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## G = C24.44D10order 320 = 26·5

### 2nd non-split extension by C24 of D10 acting via D10/D5=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — C24.44D10
 Chief series C1 — C5 — C10 — C2×C10 — C22×C10 — C22×Dic5 — C23×Dic5 — C24.44D10
 Lower central C5 — C2×C10 — C24.44D10
 Upper central C1 — C23 — C2×C22⋊C4

Generators and relations for C24.44D10
G = < a,b,c,d,e,f | a2=b2=c2=d2=1, e10=f2=b, ab=ba, ac=ca, eae-1=faf-1=ad=da, bc=cb, bd=db, be=eb, bf=fb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=ce9 >

Subgroups: 734 in 234 conjugacy classes, 87 normal (25 characteristic)
C1, C2, C2, C2, C4, C22, C22, C22, C5, C2×C4, C23, C23, C23, C10, C10, C10, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C24, Dic5, Dic5, C20, C2×C10, C2×C10, C2×C10, C2.C42, C2×C22⋊C4, C2×C22⋊C4, C2×C4⋊C4, C23×C4, C2×Dic5, C2×Dic5, C2×C20, C22×C10, C22×C10, C22×C10, C23.7Q8, C10.D4, C23.D5, C5×C22⋊C4, C22×Dic5, C22×Dic5, C22×Dic5, C22×C20, C23×C10, C10.10C42, C2×C10.D4, C2×C23.D5, C10×C22⋊C4, C23×Dic5, C24.44D10
Quotients:

Smallest permutation representation of C24.44D10
On 160 points
Generators in S160
```(1 11)(2 30)(3 13)(4 32)(5 15)(6 34)(7 17)(8 36)(9 19)(10 38)(12 40)(14 22)(16 24)(18 26)(20 28)(21 31)(23 33)(25 35)(27 37)(29 39)(41 51)(42 62)(43 53)(44 64)(45 55)(46 66)(47 57)(48 68)(49 59)(50 70)(52 72)(54 74)(56 76)(58 78)(60 80)(61 71)(63 73)(65 75)(67 77)(69 79)(81 112)(82 92)(83 114)(84 94)(85 116)(86 96)(87 118)(88 98)(89 120)(90 100)(91 102)(93 104)(95 106)(97 108)(99 110)(101 111)(103 113)(105 115)(107 117)(109 119)(121 131)(122 147)(123 133)(124 149)(125 135)(126 151)(127 137)(128 153)(129 139)(130 155)(132 157)(134 159)(136 141)(138 143)(140 145)(142 152)(144 154)(146 156)(148 158)(150 160)
(1 11)(2 12)(3 13)(4 14)(5 15)(6 16)(7 17)(8 18)(9 19)(10 20)(21 31)(22 32)(23 33)(24 34)(25 35)(26 36)(27 37)(28 38)(29 39)(30 40)(41 51)(42 52)(43 53)(44 54)(45 55)(46 56)(47 57)(48 58)(49 59)(50 60)(61 71)(62 72)(63 73)(64 74)(65 75)(66 76)(67 77)(68 78)(69 79)(70 80)(81 91)(82 92)(83 93)(84 94)(85 95)(86 96)(87 97)(88 98)(89 99)(90 100)(101 111)(102 112)(103 113)(104 114)(105 115)(106 116)(107 117)(108 118)(109 119)(110 120)(121 131)(122 132)(123 133)(124 134)(125 135)(126 136)(127 137)(128 138)(129 139)(130 140)(141 151)(142 152)(143 153)(144 154)(145 155)(146 156)(147 157)(148 158)(149 159)(150 160)
(1 59)(2 60)(3 41)(4 42)(5 43)(6 44)(7 45)(8 46)(9 47)(10 48)(11 49)(12 50)(13 51)(14 52)(15 53)(16 54)(17 55)(18 56)(19 57)(20 58)(21 71)(22 72)(23 73)(24 74)(25 75)(26 76)(27 77)(28 78)(29 79)(30 80)(31 61)(32 62)(33 63)(34 64)(35 65)(36 66)(37 67)(38 68)(39 69)(40 70)(81 141)(82 142)(83 143)(84 144)(85 145)(86 146)(87 147)(88 148)(89 149)(90 150)(91 151)(92 152)(93 153)(94 154)(95 155)(96 156)(97 157)(98 158)(99 159)(100 160)(101 125)(102 126)(103 127)(104 128)(105 129)(106 130)(107 131)(108 132)(109 133)(110 134)(111 135)(112 136)(113 137)(114 138)(115 139)(116 140)(117 121)(118 122)(119 123)(120 124)
(1 39)(2 40)(3 21)(4 22)(5 23)(6 24)(7 25)(8 26)(9 27)(10 28)(11 29)(12 30)(13 31)(14 32)(15 33)(16 34)(17 35)(18 36)(19 37)(20 38)(41 71)(42 72)(43 73)(44 74)(45 75)(46 76)(47 77)(48 78)(49 79)(50 80)(51 61)(52 62)(53 63)(54 64)(55 65)(56 66)(57 67)(58 68)(59 69)(60 70)(81 102)(82 103)(83 104)(84 105)(85 106)(86 107)(87 108)(88 109)(89 110)(90 111)(91 112)(92 113)(93 114)(94 115)(95 116)(96 117)(97 118)(98 119)(99 120)(100 101)(121 156)(122 157)(123 158)(124 159)(125 160)(126 141)(127 142)(128 143)(129 144)(130 145)(131 146)(132 147)(133 148)(134 149)(135 150)(136 151)(137 152)(138 153)(139 154)(140 155)
(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)
(1 81 11 91)(2 150 12 160)(3 99 13 89)(4 148 14 158)(5 97 15 87)(6 146 16 156)(7 95 17 85)(8 144 18 154)(9 93 19 83)(10 142 20 152)(21 120 31 110)(22 133 32 123)(23 118 33 108)(24 131 34 121)(25 116 35 106)(26 129 36 139)(27 114 37 104)(28 127 38 137)(29 112 39 102)(30 125 40 135)(41 159 51 149)(42 88 52 98)(43 157 53 147)(44 86 54 96)(45 155 55 145)(46 84 56 94)(47 153 57 143)(48 82 58 92)(49 151 59 141)(50 100 60 90)(61 134 71 124)(62 119 72 109)(63 132 73 122)(64 117 74 107)(65 130 75 140)(66 115 76 105)(67 128 77 138)(68 113 78 103)(69 126 79 136)(70 111 80 101)```

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

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

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

68 conjugacy classes

 class 1 2A ··· 2G 2H 2I 2J 2K 4A 4B 4C 4D 4E ··· 4L 4M 4N 4O 4P 5A 5B 10A ··· 10N 10O ··· 10V 20A ··· 20P order 1 2 ··· 2 2 2 2 2 4 4 4 4 4 ··· 4 4 4 4 4 5 5 10 ··· 10 10 ··· 10 20 ··· 20 size 1 1 ··· 1 2 2 2 2 4 4 4 4 10 ··· 10 20 20 20 20 2 2 2 ··· 2 4 ··· 4 4 ··· 4

68 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 4 4 type + + + + + + + + - + + + - + - image C1 C2 C2 C2 C2 C2 C4 D4 D4 Q8 D5 C4○D4 D10 D10 Dic10 C4×D5 C5⋊D4 D4×D5 D4⋊2D5 kernel C24.44D10 C10.10C42 C2×C10.D4 C2×C23.D5 C10×C22⋊C4 C23×Dic5 C22×Dic5 C2×Dic5 C22×C10 C22×C10 C2×C22⋊C4 C2×C10 C22×C4 C24 C23 C23 C23 C22 C22 # reps 1 2 2 1 1 1 8 4 2 2 2 4 4 2 8 8 8 4 4

Matrix representation of C24.44D10 in GL5(𝔽41)

 1 0 0 0 0 0 1 0 0 0 0 0 40 0 0 0 0 0 1 0 0 0 0 0 1
,
 40 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1
,
 1 0 0 0 0 0 40 0 0 0 0 0 40 0 0 0 0 0 40 0 0 0 0 0 40
,
 1 0 0 0 0 0 40 0 0 0 0 0 40 0 0 0 0 0 1 0 0 0 0 0 1
,
 32 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 3 4 0 0 0 20 40
,
 9 0 0 0 0 0 0 32 0 0 0 9 0 0 0 0 0 0 7 8 0 0 0 35 34

`G:=sub<GL(5,GF(41))| [1,0,0,0,0,0,1,0,0,0,0,0,40,0,0,0,0,0,1,0,0,0,0,0,1],[40,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,40,0,0,0,0,0,40,0,0,0,0,0,40,0,0,0,0,0,40],[1,0,0,0,0,0,40,0,0,0,0,0,40,0,0,0,0,0,1,0,0,0,0,0,1],[32,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,3,20,0,0,0,4,40],[9,0,0,0,0,0,0,9,0,0,0,32,0,0,0,0,0,0,7,35,0,0,0,8,34] >;`

C24.44D10 in GAP, Magma, Sage, TeX

`C_2^4._{44}D_{10}`
`% in TeX`

`G:=Group("C2^4.44D10");`
`// GroupNames label`

`G:=SmallGroup(320,569);`
`// by ID`

`G=gap.SmallGroup(320,569);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,112,422,387,58,12550]);`
`// Polycyclic`

`G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=1,e^10=f^2=b,a*b=b*a,a*c=c*a,e*a*e^-1=f*a*f^-1=a*d=d*a,b*c=c*b,b*d=d*b,b*e=e*b,b*f=f*b,c*d=d*c,c*e=e*c,c*f=f*c,d*e=e*d,d*f=f*d,f*e*f^-1=c*e^9>;`
`// generators/relations`

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