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## G = C22.D44order 352 = 25·11

### 3rd non-split extension by C22 of D44 acting via D44/D22=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C22 — C22.D44
 Chief series C1 — C11 — C22 — C2×C22 — C22×D11 — C2×C11⋊D4 — C22.D44
 Lower central C11 — C2×C22 — C22.D44
 Upper central C1 — C22 — C22⋊C4

Generators and relations for C22.D44
G = < a,b,c,d | a2=b2=c44=1, d2=b, cac-1=ab=ba, ad=da, bc=cb, bd=db, dcd-1=bc-1 >

Subgroups: 466 in 78 conjugacy classes, 33 normal (15 characteristic)
C1, C2, C2, C2, C4, C22, C22, C22, C2×C4, C2×C4, D4, C23, C23, C11, C22⋊C4, C22⋊C4, C4⋊C4, C22×C4, C2×D4, D11, C22, C22, C22, C22.D4, Dic11, C44, D22, C2×C22, C2×C22, C2×C22, C2×Dic11, C2×Dic11, C2×Dic11, C11⋊D4, C2×C44, C22×D11, C22×C22, C44⋊C4, D22⋊C4, C11×C22⋊C4, C22×Dic11, C2×C11⋊D4, C22.D44
Quotients: C1, C2, C22, D4, C23, C2×D4, C4○D4, D11, C22.D4, D22, D44, C22×D11, C2×D44, D42D11, C22.D44

Smallest permutation representation of C22.D44
On 176 points
Generators in S176
(1 159)(2 73)(3 161)(4 75)(5 163)(6 77)(7 165)(8 79)(9 167)(10 81)(11 169)(12 83)(13 171)(14 85)(15 173)(16 87)(17 175)(18 45)(19 133)(20 47)(21 135)(22 49)(23 137)(24 51)(25 139)(26 53)(27 141)(28 55)(29 143)(30 57)(31 145)(32 59)(33 147)(34 61)(35 149)(36 63)(37 151)(38 65)(39 153)(40 67)(41 155)(42 69)(43 157)(44 71)(46 93)(48 95)(50 97)(52 99)(54 101)(56 103)(58 105)(60 107)(62 109)(64 111)(66 113)(68 115)(70 117)(72 119)(74 121)(76 123)(78 125)(80 127)(82 129)(84 131)(86 89)(88 91)(90 174)(92 176)(94 134)(96 136)(98 138)(100 140)(102 142)(104 144)(106 146)(108 148)(110 150)(112 152)(114 154)(116 156)(118 158)(120 160)(122 162)(124 164)(126 166)(128 168)(130 170)(132 172)
(1 119)(2 120)(3 121)(4 122)(5 123)(6 124)(7 125)(8 126)(9 127)(10 128)(11 129)(12 130)(13 131)(14 132)(15 89)(16 90)(17 91)(18 92)(19 93)(20 94)(21 95)(22 96)(23 97)(24 98)(25 99)(26 100)(27 101)(28 102)(29 103)(30 104)(31 105)(32 106)(33 107)(34 108)(35 109)(36 110)(37 111)(38 112)(39 113)(40 114)(41 115)(42 116)(43 117)(44 118)(45 176)(46 133)(47 134)(48 135)(49 136)(50 137)(51 138)(52 139)(53 140)(54 141)(55 142)(56 143)(57 144)(58 145)(59 146)(60 147)(61 148)(62 149)(63 150)(64 151)(65 152)(66 153)(67 154)(68 155)(69 156)(70 157)(71 158)(72 159)(73 160)(74 161)(75 162)(76 163)(77 164)(78 165)(79 166)(80 167)(81 168)(82 169)(83 170)(84 171)(85 172)(86 173)(87 174)(88 175)
(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)
(1 118 119 44)(2 43 120 117)(3 116 121 42)(4 41 122 115)(5 114 123 40)(6 39 124 113)(7 112 125 38)(8 37 126 111)(9 110 127 36)(10 35 128 109)(11 108 129 34)(12 33 130 107)(13 106 131 32)(14 31 132 105)(15 104 89 30)(16 29 90 103)(17 102 91 28)(18 27 92 101)(19 100 93 26)(20 25 94 99)(21 98 95 24)(22 23 96 97)(45 141 176 54)(46 53 133 140)(47 139 134 52)(48 51 135 138)(49 137 136 50)(55 175 142 88)(56 87 143 174)(57 173 144 86)(58 85 145 172)(59 171 146 84)(60 83 147 170)(61 169 148 82)(62 81 149 168)(63 167 150 80)(64 79 151 166)(65 165 152 78)(66 77 153 164)(67 163 154 76)(68 75 155 162)(69 161 156 74)(70 73 157 160)(71 159 158 72)

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

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

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

64 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 4A 4B 4C 4D 4E 4F 4G 11A ··· 11E 22A ··· 22O 22P ··· 22Y 44A ··· 44T order 1 2 2 2 2 2 2 4 4 4 4 4 4 4 11 ··· 11 22 ··· 22 22 ··· 22 44 ··· 44 size 1 1 1 1 2 2 44 4 4 22 22 22 22 44 2 ··· 2 2 ··· 2 4 ··· 4 4 ··· 4

64 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 4 type + + + + + + + + + + + - image C1 C2 C2 C2 C2 C2 D4 C4○D4 D11 D22 D22 D44 D4⋊2D11 kernel C22.D44 C44⋊C4 D22⋊C4 C11×C22⋊C4 C22×Dic11 C2×C11⋊D4 C2×C22 C22 C22⋊C4 C2×C4 C23 C22 C2 # reps 1 2 2 1 1 1 2 4 5 10 5 20 10

Matrix representation of C22.D44 in GL4(𝔽89) generated by

 0 55 0 0 34 0 0 0 0 0 1 0 0 0 0 1
,
 88 0 0 0 0 88 0 0 0 0 1 0 0 0 0 1
,
 0 1 0 0 1 0 0 0 0 0 44 54 0 0 37 7
,
 0 1 0 0 88 0 0 0 0 0 65 9 0 0 35 24
G:=sub<GL(4,GF(89))| [0,34,0,0,55,0,0,0,0,0,1,0,0,0,0,1],[88,0,0,0,0,88,0,0,0,0,1,0,0,0,0,1],[0,1,0,0,1,0,0,0,0,0,44,37,0,0,54,7],[0,88,0,0,1,0,0,0,0,0,65,35,0,0,9,24] >;

C22.D44 in GAP, Magma, Sage, TeX

C_2^2.D_{44}
% in TeX

G:=Group("C2^2.D44");
// GroupNames label

G:=SmallGroup(352,81);
// by ID

G=gap.SmallGroup(352,81);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-11,217,218,188,122,11525]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^2=c^44=1,d^2=b,c*a*c^-1=a*b=b*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=b*c^-1>;
// generators/relations

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