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## G = D4.3Dic10order 320 = 26·5

### The non-split extension by D4 of Dic10 acting via Dic10/C20=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C20 — D4.3Dic10
 Chief series C1 — C5 — C10 — C20 — C2×C20 — C4⋊Dic5 — C20.6Q8 — D4.3Dic10
 Lower central C5 — C10 — C2×C20 — D4.3Dic10
 Upper central C1 — C22 — C42 — C4×D4

Generators and relations for D4.3Dic10
G = < a,b,c,d | a4=b2=c20=1, d2=a2c10, bab=dad-1=a-1, ac=ca, bc=cb, dbd-1=ab, dcd-1=a2c-1 >

Subgroups: 310 in 102 conjugacy classes, 43 normal (39 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, D4, D4, C23, C10, C10, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C22×C4, C2×D4, Dic5, C20, C20, C2×C10, C2×C10, D4⋊C4, C4⋊C8, C4.Q8, C2.D8, C4×D4, C42.C2, C52C8, C2×Dic5, C2×C20, C2×C20, C5×D4, C5×D4, C22×C10, D4.Q8, C2×C52C8, C10.D4, C4⋊Dic5, C4×C20, C5×C22⋊C4, C5×C4⋊C4, C22×C20, D4×C10, C203C8, C10.D8, C20.Q8, D4⋊Dic5, C20.6Q8, D4×C20, D4.3Dic10
Quotients: C1, C2, C22, D4, Q8, C23, D5, C2×D4, C2×Q8, C4○D4, D10, C22⋊Q8, C4○D8, C8⋊C22, Dic10, C5⋊D4, C22×D5, D4.Q8, C2×Dic10, C4○D20, C2×C5⋊D4, C20.48D4, D4.D10, D4.8D10, D4.3Dic10

Smallest permutation representation of D4.3Dic10
On 160 points
Generators in S160
(1 85 124 149)(2 86 125 150)(3 87 126 151)(4 88 127 152)(5 89 128 153)(6 90 129 154)(7 91 130 155)(8 92 131 156)(9 93 132 157)(10 94 133 158)(11 95 134 159)(12 96 135 160)(13 97 136 141)(14 98 137 142)(15 99 138 143)(16 100 139 144)(17 81 140 145)(18 82 121 146)(19 83 122 147)(20 84 123 148)(21 108 53 72)(22 109 54 73)(23 110 55 74)(24 111 56 75)(25 112 57 76)(26 113 58 77)(27 114 59 78)(28 115 60 79)(29 116 41 80)(30 117 42 61)(31 118 43 62)(32 119 44 63)(33 120 45 64)(34 101 46 65)(35 102 47 66)(36 103 48 67)(37 104 49 68)(38 105 50 69)(39 106 51 70)(40 107 52 71)
(1 95)(2 96)(3 97)(4 98)(5 99)(6 100)(7 81)(8 82)(9 83)(10 84)(11 85)(12 86)(13 87)(14 88)(15 89)(16 90)(17 91)(18 92)(19 93)(20 94)(21 31)(22 32)(23 33)(24 34)(25 35)(26 36)(27 37)(28 38)(29 39)(30 40)(41 51)(42 52)(43 53)(44 54)(45 55)(46 56)(47 57)(48 58)(49 59)(50 60)(61 107)(62 108)(63 109)(64 110)(65 111)(66 112)(67 113)(68 114)(69 115)(70 116)(71 117)(72 118)(73 119)(74 120)(75 101)(76 102)(77 103)(78 104)(79 105)(80 106)(121 156)(122 157)(123 158)(124 159)(125 160)(126 141)(127 142)(128 143)(129 144)(130 145)(131 146)(132 147)(133 148)(134 149)(135 150)(136 151)(137 152)(138 153)(139 154)(140 155)
(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 34 134 56)(2 45 135 23)(3 32 136 54)(4 43 137 21)(5 30 138 52)(6 41 139 39)(7 28 140 50)(8 59 121 37)(9 26 122 48)(10 57 123 35)(11 24 124 46)(12 55 125 33)(13 22 126 44)(14 53 127 31)(15 40 128 42)(16 51 129 29)(17 38 130 60)(18 49 131 27)(19 36 132 58)(20 47 133 25)(61 143 107 89)(62 98 108 152)(63 141 109 87)(64 96 110 150)(65 159 111 85)(66 94 112 148)(67 157 113 83)(68 92 114 146)(69 155 115 81)(70 90 116 144)(71 153 117 99)(72 88 118 142)(73 151 119 97)(74 86 120 160)(75 149 101 95)(76 84 102 158)(77 147 103 93)(78 82 104 156)(79 145 105 91)(80 100 106 154)

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

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

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

59 conjugacy classes

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 4D 4E 4F 4G 4H 4I 5A 5B 8A 8B 8C 8D 10A ··· 10F 10G ··· 10N 20A ··· 20H 20I ··· 20X order 1 2 2 2 2 2 4 4 4 4 4 4 4 4 4 5 5 8 8 8 8 10 ··· 10 10 ··· 10 20 ··· 20 20 ··· 20 size 1 1 1 1 4 4 2 2 2 2 4 4 4 40 40 2 2 20 20 20 20 2 ··· 2 4 ··· 4 2 ··· 2 4 ··· 4

59 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 4 4 4 type + + + + + + + + - + + + + - + image C1 C2 C2 C2 C2 C2 C2 D4 Q8 D5 C4○D4 D10 D10 D10 C4○D8 C5⋊D4 Dic10 C4○D20 C8⋊C22 D4.D10 D4.8D10 kernel D4.3Dic10 C20⋊3C8 C10.D8 C20.Q8 D4⋊Dic5 C20.6Q8 D4×C20 C2×C20 C5×D4 C4×D4 C20 C42 C4⋊C4 C2×D4 C10 C2×C4 D4 C4 C10 C2 C2 # reps 1 1 1 1 2 1 1 2 2 2 2 2 2 2 4 8 8 8 1 4 4

Matrix representation of D4.3Dic10 in GL4(𝔽41) generated by

 0 1 0 0 40 0 0 0 0 0 1 0 0 0 0 1
,
 0 1 0 0 1 0 0 0 0 0 40 0 0 0 0 40
,
 32 0 0 0 0 32 0 0 0 0 28 39 0 0 2 16
,
 12 29 0 0 29 29 0 0 0 0 27 39 0 0 37 14
G:=sub<GL(4,GF(41))| [0,40,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[0,1,0,0,1,0,0,0,0,0,40,0,0,0,0,40],[32,0,0,0,0,32,0,0,0,0,28,2,0,0,39,16],[12,29,0,0,29,29,0,0,0,0,27,37,0,0,39,14] >;

D4.3Dic10 in GAP, Magma, Sage, TeX

D_4._3{\rm Dic}_{10}
% in TeX

G:=Group("D4.3Dic10");
// GroupNames label

G:=SmallGroup(320,636);
// by ID

G=gap.SmallGroup(320,636);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,112,253,344,254,1123,297,136,12550]);
// Polycyclic

G:=Group<a,b,c,d|a^4=b^2=c^20=1,d^2=a^2*c^10,b*a*b=d*a*d^-1=a^-1,a*c=c*a,b*c=c*b,d*b*d^-1=a*b,d*c*d^-1=a^2*c^-1>;
// generators/relations

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