9)(2,15,10)(3,11,6)(4,12,7)(5,13,8)(16,26,21)(17,27,22)(18,28,23)(19,29,24)(20,30,25)(31,41,36)(32,42,37)(33,43,38)(34,44,39)(35,45,40)(46,56,51)(47,57,52)(48,58,53)(49,59,54)(50,60,55)(61,71,66)(62,72,67)(63,73,68)(64,74,69)(65,75,70)(76,86,81)(77,87,82)(78,88,83)(79,89,84)(80,90,85)(91,101,96)(92,102,97)(93,103,98)(94,104,99)(95,105,100)(106,116,111)(107,117,112)(108,118,113)(109,119,114)(110,120,115)(121,131,126)(122,132,127)(123,133,128)(124,134,129)(125,135,130), (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)(116,117,118,119,120)(121,122,123,124,125)(126,127,128,129,130)(131,132,133,134,135), (1,5)(2,4)(6,11)(7,15)(8,14)(9,13)(10,12)(16,31)(17,35)(18,34)(19,33)(20,32)(21,41)(22,45)(23,44)(24,43)(25,42)(26,36)(27,40)(28,39)(29,38)(30,37)(46,91)(47,95)(48,94)(49,93)(50,92)(51,101)(52,105)(53,104)(54,103)(55,102)(56,96)(57,100)(58,99)(59,98)(60,97)(61,121)(62,125)(63,124)(64,123)(65,122)(66,131)(67,135)(68,134)(69,133)(70,132)(71,126)(72,130)(73,129)(74,128)(75,127)(76,106)(77,110)(78,109)(79,108)(80,107)(81,116)(82,120)(83,119)(84,118)(85,117)(86,111)(87,115)(88,114)(89,113)(90,112)>;

G:=Group( (1,94,49)(2,95,50)(3,91,46)(4,92,47)(5,93,48)(6,96,51)(7,97,52)(8,98,53)(9,99,54)(10,100,55)(11,101,56)(12,102,57)(13,103,58)(14,104,59)(15,105,60)(16,106,61)(17,107,62)(18,108,63)(19,109,64)(20,110,65)(21,111,66)(22,112,67)(23,113,68)(24,114,69)(25,115,70)(26,116,71)(27,117,72)(28,118,73)(29,119,74)(30,120,75)(31,121,76)(32,122,77)(33,123,78)(34,124,79)(35,125,80)(36,126,81)(37,127,82)(38,128,83)(39,129,84)(40,130,85)(41,131,86)(42,132,87)(43,133,88)(44,134,89)(45,135,90), (1,34,19)(2,35,20)(3,31,16)(4,32,17)(5,33,18)(6,36,21)(7,37,22)(8,38,23)(9,39,24)(10,40,25)(11,41,26)(12,42,27)(13,43,28)(14,44,29)(15,45,30)(46,76,61)(47,77,62)(48,78,63)(49,79,64)(50,80,65)(51,81,66)(52,82,67)(53,83,68)(54,84,69)(55,85,70)(56,86,71)(57,87,72)(58,88,73)(59,89,74)(60,90,75)(91,121,106)(92,122,107)(93,123,108)(94,124,109)(95,125,110)(96,126,111)(97,127,112)(98,128,113)(99,129,114)(100,130,115)(101,131,116)(102,132,117)(103,133,118)(104,134,119)(105,135,120), (1,14,9)(2,15,10)(3,11,6)(4,12,7)(5,13,8)(16,26,21)(17,27,22)(18,28,23)(19,29,24)(20,30,25)(31,41,36)(32,42,37)(33,43,38)(34,44,39)(35,45,40)(46,56,51)(47,57,52)(48,58,53)(49,59,54)(50,60,55)(61,71,66)(62,72,67)(63,73,68)(64,74,69)(65,75,70)(76,86,81)(77,87,82)(78,88,83)(79,89,84)(80,90,85)(91,101,96)(92,102,97)(93,103,98)(94,104,99)(95,105,100)(106,116,111)(107,117,112)(108,118,113)(109,119,114)(110,120,115)(121,131,126)(122,132,127)(123,133,128)(124,134,129)(125,135,130), (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)(116,117,118,119,120)(121,122,123,124,125)(126,127,128,129,130)(131,132,133,134,135), (1,5)(2,4)(6,11)(7,15)(8,14)(9,13)(10,12)(16,31)(17,35)(18,34)(19,33)(20,32)(21,41)(22,45)(23,44)(24,43)(25,42)(26,36)(27,40)(28,39)(29,38)(30,37)(46,91)(47,95)(48,94)(49,93)(50,92)(51,101)(52,105)(53,104)(54,103)(55,102)(56,96)(57,100)(58,99)(59,98)(60,97)(61,121)(62,125)(63,124)(64,123)(65,122)(66,131)(67,135)(68,134)(69,133)(70,132)(71,126)(72,130)(73,129)(74,128)(75,127)(76,106)(77,110)(78,109)(79,108)(80,107)(81,116)(82,120)(83,119)(84,118)(85,117)(86,111)(87,115)(88,114)(89,113)(90,112) );

G=PermutationGroup([[(1,94,49),(2,95,50),(3,91,46),(4,92,47),(5,93,48),(6,96,51),(7,97,52),(8,98,53),(9,99,54),(10,100,55),(11,101,56),(12,102,57),(13,103,58),(14,104,59),(15,105,60),(16,106,61),(17,107,62),(18,108,63),(19,109,64),(20,110,65),(21,111,66),(22,112,67),(23,113,68),(24,114,69),(25,115,70),(26,116,71),(27,117,72),(28,118,73),(29,119,74),(30,120,75),(31,121,76),(32,122,77),(33,123,78),(34,124,79),(35,125,80),(36,126,81),(37,127,82),(38,128,83),(39,129,84),(40,130,85),(41,131,86),(42,132,87),(43,133,88),(44,134,89),(45,135,90)], [(1,34,19),(2,35,20),(3,31,16),(4,32,17),(5,33,18),(6,36,21),(7,37,22),(8,38,23),(9,39,24),(10,40,25),(11,41,26),(12,42,27),(13,43,28),(14,44,29),(15,45,30),(46,76,61),(47,77,62),(48,78,63),(49,79,64),(50,80,65),(51,81,66),(52,82,67),(53,83,68),(54,84,69),(55,85,70),(56,86,71),(57,87,72),(58,88,73),(59,89,74),(60,90,75),(91,121,106),(92,122,107),(93,123,108),(94,124,109),(95,125,110),(96,126,111),(97,127,112),(98,128,113),(99,129,114),(100,130,115),(101,131,116),(102,132,117),(103,133,118),(104,134,119),(105,135,120)], [(1,14,9),(2,15,10),(3,11,6),(4,12,7),(5,13,8),(16,26,21),(17,27,22),(18,28,23),(19,29,24),(20,30,25),(31,41,36),(32,42,37),(33,43,38),(34,44,39),(35,45,40),(46,56,51),(47,57,52),(48,58,53),(49,59,54),(50,60,55),(61,71,66),(62,72,67),(63,73,68),(64,74,69),(65,75,70),(76,86,81),(77,87,82),(78,88,83),(79,89,84),(80,90,85),(91,101,96),(92,102,97),(93,103,98),(94,104,99),(95,105,100),(106,116,111),(107,117,112),(108,118,113),(109,119,114),(110,120,115),(121,131,126),(122,132,127),(123,133,128),(124,134,129),(125,135,130)], [(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),(116,117,118,119,120),(121,122,123,124,125),(126,127,128,129,130),(131,132,133,134,135)], [(1,5),(2,4),(6,11),(7,15),(8,14),(9,13),(10,12),(16,31),(17,35),(18,34),(19,33),(20,32),(21,41),(22,45),(23,44),(24,43),(25,42),(26,36),(27,40),(28,39),(29,38),(30,37),(46,91),(47,95),(48,94),(49,93),(50,92),(51,101),(52,105),(53,104),(54,103),(55,102),(56,96),(57,100),(58,99),(59,98),(60,97),(61,121),(62,125),(63,124),(64,123),(65,122),(66,131),(67,135),(68,134),(69,133),(70,132),(71,126),(72,130),(73,129),(74,128),(75,127),(76,106),(77,110),(78,109),(79,108),(80,107),(81,116),(82,120),(83,119),(84,118),(85,117),(86,111),(87,115),(88,114),(89,113),(90,112)]])

69 conjugacy classes

class	1	2	3A	···	3M	5A	5B	15A	···	15AZ
order	1	2	3	···	3	5	5	15	···	15
size	1	135	2	···	2	2	2	2	···	2

69 irreducible representations

dim	1	1	2	2	2
type	+	+	+	+	+
image	C₁	C₂	S₃	D₅	D₁₅
kernel	C₃³⋊D₅	C₃²×C₁₅	C₃×C₁₅	C₃³	C₃²
# reps	1	1	13	2	52

Matrix representation of C₃³⋊D₅ ►in GL₆(𝔽₃₁)

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	12	5
0	0	0	0	12	18

11	28	0	0	0	0
3	19	0	0	0	0
0	0	19	3	0	0
0	0	28	11	0	0
0	0	0	0	18	26
0	0	0	0	19	12

11	28	0	0	0	0
3	19	0	0	0	0
0	0	11	28	0	0
0	0	3	19	0	0
0	0	0	0	12	5
0	0	0	0	12	18

0	1	0	0	0	0
30	18	0	0	0	0
0	0	0	1	0	0
0	0	30	18	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	0	30	0	0
0	0	30	0	0	0
0	0	0	0	1	0
0	0	0	0	26	30

G:=sub<GL(6,GF(31))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,12,0,0,0,0,5,18],[11,3,0,0,0,0,28,19,0,0,0,0,0,0,19,28,0,0,0,0,3,11,0,0,0,0,0,0,18,19,0,0,0,0,26,12],[11,3,0,0,0,0,28,19,0,0,0,0,0,0,11,3,0,0,0,0,28,19,0,0,0,0,0,0,12,12,0,0,0,0,5,18],[0,30,0,0,0,0,1,18,0,0,0,0,0,0,0,30,0,0,0,0,1,18,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,30,0,0,0,0,30,0,0,0,0,0,0,0,1,26,0,0,0,0,0,30] >;

C₃³⋊D₅ in GAP, Magma, Sage, TeX

C_3^3\rtimes D_5

% in TeX

G:=Group("C3^3:D5");

// GroupNames label

G:=SmallGroup(270,29);

// by ID

G=gap.SmallGroup(270,29);

# by ID

G:=PCGroup([5,-2,-3,-3,-3,-5,41,182,723,5404]);

// Polycyclic

G:=Group<a,b,c,d,e|a^3=b^3=c^3=d^5=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,e*a*e=a^-1,b*c=c*b,b*d=d*b,e*b*e=b^-1,c*d=d*c,e*c*e=c^-1,e*d*e=d^-1>;

// generators/relations

G = C33⋊D5 order 270 = 2·33·5

3rd semidirect product of C33 and D5 acting via D5/C5=C2

G = C₃³⋊D₅ order 270 = 2·3³·5

3^rd semidirect product of C₃³ and D₅ acting via D₅/C₅=C₂