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G = C2×D44⋊C2order 352 = 25·11

Direct product of C2 and D44⋊C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — C2×D44⋊C2
 Chief series C1 — C11 — C22 — D22 — C22×D11 — C2×C4×D11 — C2×D44⋊C2
 Lower central C11 — C22 — C2×D44⋊C2
 Upper central C1 — C22 — C2×Q8

Generators and relations for C2×D44⋊C2
G = < a,b,c,d | a2=b44=c2=d2=1, ab=ba, ac=ca, ad=da, cbc=b-1, dbd=b21, dcd=b42c >

Subgroups: 938 in 164 conjugacy classes, 89 normal (10 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C2×C4, C2×C4, D4, Q8, C23, C11, C22×C4, C2×D4, C2×Q8, C4○D4, D11, C22, C22, C2×C4○D4, Dic11, C44, D22, D22, C2×C22, C4×D11, D44, C2×Dic11, C2×C44, Q8×C11, C22×D11, C2×C4×D11, C2×D44, D44⋊C2, Q8×C22, C2×D44⋊C2
Quotients: C1, C2, C22, C23, C4○D4, C24, D11, C2×C4○D4, D22, C22×D11, D44⋊C2, C23×D11, C2×D44⋊C2

Smallest permutation representation of C2×D44⋊C2
On 176 points
Generators in S176
(1 94)(2 95)(3 96)(4 97)(5 98)(6 99)(7 100)(8 101)(9 102)(10 103)(11 104)(12 105)(13 106)(14 107)(15 108)(16 109)(17 110)(18 111)(19 112)(20 113)(21 114)(22 115)(23 116)(24 117)(25 118)(26 119)(27 120)(28 121)(29 122)(30 123)(31 124)(32 125)(33 126)(34 127)(35 128)(36 129)(37 130)(38 131)(39 132)(40 89)(41 90)(42 91)(43 92)(44 93)(45 155)(46 156)(47 157)(48 158)(49 159)(50 160)(51 161)(52 162)(53 163)(54 164)(55 165)(56 166)(57 167)(58 168)(59 169)(60 170)(61 171)(62 172)(63 173)(64 174)(65 175)(66 176)(67 133)(68 134)(69 135)(70 136)(71 137)(72 138)(73 139)(74 140)(75 141)(76 142)(77 143)(78 144)(79 145)(80 146)(81 147)(82 148)(83 149)(84 150)(85 151)(86 152)(87 153)(88 154)
(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 93)(2 92)(3 91)(4 90)(5 89)(6 132)(7 131)(8 130)(9 129)(10 128)(11 127)(12 126)(13 125)(14 124)(15 123)(16 122)(17 121)(18 120)(19 119)(20 118)(21 117)(22 116)(23 115)(24 114)(25 113)(26 112)(27 111)(28 110)(29 109)(30 108)(31 107)(32 106)(33 105)(34 104)(35 103)(36 102)(37 101)(38 100)(39 99)(40 98)(41 97)(42 96)(43 95)(44 94)(45 170)(46 169)(47 168)(48 167)(49 166)(50 165)(51 164)(52 163)(53 162)(54 161)(55 160)(56 159)(57 158)(58 157)(59 156)(60 155)(61 154)(62 153)(63 152)(64 151)(65 150)(66 149)(67 148)(68 147)(69 146)(70 145)(71 144)(72 143)(73 142)(74 141)(75 140)(76 139)(77 138)(78 137)(79 136)(80 135)(81 134)(82 133)(83 176)(84 175)(85 174)(86 173)(87 172)(88 171)
(1 152)(2 173)(3 150)(4 171)(5 148)(6 169)(7 146)(8 167)(9 144)(10 165)(11 142)(12 163)(13 140)(14 161)(15 138)(16 159)(17 136)(18 157)(19 134)(20 155)(21 176)(22 153)(23 174)(24 151)(25 172)(26 149)(27 170)(28 147)(29 168)(30 145)(31 166)(32 143)(33 164)(34 141)(35 162)(36 139)(37 160)(38 137)(39 158)(40 135)(41 156)(42 133)(43 154)(44 175)(45 113)(46 90)(47 111)(48 132)(49 109)(50 130)(51 107)(52 128)(53 105)(54 126)(55 103)(56 124)(57 101)(58 122)(59 99)(60 120)(61 97)(62 118)(63 95)(64 116)(65 93)(66 114)(67 91)(68 112)(69 89)(70 110)(71 131)(72 108)(73 129)(74 106)(75 127)(76 104)(77 125)(78 102)(79 123)(80 100)(81 121)(82 98)(83 119)(84 96)(85 117)(86 94)(87 115)(88 92)

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

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

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

70 conjugacy classes

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

70 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 4 type + + + + + + + + + image C1 C2 C2 C2 C2 C4○D4 D11 D22 D22 D44⋊C2 kernel C2×D44⋊C2 C2×C4×D11 C2×D44 D44⋊C2 Q8×C22 C22 C2×Q8 C2×C4 Q8 C2 # reps 1 3 3 8 1 4 5 15 20 10

Matrix representation of C2×D44⋊C2 in GL4(𝔽89) generated by

 1 0 0 0 0 1 0 0 0 0 88 0 0 0 0 88
,
 1 36 0 0 84 88 0 0 0 0 76 81 0 0 6 31
,
 1 36 0 0 0 88 0 0 0 0 30 68 0 0 64 59
,
 55 22 0 0 81 34 0 0 0 0 77 68 0 0 28 12
G:=sub<GL(4,GF(89))| [1,0,0,0,0,1,0,0,0,0,88,0,0,0,0,88],[1,84,0,0,36,88,0,0,0,0,76,6,0,0,81,31],[1,0,0,0,36,88,0,0,0,0,30,64,0,0,68,59],[55,81,0,0,22,34,0,0,0,0,77,28,0,0,68,12] >;

C2×D44⋊C2 in GAP, Magma, Sage, TeX

C_2\times D_{44}\rtimes C_2
% in TeX

G:=Group("C2xD44:C2");
// GroupNames label

G:=SmallGroup(352,181);
// by ID

G=gap.SmallGroup(352,181);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-11,86,579,159,69,11525]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^44=c^2=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c=b^-1,d*b*d=b^21,d*c*d=b^42*c>;
// generators/relations

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