D4:2D21 - GroupNames

G = D₄⋊₂D₂₁ order 336 = 2⁴·3·7

The semidirect product of D₄ and D₂₁ acting through Inn(D₄)

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D₄⋊₂D₂₁, C₄.₅D₄₂, C₂₈.₁₉D₆, Dic₄₂⋊₃C₂, C₁₂.₁₉D₁₄, C₈₄.₅C₂², C₂².₁D₄₂, C₄₂.₃₃C₂³, D₄₂.₆C₂², Dic₂₁.₁₆C₂², (C₃×D₄)⋊₃D₇, (C₇×D₄)⋊₃S₃, (C₄×D₂₁)⋊₂C₂, (D₄×C₂₁)⋊₃C₂, C₂₁⋊₇D₄⋊₂C₂, (C₂×C₆).₃D₁₄, (C₂×C₁₄).₃D₆, C₂₁⋊₁₄(C₄○D₄), C₇⋊₅(D₄⋊₂S₃), C₃⋊₅(D₄⋊₂D₇), (C₂×Dic₂₁)⋊₃C₂, (C₂×C₄₂).₁C₂², C₆.₃₃(C₂²×D₇), C₂.₇(C₂²×D₂₁), C₁₄.₃₃(C₂²×S₃), SmallGroup(336,199)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₄₂ — D₄⋊₂D₂₁

Chief series

C₁ — C₇ — C₂₁ — C₄₂ — D₄₂ — C₄×D₂₁ — D₄⋊₂D₂₁

Lower central

C₂₁ — C₄₂ — D₄⋊₂D₂₁

Upper central

C₁ — C₂ — D₄

Generators and relations for D₄⋊₂D₂₁
G = < a,b,c,d | a⁴=b²=c²¹=d²=1, bab=a^-1, ac=ca, ad=da, bc=cb, dbd=a²b, dcd=c^-1 >

Subgroups: 456 in 80 conjugacy classes, 35 normal (21 characteristic)
C₁, C₂, C₂, C₃, C₄, C₄, C₂², C₂², S₃, C₆, C₆, C₇, C₂×C₄, D₄, D₄, Q₈, Dic₃, C₁₂, D₆, C₂×C₆, D₇, C₁₄, C₁₄, C₄○D₄, C₂₁, Dic₆, C₄×S₃, C₂×Dic₃, C₃⋊D₄, C₃×D₄, Dic₇, C₂₈, D₁₄, C₂×C₁₄, D₂₁, C₄₂, C₄₂, D₄⋊₂S₃, Dic₁₄, C₄×D₇, C₂×Dic₇, C₇⋊D₄, C₇×D₄, Dic₂₁, Dic₂₁, C₈₄, D₄₂, C₂×C₄₂, D₄⋊₂D₇, Dic₄₂, C₄×D₂₁, C₂×Dic₂₁, C₂₁⋊₇D₄, D₄×C₂₁, D₄⋊₂D₂₁
Quotients: C₁, C₂, C₂², S₃, C₂³, D₆, D₇, C₄○D₄, C₂²×S₃, D₁₄, D₂₁, D₄⋊₂S₃, C₂²×D₇, D₄₂, D₄⋊₂D₇, C₂²×D₂₁, D₄⋊₂D₂₁

Smallest permutation representation of D₄⋊₂D₂₁
►On 168 points

Generators in S₁₆₈

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

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

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

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

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

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

60 conjugacy classes

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

60 irreducible representations

dim	1	1	1	1	1	1	2	2	2	2	2	2	2	2	2	2	4	4	4
type	+	+	+	+	+	+	+	+	+	+		+	+	+	+	+	-	-	-
image	C₁	C₂	C₂	C₂	C₂	C₂	S₃	D₆	D₆	D₇	C₄○D₄	D₁₄	D₁₄	D₂₁	D₄₂	D₄₂	D₄⋊₂S₃	D₄⋊₂D₇	D₄⋊₂D₂₁
kernel	D₄⋊₂D₂₁	Dic₄₂	C₄×D₂₁	C₂×Dic₂₁	C₂₁⋊₇D₄	D₄×C₂₁	C₇×D₄	C₂₈	C₂×C₁₄	C₃×D₄	C₂₁	C₁₂	C₂×C₆	D₄	C₄	C₂²	C₇	C₃	C₁
# reps	1	1	1	2	2	1	1	1	2	3	2	3	6	6	6	12	1	3	6

Matrix representation of D₄⋊₂D₂₁ ►in GL₄(𝔽₃₃₇) generated by

1	0	0	0
0	1	0	0
0	0	148	0
0	0	0	189

336	0	0	0
0	336	0	0
0	0	0	189
0	0	148	0

278	273	0	0
112	230	0	0
0	0	1	0
0	0	0	1

230	30	0	0
225	107	0	0
0	0	1	0
0	0	0	336

G:=sub<GL(4,GF(337))| [1,0,0,0,0,1,0,0,0,0,148,0,0,0,0,189],[336,0,0,0,0,336,0,0,0,0,0,148,0,0,189,0],[278,112,0,0,273,230,0,0,0,0,1,0,0,0,0,1],[230,225,0,0,30,107,0,0,0,0,1,0,0,0,0,336] >;

D₄⋊₂D₂₁ in GAP, Magma, Sage, TeX

D_4\rtimes_2D_{21}

% in TeX

G:=Group("D4:2D21");

// GroupNames label

G:=SmallGroup(336,199);

// by ID

G=gap.SmallGroup(336,199);

# by ID

G:=PCGroup([6,-2,-2,-2,-2,-3,-7,55,218,116,964,10373]);

// Polycyclic

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

// generators/relations

G = D4⋊2D21 order 336 = 24·3·7

The semidirect product of D4 and D21 acting through Inn(D4)

G = D₄⋊₂D₂₁ order 336 = 2⁴·3·7

The semidirect product of D₄ and D₂₁ acting through Inn(D₄)