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G = D103(C4⋊C4)  order 320 = 26·5

2nd semidirect product of D10 and C4⋊C4 acting via C4⋊C4/C2×C4=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C2.6(C4×D20), D103(C4⋊C4), C10.33(C4×D4), D10⋊C45C4, C10.3C22≀C2, (C2×C4).112D20, (C2×C20).234D4, C22.62(D4×D5), C22.17(Q8×D5), C2.C428D5, (C22×C4).17D10, C22.25(C2×D20), (C22×D5).13Q8, C2.2(C22⋊D20), C52(C23.8Q8), C2.4(D10⋊Q8), C2.2(D102Q8), (C2×Dic5).187D4, (C22×D5).104D4, C10.25(C22⋊Q8), C10.10C422C2, C2.8(Dic54D4), C22.36(C4○D20), (C23×D5).97C22, C23.258(C22×D5), C2.4(D10.12D4), C22.37(D42D5), (C22×C20).333C22, (C22×C10).293C23, C10.9(C22.D4), (C22×Dic5).16C22, (C2×C4)⋊3(C4×D5), C2.8(D5×C4⋊C4), (C2×C20)⋊18(C2×C4), C10.29(C2×C4⋊C4), (C2×C4⋊Dic5)⋊1C2, C22.91(C2×C4×D5), (C2×Dic5)⋊5(C2×C4), (C2×C10).68(C2×Q8), (D5×C22×C4).15C2, (C2×C10).202(C2×D4), (C2×D10⋊C4).5C2, (C2×C10.D4)⋊31C2, (C22×D5).67(C2×C4), (C2×C10).132(C4○D4), (C2×C10).153(C22×C4), (C5×C2.C42)⋊15C2, SmallGroup(320,295)

Series: Derived Chief Lower central Upper central

C1C2×C10 — D103(C4⋊C4)
C1C5C10C2×C10C22×C10C23×D5C2×D10⋊C4 — D103(C4⋊C4)
C5C2×C10 — D103(C4⋊C4)
C1C23C2.C42

Generators and relations for D103(C4⋊C4)
 G = < a,b,c,d | a10=b2=c4=d4=1, bab=cac-1=a-1, ad=da, cbc-1=a3b, bd=db, dcd-1=c-1 >

Subgroups: 910 in 234 conjugacy classes, 77 normal (51 characteristic)
C1, C2, C2, C4, C22, C22, C5, C2×C4, C2×C4, C23, C23, D5, C10, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C24, Dic5, C20, D10, D10, C2×C10, C2.C42, C2.C42, C2×C22⋊C4, C2×C4⋊C4, C23×C4, C4×D5, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C22×D5, C22×D5, C22×C10, C23.8Q8, C10.D4, C4⋊Dic5, D10⋊C4, D10⋊C4, C2×C4×D5, C22×Dic5, C22×C20, C23×D5, C10.10C42, C5×C2.C42, C2×C10.D4, C2×C4⋊Dic5, C2×D10⋊C4, D5×C22×C4, D103(C4⋊C4)
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C23, D5, C4⋊C4, C22×C4, C2×D4, C2×Q8, C4○D4, D10, C2×C4⋊C4, C4×D4, C22≀C2, C22⋊Q8, C22.D4, C4×D5, D20, C22×D5, C23.8Q8, C2×C4×D5, C2×D20, C4○D20, D4×D5, D42D5, Q8×D5, C4×D20, Dic54D4, C22⋊D20, D10.12D4, D5×C4⋊C4, D10⋊Q8, D102Q8, D103(C4⋊C4)

Smallest permutation representation of D103(C4⋊C4)
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 39)(2 38)(3 37)(4 36)(5 35)(6 34)(7 33)(8 32)(9 31)(10 40)(11 149)(12 148)(13 147)(14 146)(15 145)(16 144)(17 143)(18 142)(19 141)(20 150)(21 48)(22 47)(23 46)(24 45)(25 44)(26 43)(27 42)(28 41)(29 50)(30 49)(51 78)(52 77)(53 76)(54 75)(55 74)(56 73)(57 72)(58 71)(59 80)(60 79)(61 88)(62 87)(63 86)(64 85)(65 84)(66 83)(67 82)(68 81)(69 90)(70 89)(91 113)(92 112)(93 111)(94 120)(95 119)(96 118)(97 117)(98 116)(99 115)(100 114)(101 123)(102 122)(103 121)(104 130)(105 129)(106 128)(107 127)(108 126)(109 125)(110 124)(131 153)(132 152)(133 151)(134 160)(135 159)(136 158)(137 157)(138 156)(139 155)(140 154)
(1 105 30 95)(2 104 21 94)(3 103 22 93)(4 102 23 92)(5 101 24 91)(6 110 25 100)(7 109 26 99)(8 108 27 98)(9 107 28 97)(10 106 29 96)(11 90 155 80)(12 89 156 79)(13 88 157 78)(14 87 158 77)(15 86 159 76)(16 85 160 75)(17 84 151 74)(18 83 152 73)(19 82 153 72)(20 81 154 71)(31 124 41 114)(32 123 42 113)(33 122 43 112)(34 121 44 111)(35 130 45 120)(36 129 46 119)(37 128 47 118)(38 127 48 117)(39 126 49 116)(40 125 50 115)(51 144 61 134)(52 143 62 133)(53 142 63 132)(54 141 64 131)(55 150 65 140)(56 149 66 139)(57 148 67 138)(58 147 68 137)(59 146 69 136)(60 145 70 135)
(1 80 40 60)(2 71 31 51)(3 72 32 52)(4 73 33 53)(5 74 34 54)(6 75 35 55)(7 76 36 56)(8 77 37 57)(9 78 38 58)(10 79 39 59)(11 115 145 95)(12 116 146 96)(13 117 147 97)(14 118 148 98)(15 119 149 99)(16 120 150 100)(17 111 141 91)(18 112 142 92)(19 113 143 93)(20 114 144 94)(21 81 41 61)(22 82 42 62)(23 83 43 63)(24 84 44 64)(25 85 45 65)(26 86 46 66)(27 87 47 67)(28 88 48 68)(29 89 49 69)(30 90 50 70)(101 151 121 131)(102 152 122 132)(103 153 123 133)(104 154 124 134)(105 155 125 135)(106 156 126 136)(107 157 127 137)(108 158 128 138)(109 159 129 139)(110 160 130 140)

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

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

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

68 conjugacy classes

class 1 2A···2G2H2I2J2K4A4B4C4D4E4F4G4H4I4J4K4L4M4N4O4P5A5B10A···10N20A···20X
order12···2222244444444444444445510···1020···20
size11···110101010222244441010101020202020222···24···4

68 irreducible representations

dim111111112222222222444
type++++++++++-++++--
imageC1C2C2C2C2C2C2C4D4D4D4Q8D5C4○D4D10C4×D5D20C4○D20D4×D5D42D5Q8×D5
kernelD103(C4⋊C4)C10.10C42C5×C2.C42C2×C10.D4C2×C4⋊Dic5C2×D10⋊C4D5×C22×C4D10⋊C4C2×Dic5C2×C20C22×D5C22×D5C2.C42C2×C10C22×C4C2×C4C2×C4C22C22C22C22
# reps111112182222246888422

Matrix representation of D103(C4⋊C4) in GL6(𝔽41)

34340000
600000
001000
000100
0000400
0000040
,
34330000
670000
0040000
0004000
000010
00002240
,
770000
40340000
00403900
001100
00001436
00002327
,
3200000
0320000
0032000
009900
0000320
000079

G:=sub<GL(6,GF(41))| [34,6,0,0,0,0,34,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[34,6,0,0,0,0,33,7,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,22,0,0,0,0,0,40],[7,40,0,0,0,0,7,34,0,0,0,0,0,0,40,1,0,0,0,0,39,1,0,0,0,0,0,0,14,23,0,0,0,0,36,27],[32,0,0,0,0,0,0,32,0,0,0,0,0,0,32,9,0,0,0,0,0,9,0,0,0,0,0,0,32,7,0,0,0,0,0,9] >;

D103(C4⋊C4) in GAP, Magma, Sage, TeX

D_{10}\rtimes_3(C_4\rtimes C_4)
% in TeX

G:=Group("D10:3(C4:C4)");
// GroupNames label

G:=SmallGroup(320,295);
// by ID

G=gap.SmallGroup(320,295);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,253,344,387,58,12550]);
// Polycyclic

G:=Group<a,b,c,d|a^10=b^2=c^4=d^4=1,b*a*b=c*a*c^-1=a^-1,a*d=d*a,c*b*c^-1=a^3*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations

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