C5xD4oD12 - GroupNames

G = C₅xD₄oD₁₂ order 480 = 2⁵·3·5

Direct product of C₅ and D₄oD₁₂

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

Aliases: C₅xD₄oD₁₂, C₃₀.₉₅C₂⁴, C₆₀.₂₄₂C₂³, C₁₅:₁₄2₊¹⁺⁴, (C₅xD₄):₃₀D₆, (S₃xD₄):₅C₁₀, D₄:₈(S₃xC₁₀), (C₂xC₂₀):₂₃D₆, Q₈:₈(S₃xC₁₀), (C₅xQ₈):₂₉D₆, C₄oD₁₂:₈C₁₀, (C₂xD₁₂):₁₃C₁₀, D₁₂:₁₁(C₂xC₁₀), (C₁₀xD₁₂):₂₉C₂, (C₂xC₆₀):₃₀C₂², Q₈:₃S₃:₅C₁₀, (S₃xC₂₀):₁₅C₂², Dic₆:₁₂(C₂xC₁₀), C₃:₂(C₅x2₊¹⁺⁴), (C₅xD₁₂):₄₁C₂², (D₄xC₁₅):₄₀C₂², C₆.₁₂(C₂³xC₁₀), C₁₀.₈₀(S₃xC₂³), (Q₈xC₁₅):₃₅C₂², D₆.₆(C₂²xC₁₀), (S₃xC₁₀).₄₂C₂³, C₂₀.₂₃₉(C₂²xS₃), (C₂xC₃₀).₂₆₀C₂³, C₁₂.₂₆(C₂²xC₁₀), (C₅xDic₆):₃₉C₂², Dic₃.₈(C₂²xC₁₀), (C₅xDic₃).₄₄C₂³, (C₅xS₃xD₄):₁₂C₂, (C₂xC₄):₄(S₃xC₁₀), C₄oD₄:₅(C₅xS₃), C₄.₂₆(S₃xC₂xC₁₀), (C₄xS₃):₂(C₂xC₁₀), (C₂xC₁₂):₅(C₂xC₁₀), (C₃xC₄oD₄):₄C₁₀, (C₅xC₄oD₄):₁₂S₃, (C₃xD₄):₉(C₂xC₁₀), C₃:D₄:₅(C₂xC₁₀), (C₃xQ₈):₈(C₂xC₁₀), C₂².₄(S₃xC₂xC₁₀), (C₅xC₄oD₁₂):₁₈C₂, (C₁₅xC₄oD₄):₁₄C₂, (S₃xC₂xC₁₀):₁₆C₂², C₂.₁₃(S₃xC₂²xC₁₀), (C₂²xS₃):₄(C₂xC₁₀), (C₅xQ₈:₃S₃):₁₂C₂, (C₅xC₃:D₄):₂₁C₂², (C₂xC₆).₄(C₂²xC₁₀), (C₂xC₁₀).₂₃(C₂²xS₃), SmallGroup(480,1161)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₆ — C₅xD₄oD₁₂

Chief series

C₁ — C₃ — C₆ — C₃₀ — S₃xC₁₀ — S₃xC₂xC₁₀ — C₅xS₃xD₄ — C₅xD₄oD₁₂

Lower central

C₃ — C₆ — C₅xD₄oD₁₂

Upper central

C₁ — C₁₀ — C₅xC₄oD₄

Generators and relations for C₅xD₄oD₁₂
G = < a,b,c,d,e | a⁵=b⁴=c²=e²=1, d⁶=b², ab=ba, ac=ca, ad=da, ae=ea, cbc=b^-1, bd=db, be=eb, cd=dc, ce=ec, ede=b²d⁵ >

Subgroups: 788 in 332 conjugacy classes, 170 normal (24 characteristic)
C₁, C₂, C₂, C₃, C₄, C₄, C₄, C₂², C₂², C₅, S₃, C₆, C₆, C₂xC₄, C₂xC₄, D₄, D₄, Q₈, Q₈, C₂³, C₁₀, C₁₀, Dic₃, C₁₂, C₁₂, D₆, D₆, C₂xC₆, C₁₅, C₂xD₄, C₄oD₄, C₄oD₄, C₂₀, C₂₀, C₂₀, C₂xC₁₀, C₂xC₁₀, Dic₆, C₄xS₃, D₁₂, C₃:D₄, C₂xC₁₂, C₃xD₄, C₃xQ₈, C₂²xS₃, C₅xS₃, C₃₀, C₃₀, 2₊¹⁺⁴, C₂xC₂₀, C₂xC₂₀, C₅xD₄, C₅xD₄, C₅xQ₈, C₅xQ₈, C₂²xC₁₀, C₂xD₁₂, C₄oD₁₂, S₃xD₄, Q₈:₃S₃, C₃xC₄oD₄, C₅xDic₃, C₆₀, C₆₀, S₃xC₁₀, S₃xC₁₀, C₂xC₃₀, D₄xC₁₀, C₅xC₄oD₄, C₅xC₄oD₄, D₄oD₁₂, C₅xDic₆, S₃xC₂₀, C₅xD₁₂, C₅xC₃:D₄, C₂xC₆₀, D₄xC₁₅, Q₈xC₁₅, S₃xC₂xC₁₀, C₅x2₊¹⁺⁴, C₁₀xD₁₂, C₅xC₄oD₁₂, C₅xS₃xD₄, C₅xQ₈:₃S₃, C₁₅xC₄oD₄, C₅xD₄oD₁₂
Quotients: C₁, C₂, C₂², C₅, S₃, C₂³, C₁₀, D₆, C₂⁴, C₂xC₁₀, C₂²xS₃, C₅xS₃, 2₊¹⁺⁴, C₂²xC₁₀, S₃xC₂³, S₃xC₁₀, C₂³xC₁₀, D₄oD₁₂, S₃xC₂xC₁₀, C₅x2₊¹⁺⁴, S₃xC₂²xC₁₀, C₅xD₄oD₁₂

Smallest permutation representation of C₅xD₄oD₁₂
►On 120 points

Generators in S₁₂₀

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

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

(37 43)(38 44)(39 45)(40 46)(41 47)(42 48)(49 55)(50 56)(51 57)(52 58)(53 59)(54 60)(73 79)(74 80)(75 81)(76 82)(77 83)(78 84)(85 91)(86 92)(87 93)(88 94)(89 95)(90 96)(97 103)(98 104)(99 105)(100 106)(101 107)(102 108)

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

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

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

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

135 conjugacy classes

class	1	2A	2B	2C	2D	2E	···	2J	3	4A	4B	4C	4D	4E	4F	5A	5B	5C	5D	6A	6B	6C	6D	10A	10B	10C	10D	10E	···	10P	10Q	···	10AN	12A	12B	12C	12D	12E	15A	15B	15C	15D	20A	···	20P	20Q	···	20X	30A	30B	30C	30D	30E	···	30P	60A	···	60H	60I	···	60T
order	1	2	2	2	2	2	···	2	3	4	4	4	4	4	4	5	5	5	5	6	6	6	6	10	10	10	10	10	···	10	10	···	10	12	12	12	12	12	15	15	15	15	20	···	20	20	···	20	30	30	30	30	30	···	30	60	···	60	60	···	60
size	1	1	2	2	2	6	···	6	2	2	2	2	2	6	6	1	1	1	1	2	4	4	4	1	1	1	1	2	···	2	6	···	6	2	2	4	4	4	2	2	2	2	2	···	2	6	···	6	2	2	2	2	4	···	4	2	···	2	4	···	4

135 irreducible representations

dim	1	1	1	1	1	1	1	1	1	1	1	1	2	2	2	2	2	2	2	2	4	4	4	4
type	+	+	+	+	+	+							+	+	+	+					+	+
image	C₁	C₂	C₂	C₂	C₂	C₂	C₅	C₁₀	C₁₀	C₁₀	C₁₀	C₁₀	S₃	D₆	D₆	D₆	C₅xS₃	S₃xC₁₀	S₃xC₁₀	S₃xC₁₀	2₊¹⁺⁴	D₄oD₁₂	C₅x2₊¹⁺⁴	C₅xD₄oD₁₂
kernel	C₅xD₄oD₁₂	C₁₀xD₁₂	C₅xC₄oD₁₂	C₅xS₃xD₄	C₅xQ₈:₃S₃	C₁₅xC₄oD₄	D₄oD₁₂	C₂xD₁₂	C₄oD₁₂	S₃xD₄	Q₈:₃S₃	C₃xC₄oD₄	C₅xC₄oD₄	C₂xC₂₀	C₅xD₄	C₅xQ₈	C₄oD₄	C₂xC₄	D₄	Q₈	C₁₅	C₅	C₃	C₁
# reps	1	3	3	6	2	1	4	12	12	24	8	4	1	3	3	1	4	12	12	4	1	2	4	8

Matrix representation of C₅xD₄oD₁₂ ►in GL₄(F₆₁) generated by

34	0	0	0
0	34	0	0
0	0	34	0
0	0	0	34

1	0	2	0
0	1	0	2
60	0	60	0
0	60	0	60

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

46	23	0	0
38	23	0	0
0	0	46	23
0	0	38	23

46	23	0	0
38	15	0	0
0	0	46	23
0	0	38	15

G:=sub<GL(4,GF(61))| [34,0,0,0,0,34,0,0,0,0,34,0,0,0,0,34],[1,0,60,0,0,1,0,60,2,0,60,0,0,2,0,60],[1,0,60,0,0,1,0,60,0,0,60,0,0,0,0,60],[46,38,0,0,23,23,0,0,0,0,46,38,0,0,23,23],[46,38,0,0,23,15,0,0,0,0,46,38,0,0,23,15] >;

C₅xD₄oD₁₂ in GAP, Magma, Sage, TeX

C_5\times D_4\circ D_{12}

% in TeX

G:=Group("C5xD4oD12");

// GroupNames label

G:=SmallGroup(480,1161);

// by ID

G=gap.SmallGroup(480,1161);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-5,-2,-3,891,2467,304,15686]);

// Polycyclic

G:=Group<a,b,c,d,e|a^5=b^4=c^2=e^2=1,d^6=b^2,a*b=b*a,a*c=c*a,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=b^2*d^5>;

// generators/relations

G = C5xD4oD12 order 480 = 25·3·5

Direct product of C5 and D4oD12

G = C₅xD₄oD₁₂ order 480 = 2⁵·3·5

Direct product of C₅ and D₄oD₁₂