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G = C23.D19order 304 = 24·19

The non-split extension by C23 of D19 acting via D19/C19=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C23.D19, C38.11D4, C22⋊Dic19, C22.7D38, (C2×C38)⋊2C4, C38.9(C2×C4), C192(C22⋊C4), (C2×Dic19)⋊2C2, C2.3(C19⋊D4), (C2×C38).7C22, (C22×C38).2C2, C2.5(C2×Dic19), SmallGroup(304,18)

Series: Derived Chief Lower central Upper central

C1C38 — C23.D19
C1C19C38C2×C38C2×Dic19 — C23.D19
C19C38 — C23.D19
C1C22C23

Generators and relations for C23.D19
 G = < a,b,c,d,e | a2=b2=c2=d19=1, e2=b, ab=ba, eae-1=ac=ca, ad=da, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede-1=d-1 >

2C2
2C2
2C22
2C22
38C4
38C4
2C38
2C38
19C2×C4
19C2×C4
2Dic19
2C2×C38
2C2×C38
2Dic19
19C22⋊C4

Smallest permutation representation of C23.D19
On 152 points
Generators in S152
(77 96)(78 97)(79 98)(80 99)(81 100)(82 101)(83 102)(84 103)(85 104)(86 105)(87 106)(88 107)(89 108)(90 109)(91 110)(92 111)(93 112)(94 113)(95 114)(115 134)(116 135)(117 136)(118 137)(119 138)(120 139)(121 140)(122 141)(123 142)(124 143)(125 144)(126 145)(127 146)(128 147)(129 148)(130 149)(131 150)(132 151)(133 152)
(1 39)(2 40)(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 69)(21 70)(22 71)(23 72)(24 73)(25 74)(26 75)(27 76)(28 58)(29 59)(30 60)(31 61)(32 62)(33 63)(34 64)(35 65)(36 66)(37 67)(38 68)(77 115)(78 116)(79 117)(80 118)(81 119)(82 120)(83 121)(84 122)(85 123)(86 124)(87 125)(88 126)(89 127)(90 128)(91 129)(92 130)(93 131)(94 132)(95 133)(96 134)(97 135)(98 136)(99 137)(100 138)(101 139)(102 140)(103 141)(104 142)(105 143)(106 144)(107 145)(108 146)(109 147)(110 148)(111 149)(112 150)(113 151)(114 152)
(1 35)(2 36)(3 37)(4 38)(5 20)(6 21)(7 22)(8 23)(9 24)(10 25)(11 26)(12 27)(13 28)(14 29)(15 30)(16 31)(17 32)(18 33)(19 34)(39 65)(40 66)(41 67)(42 68)(43 69)(44 70)(45 71)(46 72)(47 73)(48 74)(49 75)(50 76)(51 58)(52 59)(53 60)(54 61)(55 62)(56 63)(57 64)(77 96)(78 97)(79 98)(80 99)(81 100)(82 101)(83 102)(84 103)(85 104)(86 105)(87 106)(88 107)(89 108)(90 109)(91 110)(92 111)(93 112)(94 113)(95 114)(115 134)(116 135)(117 136)(118 137)(119 138)(120 139)(121 140)(122 141)(123 142)(124 143)(125 144)(126 145)(127 146)(128 147)(129 148)(130 149)(131 150)(132 151)(133 152)
(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)
(1 133 39 95)(2 132 40 94)(3 131 41 93)(4 130 42 92)(5 129 43 91)(6 128 44 90)(7 127 45 89)(8 126 46 88)(9 125 47 87)(10 124 48 86)(11 123 49 85)(12 122 50 84)(13 121 51 83)(14 120 52 82)(15 119 53 81)(16 118 54 80)(17 117 55 79)(18 116 56 78)(19 115 57 77)(20 148 69 110)(21 147 70 109)(22 146 71 108)(23 145 72 107)(24 144 73 106)(25 143 74 105)(26 142 75 104)(27 141 76 103)(28 140 58 102)(29 139 59 101)(30 138 60 100)(31 137 61 99)(32 136 62 98)(33 135 63 97)(34 134 64 96)(35 152 65 114)(36 151 66 113)(37 150 67 112)(38 149 68 111)

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

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

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

82 conjugacy classes

class 1 2A2B2C2D2E4A4B4C4D19A···19I38A···38BK
order122222444419···1938···38
size111122383838382···22···2

82 irreducible representations

dim111122222
type+++++-+
imageC1C2C2C4D4D19Dic19D38C19⋊D4
kernelC23.D19C2×Dic19C22×C38C2×C38C38C23C22C22C2
# reps12142918936

Matrix representation of C23.D19 in GL3(𝔽229) generated by

22800
010
022228
,
22800
02280
00228
,
100
02280
00228
,
100
0430
06816
,
12200
0120114
0177109
G:=sub<GL(3,GF(229))| [228,0,0,0,1,22,0,0,228],[228,0,0,0,228,0,0,0,228],[1,0,0,0,228,0,0,0,228],[1,0,0,0,43,68,0,0,16],[122,0,0,0,120,177,0,114,109] >;

C23.D19 in GAP, Magma, Sage, TeX

C_2^3.D_{19}
% in TeX

G:=Group("C2^3.D19");
// GroupNames label

G:=SmallGroup(304,18);
// by ID

G=gap.SmallGroup(304,18);
# by ID

G:=PCGroup([5,-2,-2,-2,-2,-19,20,101,7204]);
// Polycyclic

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

Export

Subgroup lattice of C23.D19 in TeX

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