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## G = D20⋊2Dic3order 480 = 25·3·5

### 2nd semidirect product of D20 and Dic3 acting via Dic3/C3=C4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C60 — D20⋊2Dic3
 Chief series C1 — C5 — C15 — C30 — C3×Dic5 — D5×C12 — C12.F5 — D20⋊2Dic3
 Lower central C15 — C30 — C60 — D20⋊2Dic3
 Upper central C1 — C2 — C4 — Q8

Generators and relations for D202Dic3
G = < a,b,c,d | a20=b2=c6=1, d2=c3, bab=a-1, cac-1=a9, dad-1=a13, cbc-1=a18b, dbd-1=a7b, dcd-1=c-1 >

Subgroups: 492 in 88 conjugacy classes, 29 normal (all characteristic)
C1, C2, C2, C3, C4, C4, C22, C5, C6, C6, C8, C2×C4, D4, Q8, D5, C10, Dic3, C12, C12, C2×C6, C15, C42, M4(2), C4○D4, Dic5, C20, C20, F5, D10, D10, C3⋊C8, C2×Dic3, C2×C12, C3×D4, C3×Q8, C3×D5, C30, C4≀C2, C5⋊C8, C4×D5, C4×D5, D20, D20, C5×Q8, C2×F5, C4.Dic3, C4×Dic3, C3×C4○D4, C3×Dic5, C60, C60, C3⋊F5, C6×D5, C6×D5, C4.F5, C4×F5, Q82D5, Q83Dic3, C15⋊C8, D5×C12, D5×C12, C3×D20, C3×D20, Q8×C15, C2×C3⋊F5, Q82F5, C12.F5, C4×C3⋊F5, C3×Q82D5, D202Dic3
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, Dic3, D6, C22⋊C4, F5, C2×Dic3, C3⋊D4, C4≀C2, C2×F5, C6.D4, C3⋊F5, C22⋊F5, Q83Dic3, C2×C3⋊F5, Q82F5, D10.D6, D202Dic3

Smallest permutation representation of D202Dic3
On 120 points
Generators in S120
(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)
(1 37)(2 36)(3 35)(4 34)(5 33)(6 32)(7 31)(8 30)(9 29)(10 28)(11 27)(12 26)(13 25)(14 24)(15 23)(16 22)(17 21)(18 40)(19 39)(20 38)(41 95)(42 94)(43 93)(44 92)(45 91)(46 90)(47 89)(48 88)(49 87)(50 86)(51 85)(52 84)(53 83)(54 82)(55 81)(56 100)(57 99)(58 98)(59 97)(60 96)(61 110)(62 109)(63 108)(64 107)(65 106)(66 105)(67 104)(68 103)(69 102)(70 101)(71 120)(72 119)(73 118)(74 117)(75 116)(76 115)(77 114)(78 113)(79 112)(80 111)
(1 81 61 11 91 71)(2 90 62 20 92 80)(3 99 63 9 93 69)(4 88 64 18 94 78)(5 97 65 7 95 67)(6 86 66 16 96 76)(8 84 68 14 98 74)(10 82 70 12 100 72)(13 89 73 19 83 79)(15 87 75 17 85 77)(21 53 114 25 49 118)(22 42 115 34 50 107)(23 51 116)(24 60 117 32 52 105)(26 58 119 30 54 103)(27 47 120 39 55 112)(28 56 101)(29 45 102 37 57 110)(31 43 104 35 59 108)(33 41 106)(36 48 109 40 44 113)(38 46 111)
(1 16 11 6)(2 13 20 19)(3 10 9 12)(4 7 18 5)(8 15 14 17)(21 29 25 37)(22 26 34 30)(24 40 32 36)(27 31 39 35)(41 106)(42 103 50 119)(43 120 59 112)(44 117 48 105)(45 114 57 118)(46 111)(47 108 55 104)(49 102 53 110)(51 116)(52 113 60 109)(54 107 58 115)(56 101)(61 86 71 96)(62 83 80 89)(63 100 69 82)(64 97 78 95)(65 94 67 88)(66 91 76 81)(68 85 74 87)(70 99 72 93)(73 90 79 92)(75 84 77 98)

G:=sub<Sym(120)| (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), (1,37)(2,36)(3,35)(4,34)(5,33)(6,32)(7,31)(8,30)(9,29)(10,28)(11,27)(12,26)(13,25)(14,24)(15,23)(16,22)(17,21)(18,40)(19,39)(20,38)(41,95)(42,94)(43,93)(44,92)(45,91)(46,90)(47,89)(48,88)(49,87)(50,86)(51,85)(52,84)(53,83)(54,82)(55,81)(56,100)(57,99)(58,98)(59,97)(60,96)(61,110)(62,109)(63,108)(64,107)(65,106)(66,105)(67,104)(68,103)(69,102)(70,101)(71,120)(72,119)(73,118)(74,117)(75,116)(76,115)(77,114)(78,113)(79,112)(80,111), (1,81,61,11,91,71)(2,90,62,20,92,80)(3,99,63,9,93,69)(4,88,64,18,94,78)(5,97,65,7,95,67)(6,86,66,16,96,76)(8,84,68,14,98,74)(10,82,70,12,100,72)(13,89,73,19,83,79)(15,87,75,17,85,77)(21,53,114,25,49,118)(22,42,115,34,50,107)(23,51,116)(24,60,117,32,52,105)(26,58,119,30,54,103)(27,47,120,39,55,112)(28,56,101)(29,45,102,37,57,110)(31,43,104,35,59,108)(33,41,106)(36,48,109,40,44,113)(38,46,111), (1,16,11,6)(2,13,20,19)(3,10,9,12)(4,7,18,5)(8,15,14,17)(21,29,25,37)(22,26,34,30)(24,40,32,36)(27,31,39,35)(41,106)(42,103,50,119)(43,120,59,112)(44,117,48,105)(45,114,57,118)(46,111)(47,108,55,104)(49,102,53,110)(51,116)(52,113,60,109)(54,107,58,115)(56,101)(61,86,71,96)(62,83,80,89)(63,100,69,82)(64,97,78,95)(65,94,67,88)(66,91,76,81)(68,85,74,87)(70,99,72,93)(73,90,79,92)(75,84,77,98)>;

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), (1,37)(2,36)(3,35)(4,34)(5,33)(6,32)(7,31)(8,30)(9,29)(10,28)(11,27)(12,26)(13,25)(14,24)(15,23)(16,22)(17,21)(18,40)(19,39)(20,38)(41,95)(42,94)(43,93)(44,92)(45,91)(46,90)(47,89)(48,88)(49,87)(50,86)(51,85)(52,84)(53,83)(54,82)(55,81)(56,100)(57,99)(58,98)(59,97)(60,96)(61,110)(62,109)(63,108)(64,107)(65,106)(66,105)(67,104)(68,103)(69,102)(70,101)(71,120)(72,119)(73,118)(74,117)(75,116)(76,115)(77,114)(78,113)(79,112)(80,111), (1,81,61,11,91,71)(2,90,62,20,92,80)(3,99,63,9,93,69)(4,88,64,18,94,78)(5,97,65,7,95,67)(6,86,66,16,96,76)(8,84,68,14,98,74)(10,82,70,12,100,72)(13,89,73,19,83,79)(15,87,75,17,85,77)(21,53,114,25,49,118)(22,42,115,34,50,107)(23,51,116)(24,60,117,32,52,105)(26,58,119,30,54,103)(27,47,120,39,55,112)(28,56,101)(29,45,102,37,57,110)(31,43,104,35,59,108)(33,41,106)(36,48,109,40,44,113)(38,46,111), (1,16,11,6)(2,13,20,19)(3,10,9,12)(4,7,18,5)(8,15,14,17)(21,29,25,37)(22,26,34,30)(24,40,32,36)(27,31,39,35)(41,106)(42,103,50,119)(43,120,59,112)(44,117,48,105)(45,114,57,118)(46,111)(47,108,55,104)(49,102,53,110)(51,116)(52,113,60,109)(54,107,58,115)(56,101)(61,86,71,96)(62,83,80,89)(63,100,69,82)(64,97,78,95)(65,94,67,88)(66,91,76,81)(68,85,74,87)(70,99,72,93)(73,90,79,92)(75,84,77,98) );

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)], [(1,37),(2,36),(3,35),(4,34),(5,33),(6,32),(7,31),(8,30),(9,29),(10,28),(11,27),(12,26),(13,25),(14,24),(15,23),(16,22),(17,21),(18,40),(19,39),(20,38),(41,95),(42,94),(43,93),(44,92),(45,91),(46,90),(47,89),(48,88),(49,87),(50,86),(51,85),(52,84),(53,83),(54,82),(55,81),(56,100),(57,99),(58,98),(59,97),(60,96),(61,110),(62,109),(63,108),(64,107),(65,106),(66,105),(67,104),(68,103),(69,102),(70,101),(71,120),(72,119),(73,118),(74,117),(75,116),(76,115),(77,114),(78,113),(79,112),(80,111)], [(1,81,61,11,91,71),(2,90,62,20,92,80),(3,99,63,9,93,69),(4,88,64,18,94,78),(5,97,65,7,95,67),(6,86,66,16,96,76),(8,84,68,14,98,74),(10,82,70,12,100,72),(13,89,73,19,83,79),(15,87,75,17,85,77),(21,53,114,25,49,118),(22,42,115,34,50,107),(23,51,116),(24,60,117,32,52,105),(26,58,119,30,54,103),(27,47,120,39,55,112),(28,56,101),(29,45,102,37,57,110),(31,43,104,35,59,108),(33,41,106),(36,48,109,40,44,113),(38,46,111)], [(1,16,11,6),(2,13,20,19),(3,10,9,12),(4,7,18,5),(8,15,14,17),(21,29,25,37),(22,26,34,30),(24,40,32,36),(27,31,39,35),(41,106),(42,103,50,119),(43,120,59,112),(44,117,48,105),(45,114,57,118),(46,111),(47,108,55,104),(49,102,53,110),(51,116),(52,113,60,109),(54,107,58,115),(56,101),(61,86,71,96),(62,83,80,89),(63,100,69,82),(64,97,78,95),(65,94,67,88),(66,91,76,81),(68,85,74,87),(70,99,72,93),(73,90,79,92),(75,84,77,98)]])

39 conjugacy classes

 class 1 2A 2B 2C 3 4A 4B 4C 4D 4E 4F 4G 4H 5 6A 6B 6C 6D 8A 8B 10 12A 12B 12C 12D 12E 15A 15B 20A 20B 20C 30A 30B 60A ··· 60F order 1 2 2 2 3 4 4 4 4 4 4 4 4 5 6 6 6 6 8 8 10 12 12 12 12 12 15 15 20 20 20 30 30 60 ··· 60 size 1 1 10 20 2 2 4 5 5 30 30 30 30 4 2 20 20 20 60 60 4 4 4 4 10 10 4 4 8 8 8 4 4 8 ··· 8

39 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 4 8 8 type + + + + + + + + - - + + + + image C1 C2 C2 C2 C4 C4 S3 D4 D4 D6 Dic3 Dic3 C3⋊D4 C3⋊D4 C4≀C2 F5 C2×F5 C3⋊F5 C22⋊F5 Q8⋊3Dic3 C2×C3⋊F5 D10.D6 Q8⋊2F5 D20⋊2Dic3 kernel D20⋊2Dic3 C12.F5 C4×C3⋊F5 C3×Q8⋊2D5 C3×D20 Q8×C15 Q8⋊2D5 C3×Dic5 C6×D5 C4×D5 D20 C5×Q8 Dic5 D10 C15 C3×Q8 C12 Q8 C6 C5 C4 C2 C3 C1 # reps 1 1 1 1 2 2 1 1 1 1 1 1 2 2 4 1 1 2 2 2 2 4 1 2

Matrix representation of D202Dic3 in GL8(𝔽241)

 64 0 0 0 0 0 0 0 0 64 0 0 0 0 0 0 103 0 177 0 0 0 0 0 39 64 0 177 0 0 0 0 0 0 0 0 240 1 0 0 0 0 0 0 240 0 1 0 0 0 0 0 240 0 0 1 0 0 0 0 240 0 0 0
,
 85 0 2 0 0 0 0 0 86 240 0 2 0 0 0 0 3 0 156 0 0 0 0 0 3 0 155 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 240 0 0 0 0 1 0 240 0 0 0 0 0 1 240 0 0
,
 0 240 0 0 0 0 0 0 1 1 0 0 0 0 0 0 43 42 0 1 0 0 0 0 156 44 240 240 0 0 0 0 0 0 0 0 229 0 12 114 0 0 0 0 229 12 126 0 0 0 0 0 0 126 12 229 0 0 0 0 114 12 0 229
,
 177 0 0 0 0 0 0 0 64 64 0 0 0 0 0 0 232 0 1 0 0 0 0 0 168 153 240 240 0 0 0 0 0 0 0 0 229 12 127 0 0 0 0 0 115 12 0 229 0 0 0 0 229 0 12 115 0 0 0 0 0 127 12 229

G:=sub<GL(8,GF(241))| [64,0,103,39,0,0,0,0,0,64,0,64,0,0,0,0,0,0,177,0,0,0,0,0,0,0,0,177,0,0,0,0,0,0,0,0,240,240,240,240,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0],[85,86,3,3,0,0,0,0,0,240,0,0,0,0,0,0,2,0,156,155,0,0,0,0,0,2,0,1,0,0,0,0,0,0,0,0,1,1,1,1,0,0,0,0,0,0,0,240,0,0,0,0,0,0,240,0,0,0,0,0,0,240,0,0],[0,1,43,156,0,0,0,0,240,1,42,44,0,0,0,0,0,0,0,240,0,0,0,0,0,0,1,240,0,0,0,0,0,0,0,0,229,229,0,114,0,0,0,0,0,12,126,12,0,0,0,0,12,126,12,0,0,0,0,0,114,0,229,229],[177,64,232,168,0,0,0,0,0,64,0,153,0,0,0,0,0,0,1,240,0,0,0,0,0,0,0,240,0,0,0,0,0,0,0,0,229,115,229,0,0,0,0,0,12,12,0,127,0,0,0,0,127,0,12,12,0,0,0,0,0,229,115,229] >;

D202Dic3 in GAP, Magma, Sage, TeX

D_{20}\rtimes_2{\rm Dic}_3
% in TeX

G:=Group("D20:2Dic3");
// GroupNames label

G:=SmallGroup(480,315);
// by ID

G=gap.SmallGroup(480,315);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,28,141,120,100,675,346,80,2693,14118,4724]);
// Polycyclic

G:=Group<a,b,c,d|a^20=b^2=c^6=1,d^2=c^3,b*a*b=a^-1,c*a*c^-1=a^9,d*a*d^-1=a^13,c*b*c^-1=a^18*b,d*b*d^-1=a^7*b,d*c*d^-1=c^-1>;
// generators/relations

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