D21:Q8 - GroupNames

G = D₂₁⋊Q₈ order 336 = 2⁴·3·7

The semidirect product of D₂₁ and Q₈ acting via Q₈/C₄=C₂

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D₂₁⋊Q₈, Dic₆⋊₄D₇, C₂₈.₁₆D₆, Dic₁₄⋊₄S₃, C₁₂.₁₈D₁₄, C₄₂.₇C₂³, Dic₇.₄D₆, C₈₄.₂₈C₂², Dic₃.₃D₁₄, D₄₂.₁₀C₂², Dic₂₁.₁₂C₂², C₇⋊₁(S₃×Q₈), C₃⋊₁(Q₈×D₇), C₂₁⋊Q₈⋊₃C₂, C₂₁⋊₃(C₂×Q₈), D₂₁⋊C₄.C₂, C₄.₂₁(S₃×D₇), (C₇×Dic₆)⋊₆C₂, (C₄×D₂₁).₃C₂, C₆.₇(C₂²×D₇), (C₃×Dic₁₄)⋊₆C₂, C₁₄.₇(C₂²×S₃), (C₇×Dic₃).₃C₂², (C₃×Dic₇).₄C₂², C₂.₁₁(C₂×S₃×D₇), SmallGroup(336,143)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₄₂ — D₂₁⋊Q₈

Chief series

C₁ — C₇ — C₂₁ — C₄₂ — C₃×Dic₇ — D₂₁⋊C₄ — D₂₁⋊Q₈

Lower central

C₂₁ — C₄₂ — D₂₁⋊Q₈

Upper central

C₁ — C₂ — C₄

Generators and relations for D₂₁⋊Q₈
G = < a,b,c,d | a²¹=b²=c⁴=1, d²=c², bab=a^-1, ac=ca, dad^-1=a⁸, bc=cb, dbd^-1=a⁷b, dcd^-1=c^-1 >

Subgroups: 404 in 76 conjugacy classes, 34 normal (20 characteristic)
C₁, C₂, C₂, C₃, C₄, C₄, C₂², S₃, C₆, C₇, C₂×C₄, Q₈, Dic₃, Dic₃, C₁₂, C₁₂, D₆, D₇, C₁₄, C₂×Q₈, C₂₁, Dic₆, Dic₆, C₄×S₃, C₃×Q₈, Dic₇, Dic₇, C₂₈, C₂₈, D₁₄, D₂₁, C₄₂, S₃×Q₈, Dic₁₄, Dic₁₄, C₄×D₇, C₇×Q₈, C₇×Dic₃, C₃×Dic₇, Dic₂₁, C₈₄, D₄₂, Q₈×D₇, D₂₁⋊C₄, C₂₁⋊Q₈, C₃×Dic₁₄, C₇×Dic₆, C₄×D₂₁, D₂₁⋊Q₈
Quotients: C₁, C₂, C₂², S₃, Q₈, C₂³, D₆, D₇, C₂×Q₈, C₂²×S₃, D₁₄, S₃×Q₈, C₂²×D₇, S₃×D₇, Q₈×D₇, C₂×S₃×D₇, D₂₁⋊Q₈

Smallest permutation representation of D₂₁⋊Q₈
►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 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 42)(21 41)(43 64)(44 84)(45 83)(46 82)(47 81)(48 80)(49 79)(50 78)(51 77)(52 76)(53 75)(54 74)(55 73)(56 72)(57 71)(58 70)(59 69)(60 68)(61 67)(62 66)(63 65)(85 126)(86 125)(87 124)(88 123)(89 122)(90 121)(91 120)(92 119)(93 118)(94 117)(95 116)(96 115)(97 114)(98 113)(99 112)(100 111)(101 110)(102 109)(103 108)(104 107)(105 106)(127 151)(128 150)(129 149)(130 148)(131 168)(132 167)(133 166)(134 165)(135 164)(136 163)(137 162)(138 161)(139 160)(140 159)(141 158)(142 157)(143 156)(144 155)(145 154)(146 153)(147 152)

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

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

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

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

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

42 conjugacy classes

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

42 irreducible representations

dim	1	1	1	1	1	1	2	2	2	2	2	2	2	4	4	4	4	4
type	+	+	+	+	+	+	+	-	+	+	+	+	+	-	+	-	+
image	C₁	C₂	C₂	C₂	C₂	C₂	S₃	Q₈	D₆	D₆	D₇	D₁₄	D₁₄	S₃×Q₈	S₃×D₇	Q₈×D₇	C₂×S₃×D₇	D₂₁⋊Q₈
kernel	D₂₁⋊Q₈	D₂₁⋊C₄	C₂₁⋊Q₈	C₃×Dic₁₄	C₇×Dic₆	C₄×D₂₁	Dic₁₄	D₂₁	Dic₇	C₂₈	Dic₆	Dic₃	C₁₂	C₇	C₄	C₃	C₂	C₁
# reps	1	2	2	1	1	1	1	2	2	1	3	6	3	1	3	3	3	6

Matrix representation of D₂₁⋊Q₈ ►in GL₆(𝔽₃₃₇)

0	1	0	0	0	0
336	336	0	0	0	0
0	0	144	34	0	0
0	0	303	336	0	0
0	0	0	0	1	0
0	0	0	0	0	1

0	1	0	0	0	0
1	0	0	0	0	0
0	0	144	34	0	0
0	0	193	193	0	0
0	0	0	0	336	0
0	0	0	0	0	336

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	191	336
0	0	0	0	86	146

336	0	0	0	0	0
1	1	0	0	0	0
0	0	1	0	0	0
0	0	0	1	0	0
0	0	0	0	153	324
0	0	0	0	38	184

G:=sub<GL(6,GF(337))| [0,336,0,0,0,0,1,336,0,0,0,0,0,0,144,303,0,0,0,0,34,336,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,144,193,0,0,0,0,34,193,0,0,0,0,0,0,336,0,0,0,0,0,0,336],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,191,86,0,0,0,0,336,146],[336,1,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,153,38,0,0,0,0,324,184] >;

D₂₁⋊Q₈ in GAP, Magma, Sage, TeX

D_{21}\rtimes Q_8

% in TeX

G:=Group("D21:Q8");

// GroupNames label

G:=SmallGroup(336,143);

// by ID

G=gap.SmallGroup(336,143);

# by ID

G:=PCGroup([6,-2,-2,-2,-2,-3,-7,48,55,218,116,50,490,10373]);

// Polycyclic

G:=Group<a,b,c,d|a^21=b^2=c^4=1,d^2=c^2,b*a*b=a^-1,a*c=c*a,d*a*d^-1=a^8,b*c=c*b,d*b*d^-1=a^7*b,d*c*d^-1=c^-1>;

// generators/relations

G = D21⋊Q8 order 336 = 24·3·7

The semidirect product of D21 and Q8 acting via Q8/C4=C2

G = D₂₁⋊Q₈ order 336 = 2⁴·3·7

The semidirect product of D₂₁ and Q₈ acting via Q₈/C₄=C₂