C14xD12 - GroupNames

G = C₁₄xD₁₂ order 336 = 2⁴·3·7

Direct product of C₁₄ and D₁₂

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

Aliases: C₁₄xD₁₂, C₂₈:₉D₆, C₄₂:₆D₄, C₈₄:₁₂C₂², C₄₂.₅₀C₂³, C₆:₁(C₇xD₄), C₃:₁(D₄xC₁₄), C₄:₂(S₃xC₁₄), (C₂xC₂₈):₆S₃, C₂₁:₁₂(C₂xD₄), (C₂xC₈₄):₁₁C₂, C₁₂:₂(C₂xC₁₄), (C₂xC₁₂):₃C₁₄, D₆:₁(C₂xC₁₄), (C₂xC₁₄).₃₈D₆, (S₃xC₁₄):₉C₂², (C₂²xS₃):₁C₁₄, C₆.₃(C₂²xC₁₄), (C₂xC₄₂).₄₉C₂², C₂².₁₀(S₃xC₁₄), C₁₄.₄₀(C₂²xS₃), (S₃xC₂xC₁₄):₅C₂, (C₂xC₄):₂(S₃xC₇), C₂.₄(S₃xC₂xC₁₄), (C₂xC₆).₁₀(C₂xC₁₄), SmallGroup(336,186)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₆ — C₁₄xD₁₂

Chief series

C₁ — C₃ — C₆ — C₄₂ — S₃xC₁₄ — S₃xC₂xC₁₄ — C₁₄xD₁₂

Lower central

C₃ — C₆ — C₁₄xD₁₂

Upper central

C₁ — C₂xC₁₄ — C₂xC₂₈

Generators and relations for C₁₄xD₁₂
G = < a,b,c | a¹⁴=b¹²=c²=1, ab=ba, ac=ca, cbc=b^-1 >

Subgroups: 248 in 108 conjugacy classes, 54 normal (18 characteristic)
C₁, C₂, C₂, C₂, C₃, C₄, C₂², C₂², S₃, C₆, C₆, C₇, C₂xC₄, D₄, C₂³, C₁₂, D₆, D₆, C₂xC₆, C₁₄, C₁₄, C₁₄, C₂xD₄, C₂₁, D₁₂, C₂xC₁₂, C₂²xS₃, C₂₈, C₂xC₁₄, C₂xC₁₄, S₃xC₇, C₄₂, C₄₂, C₂xD₁₂, C₂xC₂₈, C₇xD₄, C₂²xC₁₄, C₈₄, S₃xC₁₄, S₃xC₁₄, C₂xC₄₂, D₄xC₁₄, C₇xD₁₂, C₂xC₈₄, S₃xC₂xC₁₄, C₁₄xD₁₂
Quotients: C₁, C₂, C₂², S₃, C₇, D₄, C₂³, D₆, C₁₄, C₂xD₄, D₁₂, C₂²xS₃, C₂xC₁₄, S₃xC₇, C₂xD₁₂, C₇xD₄, C₂²xC₁₄, S₃xC₁₄, D₄xC₁₄, C₇xD₁₂, S₃xC₂xC₁₄, C₁₄xD₁₂

Smallest permutation representation of C₁₄xD₁₂
►On 168 points

Generators in S₁₆₈

(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)(127 128 129 130 131 132 133 134 135 136 137 138 139 140)(141 142 143 144 145 146 147 148 149 150 151 152 153 154)(155 156 157 158 159 160 161 162 163 164 165 166 167 168)

(1 50 92 114 31 78 142 160 112 62 130 20)(2 51 93 115 32 79 143 161 99 63 131 21)(3 52 94 116 33 80 144 162 100 64 132 22)(4 53 95 117 34 81 145 163 101 65 133 23)(5 54 96 118 35 82 146 164 102 66 134 24)(6 55 97 119 36 83 147 165 103 67 135 25)(7 56 98 120 37 84 148 166 104 68 136 26)(8 43 85 121 38 71 149 167 105 69 137 27)(9 44 86 122 39 72 150 168 106 70 138 28)(10 45 87 123 40 73 151 155 107 57 139 15)(11 46 88 124 41 74 152 156 108 58 140 16)(12 47 89 125 42 75 153 157 109 59 127 17)(13 48 90 126 29 76 154 158 110 60 128 18)(14 49 91 113 30 77 141 159 111 61 129 19)

(1 20)(2 21)(3 22)(4 23)(5 24)(6 25)(7 26)(8 27)(9 28)(10 15)(11 16)(12 17)(13 18)(14 19)(29 158)(30 159)(31 160)(32 161)(33 162)(34 163)(35 164)(36 165)(37 166)(38 167)(39 168)(40 155)(41 156)(42 157)(43 137)(44 138)(45 139)(46 140)(47 127)(48 128)(49 129)(50 130)(51 131)(52 132)(53 133)(54 134)(55 135)(56 136)(57 87)(58 88)(59 89)(60 90)(61 91)(62 92)(63 93)(64 94)(65 95)(66 96)(67 97)(68 98)(69 85)(70 86)(71 149)(72 150)(73 151)(74 152)(75 153)(76 154)(77 141)(78 142)(79 143)(80 144)(81 145)(82 146)(83 147)(84 148)(99 115)(100 116)(101 117)(102 118)(103 119)(104 120)(105 121)(106 122)(107 123)(108 124)(109 125)(110 126)(111 113)(112 114)

G:=sub<Sym(168)| (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)(127,128,129,130,131,132,133,134,135,136,137,138,139,140)(141,142,143,144,145,146,147,148,149,150,151,152,153,154)(155,156,157,158,159,160,161,162,163,164,165,166,167,168), (1,50,92,114,31,78,142,160,112,62,130,20)(2,51,93,115,32,79,143,161,99,63,131,21)(3,52,94,116,33,80,144,162,100,64,132,22)(4,53,95,117,34,81,145,163,101,65,133,23)(5,54,96,118,35,82,146,164,102,66,134,24)(6,55,97,119,36,83,147,165,103,67,135,25)(7,56,98,120,37,84,148,166,104,68,136,26)(8,43,85,121,38,71,149,167,105,69,137,27)(9,44,86,122,39,72,150,168,106,70,138,28)(10,45,87,123,40,73,151,155,107,57,139,15)(11,46,88,124,41,74,152,156,108,58,140,16)(12,47,89,125,42,75,153,157,109,59,127,17)(13,48,90,126,29,76,154,158,110,60,128,18)(14,49,91,113,30,77,141,159,111,61,129,19), (1,20)(2,21)(3,22)(4,23)(5,24)(6,25)(7,26)(8,27)(9,28)(10,15)(11,16)(12,17)(13,18)(14,19)(29,158)(30,159)(31,160)(32,161)(33,162)(34,163)(35,164)(36,165)(37,166)(38,167)(39,168)(40,155)(41,156)(42,157)(43,137)(44,138)(45,139)(46,140)(47,127)(48,128)(49,129)(50,130)(51,131)(52,132)(53,133)(54,134)(55,135)(56,136)(57,87)(58,88)(59,89)(60,90)(61,91)(62,92)(63,93)(64,94)(65,95)(66,96)(67,97)(68,98)(69,85)(70,86)(71,149)(72,150)(73,151)(74,152)(75,153)(76,154)(77,141)(78,142)(79,143)(80,144)(81,145)(82,146)(83,147)(84,148)(99,115)(100,116)(101,117)(102,118)(103,119)(104,120)(105,121)(106,122)(107,123)(108,124)(109,125)(110,126)(111,113)(112,114)>;

G:=Group( (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)(127,128,129,130,131,132,133,134,135,136,137,138,139,140)(141,142,143,144,145,146,147,148,149,150,151,152,153,154)(155,156,157,158,159,160,161,162,163,164,165,166,167,168), (1,50,92,114,31,78,142,160,112,62,130,20)(2,51,93,115,32,79,143,161,99,63,131,21)(3,52,94,116,33,80,144,162,100,64,132,22)(4,53,95,117,34,81,145,163,101,65,133,23)(5,54,96,118,35,82,146,164,102,66,134,24)(6,55,97,119,36,83,147,165,103,67,135,25)(7,56,98,120,37,84,148,166,104,68,136,26)(8,43,85,121,38,71,149,167,105,69,137,27)(9,44,86,122,39,72,150,168,106,70,138,28)(10,45,87,123,40,73,151,155,107,57,139,15)(11,46,88,124,41,74,152,156,108,58,140,16)(12,47,89,125,42,75,153,157,109,59,127,17)(13,48,90,126,29,76,154,158,110,60,128,18)(14,49,91,113,30,77,141,159,111,61,129,19), (1,20)(2,21)(3,22)(4,23)(5,24)(6,25)(7,26)(8,27)(9,28)(10,15)(11,16)(12,17)(13,18)(14,19)(29,158)(30,159)(31,160)(32,161)(33,162)(34,163)(35,164)(36,165)(37,166)(38,167)(39,168)(40,155)(41,156)(42,157)(43,137)(44,138)(45,139)(46,140)(47,127)(48,128)(49,129)(50,130)(51,131)(52,132)(53,133)(54,134)(55,135)(56,136)(57,87)(58,88)(59,89)(60,90)(61,91)(62,92)(63,93)(64,94)(65,95)(66,96)(67,97)(68,98)(69,85)(70,86)(71,149)(72,150)(73,151)(74,152)(75,153)(76,154)(77,141)(78,142)(79,143)(80,144)(81,145)(82,146)(83,147)(84,148)(99,115)(100,116)(101,117)(102,118)(103,119)(104,120)(105,121)(106,122)(107,123)(108,124)(109,125)(110,126)(111,113)(112,114) );

G=PermutationGroup([[(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),(127,128,129,130,131,132,133,134,135,136,137,138,139,140),(141,142,143,144,145,146,147,148,149,150,151,152,153,154),(155,156,157,158,159,160,161,162,163,164,165,166,167,168)], [(1,50,92,114,31,78,142,160,112,62,130,20),(2,51,93,115,32,79,143,161,99,63,131,21),(3,52,94,116,33,80,144,162,100,64,132,22),(4,53,95,117,34,81,145,163,101,65,133,23),(5,54,96,118,35,82,146,164,102,66,134,24),(6,55,97,119,36,83,147,165,103,67,135,25),(7,56,98,120,37,84,148,166,104,68,136,26),(8,43,85,121,38,71,149,167,105,69,137,27),(9,44,86,122,39,72,150,168,106,70,138,28),(10,45,87,123,40,73,151,155,107,57,139,15),(11,46,88,124,41,74,152,156,108,58,140,16),(12,47,89,125,42,75,153,157,109,59,127,17),(13,48,90,126,29,76,154,158,110,60,128,18),(14,49,91,113,30,77,141,159,111,61,129,19)], [(1,20),(2,21),(3,22),(4,23),(5,24),(6,25),(7,26),(8,27),(9,28),(10,15),(11,16),(12,17),(13,18),(14,19),(29,158),(30,159),(31,160),(32,161),(33,162),(34,163),(35,164),(36,165),(37,166),(38,167),(39,168),(40,155),(41,156),(42,157),(43,137),(44,138),(45,139),(46,140),(47,127),(48,128),(49,129),(50,130),(51,131),(52,132),(53,133),(54,134),(55,135),(56,136),(57,87),(58,88),(59,89),(60,90),(61,91),(62,92),(63,93),(64,94),(65,95),(66,96),(67,97),(68,98),(69,85),(70,86),(71,149),(72,150),(73,151),(74,152),(75,153),(76,154),(77,141),(78,142),(79,143),(80,144),(81,145),(82,146),(83,147),(84,148),(99,115),(100,116),(101,117),(102,118),(103,119),(104,120),(105,121),(106,122),(107,123),(108,124),(109,125),(110,126),(111,113),(112,114)]])

126 conjugacy classes

class	1	2A	2B	2C	2D	2E	2F	2G	3	4A	4B	6A	6B	6C	7A	···	7F	12A	12B	12C	12D	14A	···	14R	14S	···	14AP	21A	···	21F	28A	···	28L	42A	···	42R	84A	···	84X
order	1	2	2	2	2	2	2	2	3	4	4	6	6	6	7	···	7	12	12	12	12	14	···	14	14	···	14	21	···	21	28	···	28	42	···	42	84	···	84
size	1	1	1	1	6	6	6	6	2	2	2	2	2	2	1	···	1	2	2	2	2	1	···	1	6	···	6	2	···	2	2	···	2	2	···	2	2	···	2

126 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	C₇	C₁₄	C₁₄	C₁₄	S₃	D₄	D₆	D₆	D₁₂	S₃xC₇	C₇xD₄	S₃xC₁₄	S₃xC₁₄	C₇xD₁₂
kernel	C₁₄xD₁₂	C₇xD₁₂	C₂xC₈₄	S₃xC₂xC₁₄	C₂xD₁₂	D₁₂	C₂xC₁₂	C₂²xS₃	C₂xC₂₈	C₄₂	C₂₈	C₂xC₁₄	C₁₄	C₂xC₄	C₆	C₄	C₂²	C₂
# reps	1	4	1	2	6	24	6	12	1	2	2	1	4	6	12	12	6	24

Matrix representation of C₁₄xD₁₂ ►in GL₅(F₃₃₇)

336	0	0	0	0
0	42	0	0	0
0	0	42	0	0
0	0	0	52	0
0	0	0	0	52

1	0	0	0	0
0	327	264	0	0
0	6	10	0	0
0	0	0	1	1
0	0	0	336	0

336	0	0	0	0
0	10	185	0	0
0	331	327	0	0
0	0	0	1	1
0	0	0	0	336

G:=sub<GL(5,GF(337))| [336,0,0,0,0,0,42,0,0,0,0,0,42,0,0,0,0,0,52,0,0,0,0,0,52],[1,0,0,0,0,0,327,6,0,0,0,264,10,0,0,0,0,0,1,336,0,0,0,1,0],[336,0,0,0,0,0,10,331,0,0,0,185,327,0,0,0,0,0,1,0,0,0,0,1,336] >;

C₁₄xD₁₂ in GAP, Magma, Sage, TeX

C_{14}\times D_{12}

% in TeX

G:=Group("C14xD12");

// GroupNames label

G:=SmallGroup(336,186);

// by ID

G=gap.SmallGroup(336,186);

# by ID

G:=PCGroup([6,-2,-2,-2,-7,-2,-3,1082,266,8069]);

// Polycyclic

G:=Group<a,b,c|a^14=b^12=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;

// generators/relations

G = C14xD12 order 336 = 24·3·7

Direct product of C14 and D12

G = C₁₄xD₁₂ order 336 = 2⁴·3·7

Direct product of C₁₄ and D₁₂