Copied to
clipboard

G = D10.7Q16order 320 = 26·5

1st non-split extension by D10 of Q16 acting via Q16/Q8=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D10.7Q16, Q8⋊C45D5, C405C410C2, (C2×C8).17D10, C2.10(D5×Q16), C4⋊C4.152D10, C10.D89C2, Q8⋊Dic58C2, (C2×Q8).17D10, C10.18(C2×Q16), D101C8.3C2, C4.56(C4○D20), (C2×C40).17C22, D103Q8.3C2, (C2×Dic5).40D4, C22.199(D4×D5), C20.162(C4○D4), C4.87(D42D5), C2.16(D40⋊C2), C10.62(C8⋊C22), (C2×C20).249C23, (C22×D5).116D4, C52(C23.48D4), C4⋊Dic5.95C22, (Q8×C10).32C22, C2.17(D10.12D4), C10.25(C22.D4), (D5×C4⋊C4).3C2, (C5×Q8⋊C4)⋊5C2, (C2×C4×D5).26C22, (C2×C10).262(C2×D4), (C5×C4⋊C4).50C22, (C2×C52C8).40C22, (C2×C4).356(C22×D5), SmallGroup(320,436)

Series: Derived Chief Lower central Upper central

C1C2×C20 — D10.7Q16
C1C5C10C2×C10C2×C20C2×C4×D5D5×C4⋊C4 — D10.7Q16
C5C10C2×C20 — D10.7Q16
C1C22C2×C4Q8⋊C4

Generators and relations for D10.7Q16
 G = < a,b,c,d | a10=b2=c8=1, d2=c4, bab=a-1, ac=ca, ad=da, cbc-1=dbd-1=a5b, dcd-1=a5c-1 >

Subgroups: 414 in 104 conjugacy classes, 39 normal (37 characteristic)
C1, C2 [×3], C2 [×2], C4 [×2], C4 [×5], C22, C22 [×4], C5, C8 [×2], C2×C4, C2×C4 [×9], Q8 [×2], C23, D5 [×2], C10 [×3], C22⋊C4, C4⋊C4, C4⋊C4 [×4], C2×C8, C2×C8, C22×C4 [×2], C2×Q8, Dic5 [×3], C20 [×2], C20 [×2], D10 [×2], D10 [×2], C2×C10, C22⋊C8, Q8⋊C4, Q8⋊C4, C2.D8 [×2], C2×C4⋊C4, C22⋊Q8, C52C8, C40, C4×D5 [×4], C2×Dic5, C2×Dic5 [×2], C2×C20, C2×C20 [×2], C5×Q8 [×2], C22×D5, C23.48D4, C2×C52C8, C10.D4 [×2], C4⋊Dic5 [×2], D10⋊C4, C5×C4⋊C4, C2×C40, C2×C4×D5, C2×C4×D5, Q8×C10, C10.D8, C405C4, D101C8, Q8⋊Dic5, C5×Q8⋊C4, D5×C4⋊C4, D103Q8, D10.7Q16
Quotients: C1, C2 [×7], C22 [×7], D4 [×2], C23, D5, Q16 [×2], C2×D4, C4○D4 [×2], D10 [×3], C22.D4, C2×Q16, C8⋊C22, C22×D5, C23.48D4, C4○D20, D4×D5, D42D5, D10.12D4, D40⋊C2, D5×Q16, D10.7Q16

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

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

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

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

47 conjugacy classes

class 1 2A2B2C2D2E4A4B4C4D4E4F4G4H4I5A5B8A8B8C8D10A···10F20A20B20C20D20E···20L40A···40H
order12222244444444455888810···102020202020···2040···40
size111110102244820202040224420202···244448···84···4

47 irreducible representations

dim1111111122222222244444
type+++++++++++-++++-++-
imageC1C2C2C2C2C2C2C2D4D4D5C4○D4Q16D10D10D10C4○D20C8⋊C22D42D5D4×D5D40⋊C2D5×Q16
kernelD10.7Q16C10.D8C405C4D101C8Q8⋊Dic5C5×Q8⋊C4D5×C4⋊C4D103Q8C2×Dic5C22×D5Q8⋊C4C20D10C4⋊C4C2×C8C2×Q8C4C10C4C22C2C2
# reps1111111111244222812244

Matrix representation of D10.7Q16 in GL6(𝔽41)

4000000
0400000
00403500
0063500
000010
000001
,
4000000
1310000
0040000
006100
0000400
0000040
,
1250000
37290000
001000
000100
00001229
00001212
,
15370000
15260000
0040000
0004000
00002138
00003820

G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,6,0,0,0,0,35,35,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[40,13,0,0,0,0,0,1,0,0,0,0,0,0,40,6,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[12,37,0,0,0,0,5,29,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,12,0,0,0,0,29,12],[15,15,0,0,0,0,37,26,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,21,38,0,0,0,0,38,20] >;

D10.7Q16 in GAP, Magma, Sage, TeX

D_{10}._7Q_{16}
% in TeX

G:=Group("D10.7Q16");
// GroupNames label

G:=SmallGroup(320,436);
// by ID

G=gap.SmallGroup(320,436);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,64,926,219,184,851,438,102,12550]);
// Polycyclic

G:=Group<a,b,c,d|a^10=b^2=c^8=1,d^2=c^4,b*a*b=a^-1,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d^-1=a^5*b,d*c*d^-1=a^5*c^-1>;
// generators/relations

׿
×
𝔽