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G = C205SD16order 320 = 26·5

5th semidirect product of C20 and SD16 acting via SD16/C4=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C205SD16, D20.24D4, C42.78D10, C4⋊Q81D5, C42(Q8⋊D5), C4.56(D4×D5), C203C833C2, C20.36(C2×D4), (C4×D20).17C2, (C2×C20).154D4, C54(D4.D4), (C2×Q8).44D10, C20.80(C4○D4), C4.5(D42D5), Q8⋊Dic523C2, C10.75(C2×SD16), C2.14(C202D4), (C2×C20).401C23, (C4×C20).130C22, (Q8×C10).62C22, C10.105(C4⋊D4), (C2×D20).255C22, C10.95(C8.C22), C4⋊Dic5.346C22, C2.16(C20.C23), (C5×C4⋊Q8)⋊1C2, (C2×Q8⋊D5).6C2, C2.13(C2×Q8⋊D5), (C2×C10).532(C2×D4), (C2×C4).187(C5⋊D4), (C2×C4).498(C22×D5), C22.204(C2×C5⋊D4), (C2×C52C8).135C22, SmallGroup(320,710)

Series: Derived Chief Lower central Upper central

C1C2×C20 — C205SD16
C1C5C10C20C2×C20C2×D20C4×D20 — C205SD16
C5C10C2×C20 — C205SD16
C1C22C42C4⋊Q8

Generators and relations for C205SD16
 G = < a,b,c | a20=b8=c2=1, bab-1=a-1, cac=a9, cbc=b3 >

Subgroups: 486 in 120 conjugacy classes, 45 normal (29 characteristic)
C1, C2 [×3], C2 [×2], C4 [×2], C4 [×2], C4 [×4], C22, C22 [×4], C5, C8 [×2], C2×C4 [×3], C2×C4 [×5], D4 [×3], Q8 [×4], C23, D5 [×2], C10 [×3], C42, C22⋊C4, C4⋊C4 [×3], C2×C8 [×2], SD16 [×4], C22×C4, C2×D4, C2×Q8 [×2], Dic5, C20 [×2], C20 [×2], C20 [×3], D10 [×4], C2×C10, Q8⋊C4 [×2], C4⋊C8, C4×D4, C4⋊Q8, C2×SD16 [×2], C52C8 [×2], C4×D5 [×2], D20 [×2], D20, C2×Dic5, C2×C20 [×3], C2×C20 [×2], C5×Q8 [×4], C22×D5, D4.D4, C2×C52C8 [×2], C4⋊Dic5, D10⋊C4, Q8⋊D5 [×4], C4×C20, C5×C4⋊C4 [×2], C2×C4×D5, C2×D20, Q8×C10 [×2], C203C8, Q8⋊Dic5 [×2], C4×D20, C2×Q8⋊D5 [×2], C5×C4⋊Q8, C205SD16
Quotients: C1, C2 [×7], C22 [×7], D4 [×4], C23, D5, SD16 [×2], C2×D4 [×2], C4○D4, D10 [×3], C4⋊D4, C2×SD16, C8.C22, C5⋊D4 [×2], C22×D5, D4.D4, Q8⋊D5 [×2], D4×D5, D42D5, C2×C5⋊D4, C202D4, C2×Q8⋊D5, C20.C23, C205SD16

Smallest permutation representation of C205SD16
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 83 52 34 153 102 137 73)(2 82 53 33 154 101 138 72)(3 81 54 32 155 120 139 71)(4 100 55 31 156 119 140 70)(5 99 56 30 157 118 121 69)(6 98 57 29 158 117 122 68)(7 97 58 28 159 116 123 67)(8 96 59 27 160 115 124 66)(9 95 60 26 141 114 125 65)(10 94 41 25 142 113 126 64)(11 93 42 24 143 112 127 63)(12 92 43 23 144 111 128 62)(13 91 44 22 145 110 129 61)(14 90 45 21 146 109 130 80)(15 89 46 40 147 108 131 79)(16 88 47 39 148 107 132 78)(17 87 48 38 149 106 133 77)(18 86 49 37 150 105 134 76)(19 85 50 36 151 104 135 75)(20 84 51 35 152 103 136 74)
(2 10)(3 19)(4 8)(5 17)(7 15)(9 13)(12 20)(14 18)(21 86)(22 95)(23 84)(24 93)(25 82)(26 91)(27 100)(28 89)(29 98)(30 87)(31 96)(32 85)(33 94)(34 83)(35 92)(36 81)(37 90)(38 99)(39 88)(40 97)(41 138)(42 127)(43 136)(44 125)(45 134)(46 123)(47 132)(48 121)(49 130)(50 139)(51 128)(52 137)(53 126)(54 135)(55 124)(56 133)(57 122)(58 131)(59 140)(60 129)(61 114)(62 103)(63 112)(64 101)(65 110)(66 119)(67 108)(68 117)(69 106)(70 115)(71 104)(72 113)(73 102)(74 111)(75 120)(76 109)(77 118)(78 107)(79 116)(80 105)(141 145)(142 154)(144 152)(146 150)(147 159)(149 157)(151 155)(156 160)

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

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

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

47 conjugacy classes

class 1 2A2B2C2D2E4A4B4C4D4E4F4G4H4I5A5B8A8B8C8D10A···10F20A···20L20M···20T
order12222244444444455888810···1020···2020···20
size111120202222488202022202020202···24···48···8

47 irreducible representations

dim1111112222222244444
type+++++++++++-++-
imageC1C2C2C2C2C2D4D4D5SD16C4○D4D10D10C5⋊D4C8.C22Q8⋊D5D4×D5D42D5C20.C23
kernelC205SD16C203C8Q8⋊Dic5C4×D20C2×Q8⋊D5C5×C4⋊Q8D20C2×C20C4⋊Q8C20C20C42C2×Q8C2×C4C10C4C4C4C2
# reps1121212224224814224

Matrix representation of C205SD16 in GL6(𝔽41)

4000000
0400000
0004000
0013400
00001618
00002925
,
15150000
26150000
003300
00243800
00002937
0000512
,
100000
0400000
0034700
0040700
000010
000001

G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,0,1,0,0,0,0,40,34,0,0,0,0,0,0,16,29,0,0,0,0,18,25],[15,26,0,0,0,0,15,15,0,0,0,0,0,0,3,24,0,0,0,0,3,38,0,0,0,0,0,0,29,5,0,0,0,0,37,12],[1,0,0,0,0,0,0,40,0,0,0,0,0,0,34,40,0,0,0,0,7,7,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;

C205SD16 in GAP, Magma, Sage, TeX

C_{20}\rtimes_5{\rm SD}_{16}
% in TeX

G:=Group("C20:5SD16");
// GroupNames label

G:=SmallGroup(320,710);
// by ID

G=gap.SmallGroup(320,710);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,120,254,219,184,1123,297,136,12550]);
// Polycyclic

G:=Group<a,b,c|a^20=b^8=c^2=1,b*a*b^-1=a^-1,c*a*c=a^9,c*b*c=b^3>;
// generators/relations

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