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G = D10.C42  order 320 = 26·5

1st non-split extension by D10 of C42 acting via C42/C22=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C20.5C42, D10.1C42, D5⋊C8⋊3C4, C4.F5⋊2C4, C4⋊C4.11F5, C4.10(C4×F5), C10.8(C8○D4), C4⋊Dic5.11C4, C10.9(C2×C42), C2.3(D4.F5), C2.1(Q8.F5), D10⋊C4.7C4, C5⋊2(C8○2M4(2)), C10.C42⋊13C2, C22.36(C22×F5), Dic5.29(C22×C4), (C4×Dic5).249C22, (C2×Dic5).328C23, (C4×C5⋊C8)⋊14C2, C5⋊C8.4(C2×C4), (C5×C4⋊C4).5C4, C2.11(C2×C4×F5), (C2×D5⋊C8).3C2, (C2×C4).59(C2×F5), (C2×C4.F5).3C2, (C2×C20).92(C2×C4), (C4×D5).16(C2×C4), (C2×C5⋊C8).25C22, C4⋊C4⋊7D5.17C2, (C2×C4×D5).190C22, (C2×C10).39(C22×C4), (C2×Dic5).54(C2×C4), (C22×D5).46(C2×C4), SmallGroup(320,1039)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C10 — D10.C42
Chief series
C1 — C5 — C10 — Dic5 — C2×Dic5 — C2×C5⋊C8 — C2×D5⋊C8 — D10.C42
Lower central
C5 — C10 — D10.C42
Upper central
C1 — C22 — C4⋊C4

Generators and relations for D10.C42
 G = < a,b,c,d | a10=b2=d4=1, c4=a5, bab=a-1, cac-1=a3, ad=da, cbc-1=a2b, dbd-1=a5b, cd=dc >

Subgroups: 378 in 130 conjugacy classes, 68 normal (26 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, C2×C4, C23, D5, C10, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, M4(2), C22×C4, Dic5, Dic5, C20, C20, D10, D10, C2×C10, C4×C8, C8⋊C4, C42⋊C2, C22×C8, C2×M4(2), C5⋊C8, C4×D5, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C22×D5, C8○2M4(2), C4×Dic5, C4⋊Dic5, D10⋊C4, C5×C4⋊C4, D5⋊C8, C4.F5, C2×C5⋊C8, C2×C5⋊C8, C2×C4×D5, C4×C5⋊C8, C10.C42, C4⋊C4⋊7D5, C2×D5⋊C8, C2×C4.F5, D10.C42
Quotients: C1, C2, C4, C22, C2×C4, C23, C42, C22×C4, F5, C2×C42, C8○D4, C2×F5, C8○2M4(2), C4×F5, C22×F5, C2×C4×F5, D4.F5, Q8.F5, D10.C42

Smallest permutation representation of D10.C42
►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 12)(2 11)(3 20)(4 19)(5 18)(6 17)(7 16)(8 15)(9 14)(10 13)(21 32)(22 31)(23 40)(24 39)(25 38)(26 37)(27 36)(28 35)(29 34)(30 33)(41 59)(42 58)(43 57)(44 56)(45 55)(46 54)(47 53)(48 52)(49 51)(50 60)(61 77)(62 76)(63 75)(64 74)(65 73)(66 72)(67 71)(68 80)(69 79)(70 78)(81 96)(82 95)(83 94)(84 93)(85 92)(86 91)(87 100)(88 99)(89 98)(90 97)(101 120)(102 119)(103 118)(104 117)(105 116)(106 115)(107 114)(108 113)(109 112)(110 111)(121 131)(122 140)(123 139)(124 138)(125 137)(126 136)(127 135)(128 134)(129 133)(130 132)(141 155)(142 154)(143 153)(144 152)(145 151)(146 160)(147 159)(148 158)(149 157)(150 156)

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

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

50 conjugacy classes

class 1 2A2B2C2D2E4A···4F4G4H4I4J4K4L4M4N 5 8A···8H8I···8T10A10B10C20A···20F
order1222224···44444444458···88···810101020···20
size111110102···255551010101045···510···104448···8

50 irreducible representations

dim11111111111244488
type++++++++-+
imageC1C2C2C2C2C2C4C4C4C4C4C8○D4F5C2×F5C4×F5D4.F5Q8.F5
kernelD10.C42C4×C5⋊C8C10.C42C4⋊C4⋊7D5C2×D5⋊C8C2×C4.F5C4⋊Dic5D10⋊C4C5×C4⋊C4D5⋊C8C4.F5C10C4⋊C4C2×C4C4C2C2
# reps12211124288813411

Matrix representation of D10.C42 ►in GL6(𝔽41)

4000000
0400000
000001
0040404040
001000
000100
,
100000
9400000
0004000
0040000
001111
0000040
,
1400000
0140000
00025625
002201616
001616022
00256250
,
9390000
0320000
009000
000900
000090
000009

G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,0,40,1,0,0,0,0,40,0,1,0,0,0,40,0,0,0,0,1,40,0,0],[1,9,0,0,0,0,0,40,0,0,0,0,0,0,0,40,1,0,0,0,40,0,1,0,0,0,0,0,1,0,0,0,0,0,1,40],[14,0,0,0,0,0,0,14,0,0,0,0,0,0,0,22,16,25,0,0,25,0,16,6,0,0,6,16,0,25,0,0,25,16,22,0],[9,0,0,0,0,0,39,32,0,0,0,0,0,0,9,0,0,0,0,0,0,9,0,0,0,0,0,0,9,0,0,0,0,0,0,9] >;
 

D10.C42 in GAP, Magma, Sage, TeX 

D_{10}.C_4^2
 
% in TeX
 
G:=Group("D10.C4^2");
 
// GroupNames label
 
G:=SmallGroup(320,1039);
 
// by ID
 
G=gap.SmallGroup(320,1039);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,232,387,100,136,6278,1595]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^10=b^2=d^4=1,c^4=a^5,b*a*b=a^-1,c*a*c^-1=a^3,a*d=d*a,c*b*c^-1=a^2*b,d*b*d^-1=a^5*b,c*d=d*c>;
 
// generators/relations
 

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