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G = C40.30C23order 320 = 26·5

23rd non-split extension by C40 of C23 acting via C23/C2=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D8.8D10, C20.59D8, C40.45D4, Q16.9D10, C40.30C23, D40.12C22, Dic20.13C22, C4○D81D5, C55(C4○D16), C5⋊D167C2, D8.D57C2, C5⋊Q327C2, C10.69(C2×D8), (C2×C10).10D8, C5⋊SD327C2, D407C25C2, C4.32(D4⋊D5), (C2×C8).253D10, C20.192(C2×D4), (C2×C20).186D4, C8.21(C5⋊D4), (C5×D8).8C22, C8.36(C22×D5), (C2×C40).43C22, C22.1(D4⋊D5), (C5×Q16).9C22, C52C16.10C22, (C5×C4○D8)⋊1C2, (C2×C52C16)⋊3C2, C2.24(C2×D4⋊D5), C4.18(C2×C5⋊D4), (C2×C4).146(C5⋊D4), SmallGroup(320,821)

Series: Derived Chief Lower central Upper central

C1C40 — C40.30C23
C1C5C10C20C40D40D407C2 — C40.30C23
C5C10C20C40 — C40.30C23
C1C4C2×C4C2×C8C4○D8

Generators and relations for C40.30C23
 G = < a,b,c,d | a40=b2=d2=1, c2=a20, bab=a-1, ac=ca, dad=a31, bc=cb, dbd=a25b, cd=dc >

Subgroups: 350 in 84 conjugacy classes, 35 normal (all characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, D4, Q8, D5, C10, C10, C16, C2×C8, D8, D8, SD16, Q16, Q16, C4○D4, Dic5, C20, C20, D10, C2×C10, C2×C10, C2×C16, D16, SD32, Q32, C4○D8, C4○D8, C40, Dic10, C4×D5, D20, C5⋊D4, C2×C20, C2×C20, C5×D4, C5×Q8, C4○D16, C52C16, C40⋊C2, D40, Dic20, C2×C40, C5×D8, C5×SD16, C5×Q16, C4○D20, C5×C4○D4, C2×C52C16, C5⋊D16, D8.D5, C5⋊SD32, C5⋊Q32, D407C2, C5×C4○D8, C40.30C23
Quotients: C1, C2, C22, D4, C23, D5, D8, C2×D4, D10, C2×D8, C5⋊D4, C22×D5, C4○D16, D4⋊D5, C2×C5⋊D4, C2×D4⋊D5, C40.30C23

Smallest permutation representation of C40.30C23
On 160 points
Generators in S160
(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)
(2 40)(3 39)(4 38)(5 37)(6 36)(7 35)(8 34)(9 33)(10 32)(11 31)(12 30)(13 29)(14 28)(15 27)(16 26)(17 25)(18 24)(19 23)(20 22)(41 74)(42 73)(43 72)(44 71)(45 70)(46 69)(47 68)(48 67)(49 66)(50 65)(51 64)(52 63)(53 62)(54 61)(55 60)(56 59)(57 58)(75 80)(76 79)(77 78)(81 87)(82 86)(83 85)(88 120)(89 119)(90 118)(91 117)(92 116)(93 115)(94 114)(95 113)(96 112)(97 111)(98 110)(99 109)(100 108)(101 107)(102 106)(103 105)(121 136)(122 135)(123 134)(124 133)(125 132)(126 131)(127 130)(128 129)(137 160)(138 159)(139 158)(140 157)(141 156)(142 155)(143 154)(144 153)(145 152)(146 151)(147 150)(148 149)
(1 84 21 104)(2 85 22 105)(3 86 23 106)(4 87 24 107)(5 88 25 108)(6 89 26 109)(7 90 27 110)(8 91 28 111)(9 92 29 112)(10 93 30 113)(11 94 31 114)(12 95 32 115)(13 96 33 116)(14 97 34 117)(15 98 35 118)(16 99 36 119)(17 100 37 120)(18 101 38 81)(19 102 39 82)(20 103 40 83)(41 132 61 152)(42 133 62 153)(43 134 63 154)(44 135 64 155)(45 136 65 156)(46 137 66 157)(47 138 67 158)(48 139 68 159)(49 140 69 160)(50 141 70 121)(51 142 71 122)(52 143 72 123)(53 144 73 124)(54 145 74 125)(55 146 75 126)(56 147 76 127)(57 148 77 128)(58 149 78 129)(59 150 79 130)(60 151 80 131)
(1 45)(2 76)(3 67)(4 58)(5 49)(6 80)(7 71)(8 62)(9 53)(10 44)(11 75)(12 66)(13 57)(14 48)(15 79)(16 70)(17 61)(18 52)(19 43)(20 74)(21 65)(22 56)(23 47)(24 78)(25 69)(26 60)(27 51)(28 42)(29 73)(30 64)(31 55)(32 46)(33 77)(34 68)(35 59)(36 50)(37 41)(38 72)(39 63)(40 54)(81 123)(82 154)(83 145)(84 136)(85 127)(86 158)(87 149)(88 140)(89 131)(90 122)(91 153)(92 144)(93 135)(94 126)(95 157)(96 148)(97 139)(98 130)(99 121)(100 152)(101 143)(102 134)(103 125)(104 156)(105 147)(106 138)(107 129)(108 160)(109 151)(110 142)(111 133)(112 124)(113 155)(114 146)(115 137)(116 128)(117 159)(118 150)(119 141)(120 132)

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

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

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

50 conjugacy classes

class 1 2A2B2C2D4A4B4C4D4E5A5B8A8B8C8D10A10B10C10D10E10F10G10H16A···16H20A20B20C20D20E20F20G20H20I20J40A···40H
order1222244444558888101010101010101016···162020202020202020202040···40
size1128401128402222222244888810···1022224488884···4

50 irreducible representations

dim1111111122222222222444
type++++++++++++++++++
imageC1C2C2C2C2C2C2C2D4D4D5D8D8D10D10D10C5⋊D4C5⋊D4C4○D16D4⋊D5D4⋊D5C40.30C23
kernelC40.30C23C2×C52C16C5⋊D16D8.D5C5⋊SD32C5⋊Q32D407C2C5×C4○D8C40C2×C20C4○D8C20C2×C10C2×C8D8Q16C8C2×C4C5C4C22C1
# reps1111111111222222448228

Matrix representation of C40.30C23 in GL4(𝔽241) generated by

1905200
190000
0021922
002300
,
51100
5119000
000219
002300
,
240000
024000
00640
00064
,
7613800
15216500
0012954
00156112
G:=sub<GL(4,GF(241))| [190,190,0,0,52,0,0,0,0,0,219,230,0,0,22,0],[51,51,0,0,1,190,0,0,0,0,0,230,0,0,219,0],[240,0,0,0,0,240,0,0,0,0,64,0,0,0,0,64],[76,152,0,0,138,165,0,0,0,0,129,156,0,0,54,112] >;

C40.30C23 in GAP, Magma, Sage, TeX

C_{40}._{30}C_2^3
% in TeX

G:=Group("C40.30C2^3");
// GroupNames label

G:=SmallGroup(320,821);
// by ID

G=gap.SmallGroup(320,821);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,232,254,675,185,192,1684,438,102,12550]);
// Polycyclic

G:=Group<a,b,c,d|a^40=b^2=d^2=1,c^2=a^20,b*a*b=a^-1,a*c=c*a,d*a*d=a^31,b*c=c*b,d*b*d=a^25*b,c*d=d*c>;
// generators/relations

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