C6xD20 - GroupNames

G = C₆xD₂₀ order 240 = 2⁴·3·5

Direct product of C₆ and D₂₀

direct product, metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C₆xD₂₀, C₃₀:₅D₄, C₁₂:₉D₁₀, C₆₀:₁₀C₂², C₃₀.₄₀C₂³, C₅:₁(C₆xD₄), C₄:₂(C₆xD₅), (C₂xC₂₀):₃C₆, C₂₀:₂(C₂xC₆), (C₂xC₆₀):₈C₂, (C₂xC₁₂):₆D₅, C₁₀:₁(C₃xD₄), C₁₅:₁₁(C₂xD₄), D₁₀:₁(C₂xC₆), (C₂xC₆).₃₈D₁₀, (C₆xD₅):₉C₂², (C₂²xD₅):₂C₆, C₁₀.₃(C₂²xC₆), C₆.₄₀(C₂²xD₅), C₂².₁₀(C₆xD₅), (C₂xC₃₀).₃₉C₂², (D₅xC₂xC₆):₅C₂, C₂.₄(D₅xC₂xC₆), (C₂xC₄):₂(C₃xD₅), (C₂xC₁₀).₁₀(C₂xC₆), SmallGroup(240,157)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₁₀ — C₆xD₂₀

Chief series

C₁ — C₅ — C₁₀ — C₃₀ — C₆xD₅ — D₅xC₂xC₆ — C₆xD₂₀

Lower central

C₅ — C₁₀ — C₆xD₂₀

Upper central

C₁ — C₂xC₆ — C₂xC₁₂

Generators and relations for C₆xD₂₀
G = < a,b,c | a⁶=b²⁰=c²=1, ab=ba, ac=ca, cbc=b^-1 >

Subgroups: 356 in 108 conjugacy classes, 54 normal (18 characteristic)
C₁, C₂, C₂, C₂, C₃, C₄, C₂², C₂², C₅, C₆, C₆, C₆, C₂xC₄, D₄, C₂³, D₅, C₁₀, C₁₀, C₁₂, C₂xC₆, C₂xC₆, C₁₅, C₂xD₄, C₂₀, D₁₀, D₁₀, C₂xC₁₀, C₂xC₁₂, C₃xD₄, C₂²xC₆, C₃xD₅, C₃₀, C₃₀, D₂₀, C₂xC₂₀, C₂²xD₅, C₆xD₄, C₆₀, C₆xD₅, C₆xD₅, C₂xC₃₀, C₂xD₂₀, C₃xD₂₀, C₂xC₆₀, D₅xC₂xC₆, C₆xD₂₀
Quotients: C₁, C₂, C₃, C₂², C₆, D₄, C₂³, D₅, C₂xC₆, C₂xD₄, D₁₀, C₃xD₄, C₂²xC₆, C₃xD₅, D₂₀, C₂²xD₅, C₆xD₄, C₆xD₅, C₂xD₂₀, C₃xD₂₀, D₅xC₂xC₆, C₆xD₂₀

Smallest permutation representation of C₆xD₂₀
►On 120 points

Generators in S₁₂₀

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

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

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

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

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

C₆xD₂₀ is a maximal subgroup of
C₆₀.₂₈D₄ C₃₀.D₈ C₆.D₄₀ (C₂xC₆₀):C₄ C₆₀.₃₆D₄ D₆₀:₃₀C₂² C₆₀.₄₄D₄ C₆₀.₈₈D₄ (C₆xD₅).D₄ Dic₁₅:D₄ Dic₃:D₂₀ D₂₀:₈Dic₃ C₆₀:₄D₄ C₁₂:D₂₀ C₆₀:₁₀D₄ C₁₂:₂D₂₀ D₆:₄D₂₀ D₂₀:₂₅D₆ C₆xD₄xD₅

78 conjugacy classes

class	1	2A	2B	2C	2D	2E	2F	2G	3A	3B	4A	4B	5A	5B	6A	···	6F	6G	···	6N	10A	···	10F	12A	12B	12C	12D	15A	15B	15C	15D	20A	···	20H	30A	···	30L	60A	···	60P
order	1	2	2	2	2	2	2	2	3	3	4	4	5	5	6	···	6	6	···	6	10	···	10	12	12	12	12	15	15	15	15	20	···	20	30	···	30	60	···	60
size	1	1	1	1	10	10	10	10	1	1	2	2	2	2	1	···	1	10	···	10	2	···	2	2	2	2	2	2	2	2	2	2	···	2	2	···	2	2	···	2

78 irreducible representations

dim	1	1	1	1	1	1	1	1	2	2	2	2	2	2	2	2	2	2
type	+	+	+	+					+	+	+	+			+
image	C₁	C₂	C₂	C₂	C₃	C₆	C₆	C₆	D₄	D₅	D₁₀	D₁₀	C₃xD₄	C₃xD₅	D₂₀	C₆xD₅	C₆xD₅	C₃xD₂₀
kernel	C₆xD₂₀	C₃xD₂₀	C₂xC₆₀	D₅xC₂xC₆	C₂xD₂₀	D₂₀	C₂xC₂₀	C₂²xD₅	C₃₀	C₂xC₁₂	C₁₂	C₂xC₆	C₁₀	C₂xC₄	C₆	C₄	C₂²	C₂
# reps	1	4	1	2	2	8	2	4	2	2	4	2	4	4	8	8	4	16

Matrix representation of C₆xD₂₀ ►in GL₃(F₆₁) generated by

14	0	0
0	14	0
0	0	14

60	0	0
0	59	29
0	32	54

60	0	0
0	54	32
0	29	7

G:=sub<GL(3,GF(61))| [14,0,0,0,14,0,0,0,14],[60,0,0,0,59,32,0,29,54],[60,0,0,0,54,29,0,32,7] >;

C₆xD₂₀ in GAP, Magma, Sage, TeX

C_6\times D_{20}

% in TeX

G:=Group("C6xD20");

// GroupNames label

G:=SmallGroup(240,157);

// by ID

G=gap.SmallGroup(240,157);

# by ID

G:=PCGroup([6,-2,-2,-2,-3,-2,-5,506,122,6917]);

// Polycyclic

G:=Group<a,b,c|a^6=b^20=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;

// generators/relations

G = C6xD20 order 240 = 24·3·5

Direct product of C6 and D20

G = C₆xD₂₀ order 240 = 2⁴·3·5

Direct product of C₆ and D₂₀