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G = C204SD16order 320 = 26·5

4th semidirect product of C20 and SD16 acting via SD16/C4=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C204SD16, C42.219D10, C52C812D4, C52(C85D4), C4.16(D4×D5), C41(D4.D5), C41D4.6D5, C20.34(C2×D4), C202Q820C2, (C2×D4).59D10, (C2×C20).150D4, C10.59(C2×SD16), C2.13(C20⋊D4), C10.22(C41D4), (C2×C20).394C23, (C4×C20).124C22, (D4×C10).75C22, (C2×Dic10).116C22, (C4×C52C8)⋊16C2, (C2×D4.D5)⋊15C2, (C5×C41D4).5C2, C2.13(C2×D4.D5), (C2×C10).525(C2×D4), (C2×C4).132(C5⋊D4), (C2×C4).492(C22×D5), C22.198(C2×C5⋊D4), (C2×C52C8).266C22, SmallGroup(320,703)

Series: Derived Chief Lower central Upper central

C1C2×C20 — C204SD16
C1C5C10C20C2×C20C2×Dic10C202Q8 — C204SD16
C5C10C2×C20 — C204SD16
C1C22C42C41D4

Generators and relations for C204SD16
 G = < a,b,c | a20=b8=c2=1, bab-1=a9, cac=a11, cbc=b3 >

Subgroups: 510 in 142 conjugacy classes, 51 normal (15 characteristic)
C1, C2, C2 [×2], C2 [×2], C4 [×6], C4 [×2], C22, C22 [×6], C5, C8 [×4], C2×C4, C2×C4 [×2], C2×C4 [×2], D4 [×8], Q8 [×4], C23 [×2], C10, C10 [×2], C10 [×2], C42, C4⋊C4 [×2], C2×C8 [×2], SD16 [×8], C2×D4 [×2], C2×D4 [×2], C2×Q8 [×2], Dic5 [×2], C20 [×6], C2×C10, C2×C10 [×6], C4×C8, C41D4, C4⋊Q8, C2×SD16 [×4], C52C8 [×4], Dic10 [×4], C2×Dic5 [×2], C2×C20, C2×C20 [×2], C5×D4 [×8], C22×C10 [×2], C85D4, C2×C52C8 [×2], C4⋊Dic5 [×2], D4.D5 [×8], C4×C20, C2×Dic10 [×2], D4×C10 [×2], D4×C10 [×2], C4×C52C8, C202Q8, C2×D4.D5 [×4], C5×C41D4, C204SD16
Quotients: C1, C2 [×7], C22 [×7], D4 [×6], C23, D5, SD16 [×4], C2×D4 [×3], D10 [×3], C41D4, C2×SD16 [×2], C5⋊D4 [×2], C22×D5, C85D4, D4.D5 [×4], D4×D5 [×2], C2×C5⋊D4, C2×D4.D5 [×2], C20⋊D4, C204SD16

Smallest permutation representation of C204SD16
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 93 116 140 36 59 150 77)(2 82 117 129 37 48 151 66)(3 91 118 138 38 57 152 75)(4 100 119 127 39 46 153 64)(5 89 120 136 40 55 154 73)(6 98 101 125 21 44 155 62)(7 87 102 134 22 53 156 71)(8 96 103 123 23 42 157 80)(9 85 104 132 24 51 158 69)(10 94 105 121 25 60 159 78)(11 83 106 130 26 49 160 67)(12 92 107 139 27 58 141 76)(13 81 108 128 28 47 142 65)(14 90 109 137 29 56 143 74)(15 99 110 126 30 45 144 63)(16 88 111 135 31 54 145 72)(17 97 112 124 32 43 146 61)(18 86 113 133 33 52 147 70)(19 95 114 122 34 41 148 79)(20 84 115 131 35 50 149 68)
(1 21)(2 32)(3 23)(4 34)(5 25)(6 36)(7 27)(8 38)(9 29)(10 40)(11 31)(12 22)(13 33)(14 24)(15 35)(16 26)(17 37)(18 28)(19 39)(20 30)(41 127)(42 138)(43 129)(44 140)(45 131)(46 122)(47 133)(48 124)(49 135)(50 126)(51 137)(52 128)(53 139)(54 130)(55 121)(56 132)(57 123)(58 134)(59 125)(60 136)(61 82)(62 93)(63 84)(64 95)(65 86)(66 97)(67 88)(68 99)(69 90)(70 81)(71 92)(72 83)(73 94)(74 85)(75 96)(76 87)(77 98)(78 89)(79 100)(80 91)(101 116)(102 107)(103 118)(104 109)(105 120)(106 111)(108 113)(110 115)(112 117)(114 119)(141 156)(142 147)(143 158)(144 149)(145 160)(146 151)(148 153)(150 155)(152 157)(154 159)

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

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

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

50 conjugacy classes

class 1 2A2B2C2D2E4A···4F4G4H5A5B8A···8H10A···10F10G···10N20A···20L
order1222224···444558···810···1010···1020···20
size1111882···240402210···102···28···84···4

50 irreducible representations

dim11111222222244
type++++++++++-+
imageC1C2C2C2C2D4D4D5SD16D10D10C5⋊D4D4.D5D4×D5
kernelC204SD16C4×C52C8C202Q8C2×D4.D5C5×C41D4C52C8C2×C20C41D4C20C42C2×D4C2×C4C4C4
# reps11141422824884

Matrix representation of C204SD16 in GL6(𝔽41)

37300000
0100000
0040000
0004000
0000402
0000401
,
40160000
510000
0003200
0093000
0000400
0000040
,
1250000
0400000
001000
00244000
0000139
0000040

G:=sub<GL(6,GF(41))| [37,0,0,0,0,0,30,10,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,40,0,0,0,0,2,1],[40,5,0,0,0,0,16,1,0,0,0,0,0,0,0,9,0,0,0,0,32,30,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[1,0,0,0,0,0,25,40,0,0,0,0,0,0,1,24,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,39,40] >;

C204SD16 in GAP, Magma, Sage, TeX

C_{20}\rtimes_4{\rm SD}_{16}
% in TeX

G:=Group("C20:4SD16");
// GroupNames label

G:=SmallGroup(320,703);
// by ID

G=gap.SmallGroup(320,703);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,224,253,120,254,219,1123,297,136,12550]);
// Polycyclic

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

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