D22:Q8 - GroupNames

G = D₂₂⋊Q₈ order 352 = 2⁵·11

1^st semidirect product of D₂₂ and Q₈ acting via Q₈/C₄=C₂

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D₂₂⋊₁Q₈, Dic₁₁.₇D₄, C₄⋊C₄⋊₄D₁₁, C₂.₆(Q₈×D₁₁), D₂₂⋊C₄.₂C₂, C₂.₁₄(D₄×D₁₁), C₂₂.₂₆(C₂×D₄), (C₂×C₄).₁₃D₂₂, C₁₁⋊₂(C₂²⋊Q₈), C₂₂.₁₃(C₂×Q₈), (C₂×Dic₂₂)⋊₄C₂, Dic₁₁⋊C₄⋊₁₂C₂, C₂₂.₁₃(C₄○D₄), (C₂×C₂₂).₃₇C₂³, (C₂×C₄₄).₅₈C₂², C₂.₁₅(D₄₄⋊₅C₂), C₂².₅₁(C₂²×D₁₁), (C₂×Dic₁₁).₁₂C₂², (C₂²×D₁₁).₂₁C₂², (C₁₁×C₄⋊C₄)⋊₇C₂, (C₂×C₄×D₁₁).₉C₂, SmallGroup(352,91)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂×C₂₂ — D₂₂⋊Q₈

Chief series

C₁ — C₁₁ — C₂₂ — C₂×C₂₂ — C₂²×D₁₁ — C₂×C₄×D₁₁ — D₂₂⋊Q₈

Lower central

C₁₁ — C₂×C₂₂ — D₂₂⋊Q₈

Upper central

C₁ — C₂² — C₄⋊C₄

Generators and relations for D₂₂⋊Q₈
G = < a,b,c,d | a²²=b²=c⁴=1, d²=c², bab=cac^-1=dad^-1=a^-1, cbc^-1=a⁹b, dbd^-1=a²⁰b, dcd^-1=c^-1 >

Subgroups: 450 in 74 conjugacy classes, 33 normal (29 characteristic)
C₁, C₂, C₂, C₄, C₂², C₂², C₂×C₄, C₂×C₄, Q₈, C₂³, C₁₁, C₂²⋊C₄, C₄⋊C₄, C₄⋊C₄, C₂²×C₄, C₂×Q₈, D₁₁, C₂₂, C₂²⋊Q₈, Dic₁₁, Dic₁₁, C₄₄, D₂₂, D₂₂, C₂×C₂₂, Dic₂₂, C₄×D₁₁, C₂×Dic₁₁, C₂×C₄₄, C₂²×D₁₁, Dic₁₁⋊C₄, D₂₂⋊C₄, C₁₁×C₄⋊C₄, C₂×Dic₂₂, C₂×C₄×D₁₁, D₂₂⋊Q₈
Quotients: C₁, C₂, C₂², D₄, Q₈, C₂³, C₂×D₄, C₂×Q₈, C₄○D₄, D₁₁, C₂²⋊Q₈, D₂₂, C₂²×D₁₁, D₄₄⋊₅C₂, D₄×D₁₁, Q₈×D₁₁, D₂₂⋊Q₈

Smallest permutation representation of D₂₂⋊Q₈
►On 176 points

Generators in S₁₇₆

(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)(155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176)

(1 162)(2 161)(3 160)(4 159)(5 158)(6 157)(7 156)(8 155)(9 176)(10 175)(11 174)(12 173)(13 172)(14 171)(15 170)(16 169)(17 168)(18 167)(19 166)(20 165)(21 164)(22 163)(23 114)(24 113)(25 112)(26 111)(27 132)(28 131)(29 130)(30 129)(31 128)(32 127)(33 126)(34 125)(35 124)(36 123)(37 122)(38 121)(39 120)(40 119)(41 118)(42 117)(43 116)(44 115)(45 89)(46 110)(47 109)(48 108)(49 107)(50 106)(51 105)(52 104)(53 103)(54 102)(55 101)(56 100)(57 99)(58 98)(59 97)(60 96)(61 95)(62 94)(63 93)(64 92)(65 91)(66 90)(67 142)(68 141)(69 140)(70 139)(71 138)(72 137)(73 136)(74 135)(75 134)(76 133)(77 154)(78 153)(79 152)(80 151)(81 150)(82 149)(83 148)(84 147)(85 146)(86 145)(87 144)(88 143)

(1 37 174 123)(2 36 175 122)(3 35 176 121)(4 34 155 120)(5 33 156 119)(6 32 157 118)(7 31 158 117)(8 30 159 116)(9 29 160 115)(10 28 161 114)(11 27 162 113)(12 26 163 112)(13 25 164 111)(14 24 165 132)(15 23 166 131)(16 44 167 130)(17 43 168 129)(18 42 169 128)(19 41 170 127)(20 40 171 126)(21 39 172 125)(22 38 173 124)(45 84 97 152)(46 83 98 151)(47 82 99 150)(48 81 100 149)(49 80 101 148)(50 79 102 147)(51 78 103 146)(52 77 104 145)(53 76 105 144)(54 75 106 143)(55 74 107 142)(56 73 108 141)(57 72 109 140)(58 71 110 139)(59 70 89 138)(60 69 90 137)(61 68 91 136)(62 67 92 135)(63 88 93 134)(64 87 94 133)(65 86 95 154)(66 85 96 153)

(1 58 174 110)(2 57 175 109)(3 56 176 108)(4 55 155 107)(5 54 156 106)(6 53 157 105)(7 52 158 104)(8 51 159 103)(9 50 160 102)(10 49 161 101)(11 48 162 100)(12 47 163 99)(13 46 164 98)(14 45 165 97)(15 66 166 96)(16 65 167 95)(17 64 168 94)(18 63 169 93)(19 62 170 92)(20 61 171 91)(21 60 172 90)(22 59 173 89)(23 153 131 85)(24 152 132 84)(25 151 111 83)(26 150 112 82)(27 149 113 81)(28 148 114 80)(29 147 115 79)(30 146 116 78)(31 145 117 77)(32 144 118 76)(33 143 119 75)(34 142 120 74)(35 141 121 73)(36 140 122 72)(37 139 123 71)(38 138 124 70)(39 137 125 69)(40 136 126 68)(41 135 127 67)(42 134 128 88)(43 133 129 87)(44 154 130 86)

G:=sub<Sym(176)| (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)(155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176), (1,162)(2,161)(3,160)(4,159)(5,158)(6,157)(7,156)(8,155)(9,176)(10,175)(11,174)(12,173)(13,172)(14,171)(15,170)(16,169)(17,168)(18,167)(19,166)(20,165)(21,164)(22,163)(23,114)(24,113)(25,112)(26,111)(27,132)(28,131)(29,130)(30,129)(31,128)(32,127)(33,126)(34,125)(35,124)(36,123)(37,122)(38,121)(39,120)(40,119)(41,118)(42,117)(43,116)(44,115)(45,89)(46,110)(47,109)(48,108)(49,107)(50,106)(51,105)(52,104)(53,103)(54,102)(55,101)(56,100)(57,99)(58,98)(59,97)(60,96)(61,95)(62,94)(63,93)(64,92)(65,91)(66,90)(67,142)(68,141)(69,140)(70,139)(71,138)(72,137)(73,136)(74,135)(75,134)(76,133)(77,154)(78,153)(79,152)(80,151)(81,150)(82,149)(83,148)(84,147)(85,146)(86,145)(87,144)(88,143), (1,37,174,123)(2,36,175,122)(3,35,176,121)(4,34,155,120)(5,33,156,119)(6,32,157,118)(7,31,158,117)(8,30,159,116)(9,29,160,115)(10,28,161,114)(11,27,162,113)(12,26,163,112)(13,25,164,111)(14,24,165,132)(15,23,166,131)(16,44,167,130)(17,43,168,129)(18,42,169,128)(19,41,170,127)(20,40,171,126)(21,39,172,125)(22,38,173,124)(45,84,97,152)(46,83,98,151)(47,82,99,150)(48,81,100,149)(49,80,101,148)(50,79,102,147)(51,78,103,146)(52,77,104,145)(53,76,105,144)(54,75,106,143)(55,74,107,142)(56,73,108,141)(57,72,109,140)(58,71,110,139)(59,70,89,138)(60,69,90,137)(61,68,91,136)(62,67,92,135)(63,88,93,134)(64,87,94,133)(65,86,95,154)(66,85,96,153), (1,58,174,110)(2,57,175,109)(3,56,176,108)(4,55,155,107)(5,54,156,106)(6,53,157,105)(7,52,158,104)(8,51,159,103)(9,50,160,102)(10,49,161,101)(11,48,162,100)(12,47,163,99)(13,46,164,98)(14,45,165,97)(15,66,166,96)(16,65,167,95)(17,64,168,94)(18,63,169,93)(19,62,170,92)(20,61,171,91)(21,60,172,90)(22,59,173,89)(23,153,131,85)(24,152,132,84)(25,151,111,83)(26,150,112,82)(27,149,113,81)(28,148,114,80)(29,147,115,79)(30,146,116,78)(31,145,117,77)(32,144,118,76)(33,143,119,75)(34,142,120,74)(35,141,121,73)(36,140,122,72)(37,139,123,71)(38,138,124,70)(39,137,125,69)(40,136,126,68)(41,135,127,67)(42,134,128,88)(43,133,129,87)(44,154,130,86)>;

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)(155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176), (1,162)(2,161)(3,160)(4,159)(5,158)(6,157)(7,156)(8,155)(9,176)(10,175)(11,174)(12,173)(13,172)(14,171)(15,170)(16,169)(17,168)(18,167)(19,166)(20,165)(21,164)(22,163)(23,114)(24,113)(25,112)(26,111)(27,132)(28,131)(29,130)(30,129)(31,128)(32,127)(33,126)(34,125)(35,124)(36,123)(37,122)(38,121)(39,120)(40,119)(41,118)(42,117)(43,116)(44,115)(45,89)(46,110)(47,109)(48,108)(49,107)(50,106)(51,105)(52,104)(53,103)(54,102)(55,101)(56,100)(57,99)(58,98)(59,97)(60,96)(61,95)(62,94)(63,93)(64,92)(65,91)(66,90)(67,142)(68,141)(69,140)(70,139)(71,138)(72,137)(73,136)(74,135)(75,134)(76,133)(77,154)(78,153)(79,152)(80,151)(81,150)(82,149)(83,148)(84,147)(85,146)(86,145)(87,144)(88,143), (1,37,174,123)(2,36,175,122)(3,35,176,121)(4,34,155,120)(5,33,156,119)(6,32,157,118)(7,31,158,117)(8,30,159,116)(9,29,160,115)(10,28,161,114)(11,27,162,113)(12,26,163,112)(13,25,164,111)(14,24,165,132)(15,23,166,131)(16,44,167,130)(17,43,168,129)(18,42,169,128)(19,41,170,127)(20,40,171,126)(21,39,172,125)(22,38,173,124)(45,84,97,152)(46,83,98,151)(47,82,99,150)(48,81,100,149)(49,80,101,148)(50,79,102,147)(51,78,103,146)(52,77,104,145)(53,76,105,144)(54,75,106,143)(55,74,107,142)(56,73,108,141)(57,72,109,140)(58,71,110,139)(59,70,89,138)(60,69,90,137)(61,68,91,136)(62,67,92,135)(63,88,93,134)(64,87,94,133)(65,86,95,154)(66,85,96,153), (1,58,174,110)(2,57,175,109)(3,56,176,108)(4,55,155,107)(5,54,156,106)(6,53,157,105)(7,52,158,104)(8,51,159,103)(9,50,160,102)(10,49,161,101)(11,48,162,100)(12,47,163,99)(13,46,164,98)(14,45,165,97)(15,66,166,96)(16,65,167,95)(17,64,168,94)(18,63,169,93)(19,62,170,92)(20,61,171,91)(21,60,172,90)(22,59,173,89)(23,153,131,85)(24,152,132,84)(25,151,111,83)(26,150,112,82)(27,149,113,81)(28,148,114,80)(29,147,115,79)(30,146,116,78)(31,145,117,77)(32,144,118,76)(33,143,119,75)(34,142,120,74)(35,141,121,73)(36,140,122,72)(37,139,123,71)(38,138,124,70)(39,137,125,69)(40,136,126,68)(41,135,127,67)(42,134,128,88)(43,133,129,87)(44,154,130,86) );

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),(155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176)], [(1,162),(2,161),(3,160),(4,159),(5,158),(6,157),(7,156),(8,155),(9,176),(10,175),(11,174),(12,173),(13,172),(14,171),(15,170),(16,169),(17,168),(18,167),(19,166),(20,165),(21,164),(22,163),(23,114),(24,113),(25,112),(26,111),(27,132),(28,131),(29,130),(30,129),(31,128),(32,127),(33,126),(34,125),(35,124),(36,123),(37,122),(38,121),(39,120),(40,119),(41,118),(42,117),(43,116),(44,115),(45,89),(46,110),(47,109),(48,108),(49,107),(50,106),(51,105),(52,104),(53,103),(54,102),(55,101),(56,100),(57,99),(58,98),(59,97),(60,96),(61,95),(62,94),(63,93),(64,92),(65,91),(66,90),(67,142),(68,141),(69,140),(70,139),(71,138),(72,137),(73,136),(74,135),(75,134),(76,133),(77,154),(78,153),(79,152),(80,151),(81,150),(82,149),(83,148),(84,147),(85,146),(86,145),(87,144),(88,143)], [(1,37,174,123),(2,36,175,122),(3,35,176,121),(4,34,155,120),(5,33,156,119),(6,32,157,118),(7,31,158,117),(8,30,159,116),(9,29,160,115),(10,28,161,114),(11,27,162,113),(12,26,163,112),(13,25,164,111),(14,24,165,132),(15,23,166,131),(16,44,167,130),(17,43,168,129),(18,42,169,128),(19,41,170,127),(20,40,171,126),(21,39,172,125),(22,38,173,124),(45,84,97,152),(46,83,98,151),(47,82,99,150),(48,81,100,149),(49,80,101,148),(50,79,102,147),(51,78,103,146),(52,77,104,145),(53,76,105,144),(54,75,106,143),(55,74,107,142),(56,73,108,141),(57,72,109,140),(58,71,110,139),(59,70,89,138),(60,69,90,137),(61,68,91,136),(62,67,92,135),(63,88,93,134),(64,87,94,133),(65,86,95,154),(66,85,96,153)], [(1,58,174,110),(2,57,175,109),(3,56,176,108),(4,55,155,107),(5,54,156,106),(6,53,157,105),(7,52,158,104),(8,51,159,103),(9,50,160,102),(10,49,161,101),(11,48,162,100),(12,47,163,99),(13,46,164,98),(14,45,165,97),(15,66,166,96),(16,65,167,95),(17,64,168,94),(18,63,169,93),(19,62,170,92),(20,61,171,91),(21,60,172,90),(22,59,173,89),(23,153,131,85),(24,152,132,84),(25,151,111,83),(26,150,112,82),(27,149,113,81),(28,148,114,80),(29,147,115,79),(30,146,116,78),(31,145,117,77),(32,144,118,76),(33,143,119,75),(34,142,120,74),(35,141,121,73),(36,140,122,72),(37,139,123,71),(38,138,124,70),(39,137,125,69),(40,136,126,68),(41,135,127,67),(42,134,128,88),(43,133,129,87),(44,154,130,86)]])

64 conjugacy classes

class	1	2A	2B	2C	2D	2E	4A	4B	4C	4D	4E	4F	4G	4H	11A	···	11E	22A	···	22O	44A	···	44AD
order	1	2	2	2	2	2	4	4	4	4	4	4	4	4	11	···	11	22	···	22	44	···	44
size	1	1	1	1	22	22	2	2	4	4	22	22	44	44	2	···	2	2	···	2	4	···	4

64 irreducible representations

dim	1	1	1	1	1	1	2	2	2	2	2	2	4	4
type	+	+	+	+	+	+	+	-		+	+		+	-
image	C₁	C₂	C₂	C₂	C₂	C₂	D₄	Q₈	C₄○D₄	D₁₁	D₂₂	D₄₄⋊₅C₂	D₄×D₁₁	Q₈×D₁₁
kernel	D₂₂⋊Q₈	Dic₁₁⋊C₄	D₂₂⋊C₄	C₁₁×C₄⋊C₄	C₂×Dic₂₂	C₂×C₄×D₁₁	Dic₁₁	D₂₂	C₂₂	C₄⋊C₄	C₂×C₄	C₂	C₂	C₂
# reps	1	2	2	1	1	1	2	2	2	5	15	20	5	5

Matrix representation of D₂₂⋊Q₈ ►in GL₆(𝔽₈₉)

88	0	0	0	0	0
0	88	0	0	0	0
0	0	36	36	0	0
0	0	53	11	0	0
0	0	0	0	1	0
0	0	0	0	0	1

88	0	0	0	0	0
70	1	0	0	0	0
0	0	36	36	0	0
0	0	11	53	0	0
0	0	0	0	88	0
0	0	0	0	0	88

15	64	0	0	0	0
66	74	0	0	0	0
0	0	1	0	0	0
0	0	71	88	0	0
0	0	0	0	0	88
0	0	0	0	1	0

34	0	0	0	0	0
23	55	0	0	0	0
0	0	88	0	0	0
0	0	18	1	0	0
0	0	0	0	20	20
0	0	0	0	20	69

G:=sub<GL(6,GF(89))| [88,0,0,0,0,0,0,88,0,0,0,0,0,0,36,53,0,0,0,0,36,11,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[88,70,0,0,0,0,0,1,0,0,0,0,0,0,36,11,0,0,0,0,36,53,0,0,0,0,0,0,88,0,0,0,0,0,0,88],[15,66,0,0,0,0,64,74,0,0,0,0,0,0,1,71,0,0,0,0,0,88,0,0,0,0,0,0,0,1,0,0,0,0,88,0],[34,23,0,0,0,0,0,55,0,0,0,0,0,0,88,18,0,0,0,0,0,1,0,0,0,0,0,0,20,20,0,0,0,0,20,69] >;

D₂₂⋊Q₈ in GAP, Magma, Sage, TeX

D_{22}\rtimes Q_8

% in TeX

G:=Group("D22:Q8");

// GroupNames label

G:=SmallGroup(352,91);

// by ID

G=gap.SmallGroup(352,91);

# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-11,55,506,188,86,11525]);

// Polycyclic

G:=Group<a,b,c,d|a^22=b^2=c^4=1,d^2=c^2,b*a*b=c*a*c^-1=d*a*d^-1=a^-1,c*b*c^-1=a^9*b,d*b*d^-1=a^20*b,d*c*d^-1=c^-1>;

// generators/relations

G = D22⋊Q8 order 352 = 25·11

1st semidirect product of D22 and Q8 acting via Q8/C4=C2

G = D₂₂⋊Q₈ order 352 = 2⁵·11

1^st semidirect product of D₂₂ and Q₈ acting via Q₈/C₄=C₂