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G = C20⋊SD16order 320 = 26·5

1st semidirect product of C20 and SD16 acting via SD16/C4=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C201SD16, D20.18D4, C42.32D10, C4⋊C88D5, C43(C40⋊C2), C4.129(D4×D5), C202Q811C2, (C4×D20).10C2, C20.338(C2×D4), (C2×C8).128D10, (C2×C4).131D20, (C2×C20).120D4, C52(D4.D4), (C4×C20).67C22, C10.10(C2×SD16), C20.327(C4○D4), C20.44D412C2, C2.10(C4⋊D20), C10.37(C4⋊D4), (C2×C40).138C22, (C2×C20).751C23, C4.43(Q82D5), C22.114(C2×D20), C2.17(C8.D10), (C2×D20).200C22, C10.14(C8.C22), C4⋊Dic5.272C22, (C2×Dic10).16C22, (C5×C4⋊C8)⋊10C2, (C2×C40⋊C2).5C2, C2.13(C2×C40⋊C2), (C2×C10).134(C2×D4), (C2×C4).696(C22×D5), SmallGroup(320,468)

Series: Derived Chief Lower central Upper central

C1C2×C20 — C20⋊SD16
C1C5C10C20C2×C20C2×D20C4×D20 — C20⋊SD16
C5C10C2×C20 — C20⋊SD16
C1C22C42C4⋊C8

Generators and relations for C20⋊SD16
 G = < a,b,c | a20=b8=c2=1, bab-1=a11, cac=a9, cbc=b3 >

Subgroups: 566 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 [×3], C20 [×2], C20 [×2], C20, D10 [×4], C2×C10, Q8⋊C4 [×2], C4⋊C8, C4×D4, C4⋊Q8, C2×SD16 [×2], C40 [×2], Dic10 [×4], C4×D5 [×2], D20 [×2], D20, C2×Dic5 [×3], C2×C20 [×3], C22×D5, D4.D4, C40⋊C2 [×4], C4⋊Dic5, C4⋊Dic5 [×2], D10⋊C4, C4×C20, C2×C40 [×2], C2×Dic10 [×2], C2×C4×D5, C2×D20, C20.44D4 [×2], C5×C4⋊C8, C202Q8, C4×D20, C2×C40⋊C2 [×2], C20⋊SD16
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, D20 [×2], C22×D5, D4.D4, C40⋊C2 [×2], C2×D20, D4×D5, Q82D5, C4⋊D20, C2×C40⋊C2, C8.D10, C20⋊SD16

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

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

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

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

59 conjugacy classes

class 1 2A2B2C2D2E4A4B4C4D4E4F4G4H4I5A5B8A8B8C8D10A···10F20A···20H20I···20P40A···40P
order12222244444444455888810···1020···2020···2040···40
size1111202022224202040402244442···22···24···44···4

59 irreducible representations

dim1111112222222224444
type++++++++++++-++-
imageC1C2C2C2C2C2D4D4D5SD16C4○D4D10D10D20C40⋊C2C8.C22D4×D5Q82D5C8.D10
kernelC20⋊SD16C20.44D4C5×C4⋊C8C202Q8C4×D20C2×C40⋊C2D20C2×C20C4⋊C8C20C20C42C2×C8C2×C4C4C10C4C4C2
# reps12111222242248161224

Matrix representation of C20⋊SD16 in GL6(𝔽41)

100000
010000
00344000
001000
00004024
0000171
,
36190000
34350000
0040000
0004000
000001
000010
,
100000
28400000
001000
00344000
000010
000001

G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,34,1,0,0,0,0,40,0,0,0,0,0,0,0,40,17,0,0,0,0,24,1],[36,34,0,0,0,0,19,35,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[1,28,0,0,0,0,0,40,0,0,0,0,0,0,1,34,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;

C20⋊SD16 in GAP, Magma, Sage, TeX

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

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

G:=SmallGroup(320,468);
// by ID

G=gap.SmallGroup(320,468);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,120,254,219,58,1123,136,12550]);
// Polycyclic

G:=Group<a,b,c|a^20=b^8=c^2=1,b*a*b^-1=a^11,c*a*c=a^9,c*b*c=b^3>;
// generators/relations

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