C3^2:3D20 - GroupNames

G = C₃²:₃D₂₀ order 360 = 2³·3²·5

2^nd semidirect product of C₃² and D₂₀ acting via D₂₀/C₁₀=C₂²

Aliases: D₃₀:₄S₃, C₃²:₃D₂₀, C₃₀.₂₀D₆, C₁₀.₁₁S₃², (C₃xC₁₅):₁₄D₄, (C₆xD₁₅):₈C₂, C₃:Dic₃:₃D₅, C₆.₂₆(S₃xD₅), C₅:₁(D₆:S₃), C₁₅:₆(C₃:D₄), C₃:₃(C₃:D₂₀), (C₃xC₆).₁₁D₁₀, C₂.₄(D₁₅:S₃), (C₃xC₃₀).₂₅C₂², (C₅xC₃:Dic₃):₅C₂, SmallGroup(360,87)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₃xC₃₀ — C₃²:₃D₂₀

Chief series

C₁ — C₅ — C₁₅ — C₃xC₁₅ — C₃xC₃₀ — C₆xD₁₅ — C₃²:₃D₂₀

Lower central

C₃xC₁₅ — C₃xC₃₀ — C₃²:₃D₂₀

Upper central

C₁ — C₂

Generators and relations for C₃²:₃D₂₀
G = < a,b,c,d | a³=b³=c²⁰=d²=1, ab=ba, cac^-1=a^-1, ad=da, cbc^-1=dbd=b^-1, dcd=c^-1 >

Subgroups: 452 in 70 conjugacy classes, 23 normal (11 characteristic)
C₁, C₂, C₂, C₃, C₃, C₄, C₂², C₅, S₃, C₆, C₆, D₄, C₃², D₅, C₁₀, Dic₃, D₆, C₂xC₆, C₁₅, C₁₅, C₃xS₃, C₃xC₆, C₂₀, D₁₀, C₃:D₄, C₃xD₅, D₁₅, C₃₀, C₃₀, C₃:Dic₃, S₃xC₆, D₂₀, C₃xC₁₅, C₅xDic₃, C₆xD₅, D₃₀, D₆:S₃, C₃xD₁₅, C₃xC₃₀, C₃:D₂₀, C₅xC₃:Dic₃, C₆xD₁₅, C₃²:₃D₂₀
Quotients: C₁, C₂, C₂², S₃, D₄, D₅, D₆, D₁₀, C₃:D₄, S₃², D₂₀, S₃xD₅, D₆:S₃, C₃:D₂₀, D₁₅:S₃, C₃²:₃D₂₀

Smallest permutation representation of C₃²:₃D₂₀
►On 120 points

Generators in S₁₂₀

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

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

(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)

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

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

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

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

39 conjugacy classes

class	1	2A	2B	2C	3A	3B	3C	4	5A	5B	6A	6B	6C	6D	6E	6F	6G	10A	10B	15A	···	15H	20A	20B	20C	20D	30A	···	30H
order	1	2	2	2	3	3	3	4	5	5	6	6	6	6	6	6	6	10	10	15	···	15	20	20	20	20	30	···	30
size	1	1	30	30	2	2	4	18	2	2	2	2	4	30	30	30	30	2	2	4	···	4	18	18	18	18	4	···	4

39 irreducible representations

dim	1	1	1	2	2	2	2	2	2	2	4	4	4	4	4	4
type	+	+	+	+	+	+	+	+		+	+	+	-	+
image	C₁	C₂	C₂	S₃	D₄	D₅	D₆	D₁₀	C₃:D₄	D₂₀	S₃²	S₃xD₅	D₆:S₃	C₃:D₂₀	D₁₅:S₃	C₃²:₃D₂₀
kernel	C₃²:₃D₂₀	C₅xC₃:Dic₃	C₆xD₁₅	D₃₀	C₃xC₁₅	C₃:Dic₃	C₃₀	C₃xC₆	C₁₅	C₃²	C₁₀	C₆	C₅	C₃	C₂	C₁
# reps	1	1	2	2	1	2	2	2	4	4	1	4	1	4	4	4

Matrix representation of C₃²:₃D₂₀ ►in GL₆(F₆₁)

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	60	60
0	0	0	0	1	0

1	0	0	0	0	0
0	1	0	0	0	0
0	0	0	60	0	0
0	0	1	60	0	0
0	0	0	0	1	0
0	0	0	0	0	1

27	36	0	0	0	0
25	4	0	0	0	0
0	0	0	60	0	0
0	0	60	0	0	0
0	0	0	0	1	0
0	0	0	0	60	60

52	45	0	0	0	0
5	9	0	0	0	0
0	0	0	1	0	0
0	0	1	0	0	0
0	0	0	0	1	0
0	0	0	0	0	1

G:=sub<GL(6,GF(61))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,60,1,0,0,0,0,60,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,60,60,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[27,25,0,0,0,0,36,4,0,0,0,0,0,0,0,60,0,0,0,0,60,0,0,0,0,0,0,0,1,60,0,0,0,0,0,60],[52,5,0,0,0,0,45,9,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;

C₃²:₃D₂₀ in GAP, Magma, Sage, TeX

C_3^2\rtimes_3D_{20}

% in TeX

G:=Group("C3^2:3D20");

// GroupNames label

G:=SmallGroup(360,87);

// by ID

G=gap.SmallGroup(360,87);

# by ID

G:=PCGroup([6,-2,-2,-2,-3,-3,-5,73,31,387,201,730,10373]);

// Polycyclic

G:=Group<a,b,c,d|a^3=b^3=c^20=d^2=1,a*b=b*a,c*a*c^-1=a^-1,a*d=d*a,c*b*c^-1=d*b*d=b^-1,d*c*d=c^-1>;

// generators/relations

G = C32:3D20 order 360 = 23·32·5

2nd semidirect product of C32 and D20 acting via D20/C10=C22

G = C₃²:₃D₂₀ order 360 = 2³·3²·5

2^nd semidirect product of C₃² and D₂₀ acting via D₂₀/C₁₀=C₂²