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G = D22⋊D4order 352 = 25·11

1st semidirect product of D22 and D4 acting via D4/C4=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D221D4, Dic112D4, C23.5D22, (C2×D44)⋊3C2, C2.9(D4×D11), (C2×C4).7D22, D22⋊C411C2, C111(C4⋊D4), C22⋊C44D11, C22.20(C2×D4), Dic11⋊C45C2, C22.9(C4○D4), (C2×C22).25C23, (C2×C44).53C22, C2.11(D445C2), (C22×C22).14C22, (C2×Dic11).7C22, C22.43(C22×D11), (C22×D11).18C22, (C2×C4×D11)⋊11C2, (C2×C11⋊D4)⋊2C2, (C11×C22⋊C4)⋊6C2, SmallGroup(352,79)

Series: Derived Chief Lower central Upper central

C1C2×C22 — D22⋊D4
C1C11C22C2×C22C22×D11C2×C4×D11 — D22⋊D4
C11C2×C22 — D22⋊D4
C1C22C22⋊C4

Generators and relations for D22⋊D4
 G = < a,b,c,d | a22=b2=c4=d2=1, bab=cac-1=dad=a-1, cbc-1=a20b, dbd=a9b, dcd=c-1 >

Subgroups: 658 in 94 conjugacy classes, 33 normal (29 characteristic)
C1, C2, C2, C4, C22, C22, C2×C4, C2×C4, D4, C23, C23, C11, C22⋊C4, C22⋊C4, C4⋊C4, C22×C4, C2×D4, D11, C22, C22, C4⋊D4, Dic11, Dic11, C44, D22, D22, C2×C22, C2×C22, C4×D11, D44, C2×Dic11, C11⋊D4, C2×C44, C22×D11, C22×C22, Dic11⋊C4, D22⋊C4, C11×C22⋊C4, C2×C4×D11, C2×D44, C2×C11⋊D4, D22⋊D4
Quotients: C1, C2, C22, D4, C23, C2×D4, C4○D4, D11, C4⋊D4, D22, C22×D11, D445C2, D4×D11, D22⋊D4

Smallest permutation representation of D22⋊D4
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)
(1 135)(2 134)(3 133)(4 154)(5 153)(6 152)(7 151)(8 150)(9 149)(10 148)(11 147)(12 146)(13 145)(14 144)(15 143)(16 142)(17 141)(18 140)(19 139)(20 138)(21 137)(22 136)(23 115)(24 114)(25 113)(26 112)(27 111)(28 132)(29 131)(30 130)(31 129)(32 128)(33 127)(34 126)(35 125)(36 124)(37 123)(38 122)(39 121)(40 120)(41 119)(42 118)(43 117)(44 116)(45 99)(46 98)(47 97)(48 96)(49 95)(50 94)(51 93)(52 92)(53 91)(54 90)(55 89)(56 110)(57 109)(58 108)(59 107)(60 106)(61 105)(62 104)(63 103)(64 102)(65 101)(66 100)(67 165)(68 164)(69 163)(70 162)(71 161)(72 160)(73 159)(74 158)(75 157)(76 156)(77 155)(78 176)(79 175)(80 174)(81 173)(82 172)(83 171)(84 170)(85 169)(86 168)(87 167)(88 166)
(1 61 157 25)(2 60 158 24)(3 59 159 23)(4 58 160 44)(5 57 161 43)(6 56 162 42)(7 55 163 41)(8 54 164 40)(9 53 165 39)(10 52 166 38)(11 51 167 37)(12 50 168 36)(13 49 169 35)(14 48 170 34)(15 47 171 33)(16 46 172 32)(17 45 173 31)(18 66 174 30)(19 65 175 29)(20 64 176 28)(21 63 155 27)(22 62 156 26)(67 123 149 93)(68 122 150 92)(69 121 151 91)(70 120 152 90)(71 119 153 89)(72 118 154 110)(73 117 133 109)(74 116 134 108)(75 115 135 107)(76 114 136 106)(77 113 137 105)(78 112 138 104)(79 111 139 103)(80 132 140 102)(81 131 141 101)(82 130 142 100)(83 129 143 99)(84 128 144 98)(85 127 145 97)(86 126 146 96)(87 125 147 95)(88 124 148 94)
(1 168)(2 167)(3 166)(4 165)(5 164)(6 163)(7 162)(8 161)(9 160)(10 159)(11 158)(12 157)(13 156)(14 155)(15 176)(16 175)(17 174)(18 173)(19 172)(20 171)(21 170)(22 169)(23 38)(24 37)(25 36)(26 35)(27 34)(28 33)(29 32)(30 31)(39 44)(40 43)(41 42)(45 66)(46 65)(47 64)(48 63)(49 62)(50 61)(51 60)(52 59)(53 58)(54 57)(55 56)(67 145)(68 144)(69 143)(70 142)(71 141)(72 140)(73 139)(74 138)(75 137)(76 136)(77 135)(78 134)(79 133)(80 154)(81 153)(82 152)(83 151)(84 150)(85 149)(86 148)(87 147)(88 146)(89 101)(90 100)(91 99)(92 98)(93 97)(94 96)(102 110)(103 109)(104 108)(105 107)(111 117)(112 116)(113 115)(118 132)(119 131)(120 130)(121 129)(122 128)(123 127)(124 126)

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

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

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

64 conjugacy classes

class 1 2A2B2C2D2E2F2G4A4B4C4D4E4F11A···11E22A···22O22P···22Y44A···44T
order1222222244444411···1122···2222···2244···44
size111142222442242222442···22···24···44···4

64 irreducible representations

dim111111122222224
type+++++++++++++
imageC1C2C2C2C2C2C2D4D4C4○D4D11D22D22D445C2D4×D11
kernelD22⋊D4Dic11⋊C4D22⋊C4C11×C22⋊C4C2×C4×D11C2×D44C2×C11⋊D4Dic11D22C22C22⋊C4C2×C4C23C2C2
# reps111111222251052010

Matrix representation of D22⋊D4 in GL4(𝔽89) generated by

625500
386100
0010
0001
,
595000
553000
00880
00088
,
55400
168400
003887
005551
,
692100
702000
00880
00511
G:=sub<GL(4,GF(89))| [62,38,0,0,55,61,0,0,0,0,1,0,0,0,0,1],[59,55,0,0,50,30,0,0,0,0,88,0,0,0,0,88],[5,16,0,0,54,84,0,0,0,0,38,55,0,0,87,51],[69,70,0,0,21,20,0,0,0,0,88,51,0,0,0,1] >;

D22⋊D4 in GAP, Magma, Sage, TeX

D_{22}\rtimes D_4
% in TeX

G:=Group("D22:D4");
// GroupNames label

G:=SmallGroup(352,79);
// by ID

G=gap.SmallGroup(352,79);
# by ID

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

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

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