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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 [×3], C4 [×2], C4 [×2], C22, C22 [×2], C5, C8 [×2], C2×C4, C2×C4 [×2], D4 [×4], Q8 [×2], D5, C10, C10 [×2], C16 [×2], C2×C8, D8, D8, SD16 [×2], Q16, Q16, C4○D4 [×2], Dic5, C20 [×2], C20, D10, C2×C10, C2×C10, C2×C16, D16, SD32 [×2], Q32, C4○D8, C4○D8, C40 [×2], Dic10, C4×D5, D20, C5⋊D4, C2×C20, C2×C20, C5×D4 [×2], C5×Q8, C4○D16, C52C16 [×2], 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 [×7], C22 [×7], D4 [×2], C23, D5, D8 [×2], C2×D4, D10 [×3], C2×D8, C5⋊D4 [×2], C22×D5, C4○D16, D4⋊D5 [×2], 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 70)(42 69)(43 68)(44 67)(45 66)(46 65)(47 64)(48 63)(49 62)(50 61)(51 60)(52 59)(53 58)(54 57)(55 56)(71 80)(72 79)(73 78)(74 77)(75 76)(81 105)(82 104)(83 103)(84 102)(85 101)(86 100)(87 99)(88 98)(89 97)(90 96)(91 95)(92 94)(106 120)(107 119)(108 118)(109 117)(110 116)(111 115)(112 114)(121 138)(122 137)(123 136)(124 135)(125 134)(126 133)(127 132)(128 131)(129 130)(139 160)(140 159)(141 158)(142 157)(143 156)(144 155)(145 154)(146 153)(147 152)(148 151)(149 150)
(1 113 21 93)(2 114 22 94)(3 115 23 95)(4 116 24 96)(5 117 25 97)(6 118 26 98)(7 119 27 99)(8 120 28 100)(9 81 29 101)(10 82 30 102)(11 83 31 103)(12 84 32 104)(13 85 33 105)(14 86 34 106)(15 87 35 107)(16 88 36 108)(17 89 37 109)(18 90 38 110)(19 91 39 111)(20 92 40 112)(41 135 61 155)(42 136 62 156)(43 137 63 157)(44 138 64 158)(45 139 65 159)(46 140 66 160)(47 141 67 121)(48 142 68 122)(49 143 69 123)(50 144 70 124)(51 145 71 125)(52 146 72 126)(53 147 73 127)(54 148 74 128)(55 149 75 129)(56 150 76 130)(57 151 77 131)(58 152 78 132)(59 153 79 133)(60 154 80 134)
(1 43)(2 74)(3 65)(4 56)(5 47)(6 78)(7 69)(8 60)(9 51)(10 42)(11 73)(12 64)(13 55)(14 46)(15 77)(16 68)(17 59)(18 50)(19 41)(20 72)(21 63)(22 54)(23 45)(24 76)(25 67)(26 58)(27 49)(28 80)(29 71)(30 62)(31 53)(32 44)(33 75)(34 66)(35 57)(36 48)(37 79)(38 70)(39 61)(40 52)(81 145)(82 136)(83 127)(84 158)(85 149)(86 140)(87 131)(88 122)(89 153)(90 144)(91 135)(92 126)(93 157)(94 148)(95 139)(96 130)(97 121)(98 152)(99 143)(100 134)(101 125)(102 156)(103 147)(104 138)(105 129)(106 160)(107 151)(108 142)(109 133)(110 124)(111 155)(112 146)(113 137)(114 128)(115 159)(116 150)(117 141)(118 132)(119 123)(120 154)

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

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

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

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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