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G = C23.38D20order 320 = 26·5

9th non-split extension by C23 of D20 acting via D20/D10=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C23.38D20, C22⋊C89D5, C406C48C2, (C2×C20).44D4, (C2×C4).33D20, (C2×C8).109D10, C207D4.2C2, D205C410C2, C10.8(C2×SD16), (C2×C10).14SD16, (C22×C4).85D10, (C22×C10).55D4, C20.282(C4○D4), C2.14(C8⋊D10), C10.11(C8⋊C22), (C2×C40).120C22, (C2×C20).745C23, (C2×D20).12C22, C22.108(C2×D20), C22.3(C40⋊C2), C51(C23.46D4), C4.106(D42D5), C4⋊Dic5.270C22, (C22×C20).52C22, C10.17(C22.D4), C2.13(C22.D20), (C2×C4⋊Dic5)⋊5C2, (C5×C22⋊C8)⋊11C2, C2.11(C2×C40⋊C2), (C2×C10).128(C2×D4), (C2×C4).690(C22×D5), SmallGroup(320,362)

Series: Derived Chief Lower central Upper central

C1C2×C20 — C23.38D20
C1C5C10C20C2×C20C2×D20C207D4 — C23.38D20
C5C10C2×C20 — C23.38D20
C1C22C22×C4C22⋊C8

Generators and relations for C23.38D20
 G = < a,b,c,d,e | a2=b2=c2=1, d20=c, e2=cb=bc, dad-1=ab=ba, ac=ca, ae=ea, bd=db, be=eb, cd=dc, ce=ec, ede-1=bd19 >

Subgroups: 542 in 114 conjugacy classes, 43 normal (25 characteristic)
C1, C2 [×3], C2 [×3], C4 [×2], C4 [×4], C22, C22 [×2], C22 [×5], C5, C8 [×2], C2×C4 [×2], C2×C4 [×7], D4 [×4], C23, C23, D5, C10 [×3], C10 [×2], C22⋊C4, C4⋊C4 [×4], C2×C8 [×2], C22×C4, C22×C4, C2×D4 [×2], Dic5 [×3], C20 [×2], C20, D10 [×3], C2×C10, C2×C10 [×2], C2×C10 [×2], C22⋊C8, D4⋊C4 [×2], C4.Q8 [×2], C2×C4⋊C4, C4⋊D4, C40 [×2], D20 [×2], C2×Dic5 [×5], C5⋊D4 [×2], C2×C20 [×2], C2×C20 [×2], C22×D5, C22×C10, C23.46D4, C4⋊Dic5, C4⋊Dic5 [×2], C4⋊Dic5, D10⋊C4, C2×C40 [×2], C2×D20, C22×Dic5, C2×C5⋊D4, C22×C20, C406C4 [×2], D205C4 [×2], C5×C22⋊C8, C2×C4⋊Dic5, C207D4, C23.38D20
Quotients: C1, C2 [×7], C22 [×7], D4 [×2], C23, D5, SD16 [×2], C2×D4, C4○D4 [×2], D10 [×3], C22.D4, C2×SD16, C8⋊C22, D20 [×2], C22×D5, C23.46D4, C40⋊C2 [×2], C2×D20, D42D5 [×2], C22.D20, C2×C40⋊C2, C8⋊D10, C23.38D20

Smallest permutation representation of C23.38D20
On 160 points
Generators in S160
(1 159)(2 88)(3 121)(4 90)(5 123)(6 92)(7 125)(8 94)(9 127)(10 96)(11 129)(12 98)(13 131)(14 100)(15 133)(16 102)(17 135)(18 104)(19 137)(20 106)(21 139)(22 108)(23 141)(24 110)(25 143)(26 112)(27 145)(28 114)(29 147)(30 116)(31 149)(32 118)(33 151)(34 120)(35 153)(36 82)(37 155)(38 84)(39 157)(40 86)(41 136)(42 105)(43 138)(44 107)(45 140)(46 109)(47 142)(48 111)(49 144)(50 113)(51 146)(52 115)(53 148)(54 117)(55 150)(56 119)(57 152)(58 81)(59 154)(60 83)(61 156)(62 85)(63 158)(64 87)(65 160)(66 89)(67 122)(68 91)(69 124)(70 93)(71 126)(72 95)(73 128)(74 97)(75 130)(76 99)(77 132)(78 101)(79 134)(80 103)
(1 64)(2 65)(3 66)(4 67)(5 68)(6 69)(7 70)(8 71)(9 72)(10 73)(11 74)(12 75)(13 76)(14 77)(15 78)(16 79)(17 80)(18 41)(19 42)(20 43)(21 44)(22 45)(23 46)(24 47)(25 48)(26 49)(27 50)(28 51)(29 52)(30 53)(31 54)(32 55)(33 56)(34 57)(35 58)(36 59)(37 60)(38 61)(39 62)(40 63)(81 153)(82 154)(83 155)(84 156)(85 157)(86 158)(87 159)(88 160)(89 121)(90 122)(91 123)(92 124)(93 125)(94 126)(95 127)(96 128)(97 129)(98 130)(99 131)(100 132)(101 133)(102 134)(103 135)(104 136)(105 137)(106 138)(107 139)(108 140)(109 141)(110 142)(111 143)(112 144)(113 145)(114 146)(115 147)(116 148)(117 149)(118 150)(119 151)(120 152)
(1 21)(2 22)(3 23)(4 24)(5 25)(6 26)(7 27)(8 28)(9 29)(10 30)(11 31)(12 32)(13 33)(14 34)(15 35)(16 36)(17 37)(18 38)(19 39)(20 40)(41 61)(42 62)(43 63)(44 64)(45 65)(46 66)(47 67)(48 68)(49 69)(50 70)(51 71)(52 72)(53 73)(54 74)(55 75)(56 76)(57 77)(58 78)(59 79)(60 80)(81 101)(82 102)(83 103)(84 104)(85 105)(86 106)(87 107)(88 108)(89 109)(90 110)(91 111)(92 112)(93 113)(94 114)(95 115)(96 116)(97 117)(98 118)(99 119)(100 120)(121 141)(122 142)(123 143)(124 144)(125 145)(126 146)(127 147)(128 148)(129 149)(130 150)(131 151)(132 152)(133 153)(134 154)(135 155)(136 156)(137 157)(138 158)(139 159)(140 160)
(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 43 44 40)(2 39 45 42)(3 41 46 38)(4 37 47 80)(5 79 48 36)(6 35 49 78)(7 77 50 34)(8 33 51 76)(9 75 52 32)(10 31 53 74)(11 73 54 30)(12 29 55 72)(13 71 56 28)(14 27 57 70)(15 69 58 26)(16 25 59 68)(17 67 60 24)(18 23 61 66)(19 65 62 22)(20 21 63 64)(81 112 133 124)(82 123 134 111)(83 110 135 122)(84 121 136 109)(85 108 137 160)(86 159 138 107)(87 106 139 158)(88 157 140 105)(89 104 141 156)(90 155 142 103)(91 102 143 154)(92 153 144 101)(93 100 145 152)(94 151 146 99)(95 98 147 150)(96 149 148 97)(113 120 125 132)(114 131 126 119)(115 118 127 130)(116 129 128 117)

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

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

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

59 conjugacy classes

class 1 2A2B2C2D2E2F4A4B4C4D4E4F4G4H5A5B8A8B8C8D10A···10F10G10H10I10J20A···20H20I20J20K20L40A···40P
order12222224444444455888810···101010101020···202020202040···40
size1111224022420202020402244442···244442···244444···4

59 irreducible representations

dim1111112222222222444
type++++++++++++++-+
imageC1C2C2C2C2C2D4D4D5C4○D4SD16D10D10D20D20C40⋊C2C8⋊C22D42D5C8⋊D10
kernelC23.38D20C406C4D205C4C5×C22⋊C8C2×C4⋊Dic5C207D4C2×C20C22×C10C22⋊C8C20C2×C10C2×C8C22×C4C2×C4C23C22C10C4C2
# reps12211111244424416144

Matrix representation of C23.38D20 in GL4(𝔽41) generated by

40000
04000
00400
0051
,
1000
0100
00400
00040
,
40000
04000
0010
0001
,
131400
272900
003221
0009
,
291600
141200
00320
00032
G:=sub<GL(4,GF(41))| [40,0,0,0,0,40,0,0,0,0,40,5,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,40,0,0,0,0,40],[40,0,0,0,0,40,0,0,0,0,1,0,0,0,0,1],[13,27,0,0,14,29,0,0,0,0,32,0,0,0,21,9],[29,14,0,0,16,12,0,0,0,0,32,0,0,0,0,32] >;

C23.38D20 in GAP, Magma, Sage, TeX

C_2^3._{38}D_{20}
% in TeX

G:=Group("C2^3.38D20");
// GroupNames label

G:=SmallGroup(320,362);
// by ID

G=gap.SmallGroup(320,362);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,253,254,219,142,1123,136,12550]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^2=c^2=1,d^20=c,e^2=c*b=b*c,d*a*d^-1=a*b=b*a,a*c=c*a,a*e=e*a,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=b*d^19>;
// generators/relations

׿
×
𝔽