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G = C4014D4order 320 = 26·5

14th semidirect product of C40 and D4 acting via D4/C2=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C4014D4, D104SD16, C57(C88D4), C811(C5⋊D4), C406C427C2, D103Q84C2, (C10×SD16)⋊8C2, (C2×SD16)⋊12D5, (C2×D4).73D10, (C2×C8).263D10, C202D4.9C2, C20.176(C2×D4), (C2×Q8).54D10, C2.30(D5×SD16), C10.63(C4○D8), Q8⋊Dic529C2, D4⋊Dic534C2, C10.47(C2×SD16), (C22×D5).90D4, C22.268(D4×D5), C20.101(C4○D4), C4.32(D42D5), C2.19(C202D4), (C2×C40).164C22, (C2×C20).448C23, (C2×Dic5).159D4, (D4×C10).97C22, (Q8×C10).77C22, C10.116(C4⋊D4), C4⋊Dic5.175C22, C2.29(SD163D5), (D5×C2×C8)⋊8C2, C4.82(C2×C5⋊D4), (C2×C10).360(C2×D4), (C2×C4×D5).310C22, (C2×C4).537(C22×D5), (C2×C52C8).282C22, SmallGroup(320,798)

Series: Derived Chief Lower central Upper central

C1C2×C20 — C4014D4
C1C5C10C2×C10C2×C20C2×C4×D5D5×C2×C8 — C4014D4
C5C10C2×C20 — C4014D4
C1C22C2×C4C2×SD16

Generators and relations for C4014D4
 G = < a,b,c | a40=b4=c2=1, bab-1=a19, cac=a9, cbc=b-1 >

Subgroups: 486 in 124 conjugacy classes, 43 normal (37 characteristic)
C1, C2 [×3], C2 [×3], C4 [×2], C4 [×4], C22, C22 [×7], C5, C8 [×2], C8, C2×C4, C2×C4 [×6], D4 [×4], Q8 [×2], C23 [×2], D5 [×2], C10 [×3], C10, C22⋊C4 [×2], C4⋊C4 [×3], C2×C8, C2×C8 [×3], SD16 [×2], C22×C4, C2×D4, C2×D4, C2×Q8, Dic5 [×3], C20 [×2], C20, D10 [×2], D10 [×2], C2×C10, C2×C10 [×3], D4⋊C4, Q8⋊C4, C4.Q8, C4⋊D4, C22⋊Q8, C22×C8, C2×SD16, C52C8, C40 [×2], C4×D5 [×2], C2×Dic5, C2×Dic5 [×2], C5⋊D4 [×2], C2×C20, C2×C20, C5×D4 [×2], C5×Q8 [×2], C22×D5, C22×C10, C88D4, C8×D5 [×2], C2×C52C8, C10.D4, C4⋊Dic5 [×2], D10⋊C4, C23.D5, C2×C40, C5×SD16 [×2], C2×C4×D5, C2×C5⋊D4, D4×C10, Q8×C10, C406C4, D4⋊Dic5, Q8⋊Dic5, D5×C2×C8, C202D4, D103Q8, C10×SD16, C4014D4
Quotients: C1, C2 [×7], C22 [×7], D4 [×4], C23, D5, SD16 [×2], C2×D4 [×2], C4○D4, D10 [×3], C4⋊D4, C2×SD16, C4○D8, C5⋊D4 [×2], C22×D5, C88D4, D4×D5, D42D5, C2×C5⋊D4, D5×SD16, SD163D5, C202D4, C4014D4

Smallest permutation representation of C4014D4
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)
(1 73 149 84)(2 52 150 103)(3 71 151 82)(4 50 152 101)(5 69 153 120)(6 48 154 99)(7 67 155 118)(8 46 156 97)(9 65 157 116)(10 44 158 95)(11 63 159 114)(12 42 160 93)(13 61 121 112)(14 80 122 91)(15 59 123 110)(16 78 124 89)(17 57 125 108)(18 76 126 87)(19 55 127 106)(20 74 128 85)(21 53 129 104)(22 72 130 83)(23 51 131 102)(24 70 132 81)(25 49 133 100)(26 68 134 119)(27 47 135 98)(28 66 136 117)(29 45 137 96)(30 64 138 115)(31 43 139 94)(32 62 140 113)(33 41 141 92)(34 60 142 111)(35 79 143 90)(36 58 144 109)(37 77 145 88)(38 56 146 107)(39 75 147 86)(40 54 148 105)
(2 10)(3 19)(4 28)(5 37)(7 15)(8 24)(9 33)(12 20)(13 29)(14 38)(17 25)(18 34)(22 30)(23 39)(27 35)(32 40)(41 116)(42 85)(43 94)(44 103)(45 112)(46 81)(47 90)(48 99)(49 108)(50 117)(51 86)(52 95)(53 104)(54 113)(55 82)(56 91)(57 100)(58 109)(59 118)(60 87)(61 96)(62 105)(63 114)(64 83)(65 92)(66 101)(67 110)(68 119)(69 88)(70 97)(71 106)(72 115)(73 84)(74 93)(75 102)(76 111)(77 120)(78 89)(79 98)(80 107)(121 137)(122 146)(123 155)(125 133)(126 142)(127 151)(128 160)(130 138)(131 147)(132 156)(135 143)(136 152)(140 148)(141 157)(145 153)(150 158)

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

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

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

50 conjugacy classes

class 1 2A2B2C2D2E2F4A4B4C4D4E4F4G5A5B8A8B8C8D8E8F8G8H10A···10F10G10H10I10J20A20B20C20D20E20F20G20H40A···40H
order12222224444444558888888810···1010101010202020202020202040···40
size11118101022810104040222222101010102···28888444488884···4

50 irreducible representations

dim11111111222222222224444
type+++++++++++++++-+
imageC1C2C2C2C2C2C2C2D4D4D4D5C4○D4SD16D10D10D10C4○D8C5⋊D4D42D5D4×D5D5×SD16SD163D5
kernelC4014D4C406C4D4⋊Dic5Q8⋊Dic5D5×C2×C8C202D4D103Q8C10×SD16C40C2×Dic5C22×D5C2×SD16C20D10C2×C8C2×D4C2×Q8C10C8C4C22C2C2
# reps11111111211224222482244

Matrix representation of C4014D4 in GL4(𝔽41) generated by

32000
02700
00734
00740
,
121100
392900
0033
002438
,
11200
04000
00347
00407
G:=sub<GL(4,GF(41))| [3,0,0,0,20,27,0,0,0,0,7,7,0,0,34,40],[12,39,0,0,11,29,0,0,0,0,3,24,0,0,3,38],[1,0,0,0,12,40,0,0,0,0,34,40,0,0,7,7] >;

C4014D4 in GAP, Magma, Sage, TeX

C_{40}\rtimes_{14}D_4
% in TeX

G:=Group("C40:14D4");
// GroupNames label

G:=SmallGroup(320,798);
// by ID

G=gap.SmallGroup(320,798);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,120,254,555,438,102,12550]);
// Polycyclic

G:=Group<a,b,c|a^40=b^4=c^2=1,b*a*b^-1=a^19,c*a*c=a^9,c*b*c=b^-1>;
// generators/relations

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