D12:5D7 - GroupNames

G = D₁₂⋊₅D₇ order 336 = 2⁴·3·7

The semidirect product of D₁₂ and D₇ acting through Inn(D₁₂)

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D₁₂⋊₅D₇, D₁₄.₆D₆, D₆.₂D₁₄, C₂₈.₁₇D₆, C₁₂.₂₇D₁₄, C₄₂.₉C₂³, Dic₄₂⋊₁₀C₂, C₈₄.₂₀C₂², Dic₇.₁₁D₆, Dic₂₁.₄C₂², (C₄×D₇)⋊₁S₃, (C₇×D₁₂)⋊₂C₂, (C₁₂×D₇)⋊₁C₂, C₇⋊₃(C₄○D₁₂), C₂₁⋊₆(C₄○D₄), C₂₁⋊D₄⋊₂C₂, C₄.₁₃(S₃×D₇), C₃⋊₂(D₄⋊₂D₇), (S₃×Dic₇)⋊₂C₂, C₆.₉(C₂²×D₇), C₁₄.₉(C₂²×S₃), (C₆×D₇).₇C₂², (S₃×C₁₄).₂C₂², (C₃×Dic₇).₉C₂², C₂.₁₃(C₂×S₃×D₇), SmallGroup(336,145)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₄₂ — D₁₂⋊₅D₇

Chief series

C₁ — C₇ — C₂₁ — C₄₂ — C₆×D₇ — C₂₁⋊D₄ — D₁₂⋊₅D₇

Lower central

C₂₁ — C₄₂ — D₁₂⋊₅D₇

Upper central

C₁ — C₂ — C₄

Generators and relations for D₁₂⋊₅D₇
G = < a,b,c,d | a¹²=b²=c⁷=d²=1, bab=a^-1, ac=ca, ad=da, bc=cb, dbd=a⁶b, dcd=c^-1 >

Subgroups: 404 in 80 conjugacy classes, 32 normal (22 characteristic)
C₁, C₂, C₂, C₃, C₄, C₄, C₂², S₃, C₆, C₆, C₇, C₂×C₄, D₄, Q₈, Dic₃, C₁₂, C₁₂, D₆, C₂×C₆, D₇, C₁₄, C₁₄, C₄○D₄, C₂₁, Dic₆, C₄×S₃, D₁₂, C₃⋊D₄, C₂×C₁₂, Dic₇, Dic₇, C₂₈, D₁₄, C₂×C₁₄, S₃×C₇, C₃×D₇, C₄₂, C₄○D₁₂, Dic₁₄, C₄×D₇, C₂×Dic₇, C₇⋊D₄, C₇×D₄, C₃×Dic₇, Dic₂₁, C₈₄, C₆×D₇, S₃×C₁₄, D₄⋊₂D₇, S₃×Dic₇, C₂₁⋊D₄, C₁₂×D₇, C₇×D₁₂, Dic₄₂, D₁₂⋊₅D₇
Quotients: C₁, C₂, C₂², S₃, C₂³, D₆, D₇, C₄○D₄, C₂²×S₃, D₁₄, C₄○D₁₂, C₂²×D₇, S₃×D₇, D₄⋊₂D₇, C₂×S₃×D₇, D₁₂⋊₅D₇

Smallest permutation representation of D₁₂⋊₅D₇
►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 39)(2 38)(3 37)(4 48)(5 47)(6 46)(7 45)(8 44)(9 43)(10 42)(11 41)(12 40)(13 124)(14 123)(15 122)(16 121)(17 132)(18 131)(19 130)(20 129)(21 128)(22 127)(23 126)(24 125)(25 92)(26 91)(27 90)(28 89)(29 88)(30 87)(31 86)(32 85)(33 96)(34 95)(35 94)(36 93)(49 136)(50 135)(51 134)(52 133)(53 144)(54 143)(55 142)(56 141)(57 140)(58 139)(59 138)(60 137)(61 105)(62 104)(63 103)(64 102)(65 101)(66 100)(67 99)(68 98)(69 97)(70 108)(71 107)(72 106)(73 166)(74 165)(75 164)(76 163)(77 162)(78 161)(79 160)(80 159)(81 158)(82 157)(83 168)(84 167)(109 150)(110 149)(111 148)(112 147)(113 146)(114 145)(115 156)(116 155)(117 154)(118 153)(119 152)(120 151)

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

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

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

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

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

45 conjugacy classes

class	1	2A	2B	2C	2D	3	4A	4B	4C	4D	4E	6A	6B	6C	7A	7B	7C	12A	12B	12C	12D	14A	14B	14C	14D	···	14I	21A	21B	21C	28A	28B	28C	42A	42B	42C	84A	···	84F
order	1	2	2	2	2	3	4	4	4	4	4	6	6	6	7	7	7	12	12	12	12	14	14	14	14	···	14	21	21	21	28	28	28	42	42	42	84	···	84
size	1	1	6	6	14	2	2	7	7	42	42	2	14	14	2	2	2	2	2	14	14	2	2	2	12	···	12	4	4	4	4	4	4	4	4	4	4	···	4

45 irreducible representations

dim	1	1	1	1	1	1	2	2	2	2	2	2	2	2	2	4	4	4	4
type	+	+	+	+	+	+	+	+	+	+	+		+	+		+	-	+	-
image	C₁	C₂	C₂	C₂	C₂	C₂	S₃	D₆	D₆	D₆	D₇	C₄○D₄	D₁₄	D₁₄	C₄○D₁₂	S₃×D₇	D₄⋊₂D₇	C₂×S₃×D₇	D₁₂⋊₅D₇
kernel	D₁₂⋊₅D₇	S₃×Dic₇	C₂₁⋊D₄	C₁₂×D₇	C₇×D₁₂	Dic₄₂	C₄×D₇	Dic₇	C₂₈	D₁₄	D₁₂	C₂₁	C₁₂	D₆	C₇	C₄	C₃	C₂	C₁
# reps	1	2	2	1	1	1	1	1	1	1	3	2	3	6	4	3	3	3	6

Matrix representation of D₁₂⋊₅D₇ ►in GL₆(𝔽₃₃₇)

336	0	0	0	0	0
0	336	0	0	0	0
0	0	2	196	0	0
0	0	294	336	0	0
0	0	0	0	189	0
0	0	0	0	189	148

1	0	0	0	0	0
0	1	0	0	0	0
0	0	0	93	0	0
0	0	29	0	0	0
0	0	0	0	148	41
0	0	0	0	148	189

34	1	0	0	0	0
335	109	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	1

109	336	0	0	0	0
85	228	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	1	336

G:=sub<GL(6,GF(337))| [336,0,0,0,0,0,0,336,0,0,0,0,0,0,2,294,0,0,0,0,196,336,0,0,0,0,0,0,189,189,0,0,0,0,0,148],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,29,0,0,0,0,93,0,0,0,0,0,0,0,148,148,0,0,0,0,41,189],[34,335,0,0,0,0,1,109,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[109,85,0,0,0,0,336,228,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,1,0,0,0,0,0,336] >;

D₁₂⋊₅D₇ in GAP, Magma, Sage, TeX

D_{12}\rtimes_5D_7

% in TeX

G:=Group("D12:5D7");

// GroupNames label

G:=SmallGroup(336,145);

// by ID

G=gap.SmallGroup(336,145);

# by ID

G:=PCGroup([6,-2,-2,-2,-2,-3,-7,121,116,50,490,10373]);

// Polycyclic

G:=Group<a,b,c,d|a^12=b^2=c^7=d^2=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d=a^6*b,d*c*d=c^-1>;

// generators/relations

G = D12⋊5D7 order 336 = 24·3·7

The semidirect product of D12 and D7 acting through Inn(D12)

G = D₁₂⋊₅D₇ order 336 = 2⁴·3·7

The semidirect product of D₁₂ and D₇ acting through Inn(D₁₂)