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## G = Dic10.A4order 480 = 25·3·5

### The non-split extension by Dic10 of A4 acting through Inn(Dic10)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — C5×Q8 — Dic10.A4
 Chief series C1 — C2 — C10 — C5×Q8 — C5×SL2(𝔽3) — Dic5.A4 — Dic10.A4
 Lower central C5×Q8 — Dic10.A4
 Upper central C1 — C2 — C4

Generators and relations for Dic10.A4
G = < a,b,c,d,e | a20=e3=1, b2=c2=d2=a10, bab-1=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, dcd-1=a10c, ece-1=a10cd, ede-1=c >

Subgroups: 686 in 92 conjugacy classes, 23 normal (17 characteristic)
C1, C2, C2 [×3], C3, C4, C4 [×3], C22 [×5], C5, C6, C2×C4 [×3], D4 [×6], Q8, Q8, C23 [×2], D5 [×2], C10, C10, C12 [×3], C15, C2×D4 [×3], C4○D4, C4○D4 [×3], Dic5 [×2], C20, C20, D10 [×4], C2×C10, SL2(𝔽3), C3×Q8, C30, 2+ 1+4, Dic10, C4×D5 [×2], D20 [×3], C5⋊D4 [×2], C2×C20, C5×D4, C5×Q8, C22×D5 [×2], C4.A4, C4.A4 [×2], C3×Dic5 [×2], C60, C2×D20, C4○D20, D4×D5 [×2], Q82D5 [×2], C5×C4○D4, Q8.A4, C5×SL2(𝔽3), C3×Dic10, D48D10, Dic5.A4 [×2], C5×C4.A4, Dic10.A4
Quotients: C1, C2 [×3], C3, C22, C6 [×3], D5, A4, C2×C6, D10, C2×A4 [×3], C3×D5, C22×A4, C6×D5, Q8.A4, D5×A4, C2×D5×A4, Dic10.A4

Smallest permutation representation of Dic10.A4
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)
(1 110 11 120)(2 109 12 119)(3 108 13 118)(4 107 14 117)(5 106 15 116)(6 105 16 115)(7 104 17 114)(8 103 18 113)(9 102 19 112)(10 101 20 111)(21 74 31 64)(22 73 32 63)(23 72 33 62)(24 71 34 61)(25 70 35 80)(26 69 36 79)(27 68 37 78)(28 67 38 77)(29 66 39 76)(30 65 40 75)(41 88 51 98)(42 87 52 97)(43 86 53 96)(44 85 54 95)(45 84 55 94)(46 83 56 93)(47 82 57 92)(48 81 58 91)(49 100 59 90)(50 99 60 89)
(1 120 11 110)(2 101 12 111)(3 102 13 112)(4 103 14 113)(5 104 15 114)(6 105 16 115)(7 106 17 116)(8 107 18 117)(9 108 19 118)(10 109 20 119)(21 36 31 26)(22 37 32 27)(23 38 33 28)(24 39 34 29)(25 40 35 30)(41 85 51 95)(42 86 52 96)(43 87 53 97)(44 88 54 98)(45 89 55 99)(46 90 56 100)(47 91 57 81)(48 92 58 82)(49 93 59 83)(50 94 60 84)(61 66 71 76)(62 67 72 77)(63 68 73 78)(64 69 74 79)(65 70 75 80)
(1 6 11 16)(2 7 12 17)(3 8 13 18)(4 9 14 19)(5 10 15 20)(21 79 31 69)(22 80 32 70)(23 61 33 71)(24 62 34 72)(25 63 35 73)(26 64 36 74)(27 65 37 75)(28 66 38 76)(29 67 39 77)(30 68 40 78)(41 90 51 100)(42 91 52 81)(43 92 53 82)(44 93 54 83)(45 94 55 84)(46 95 56 85)(47 96 57 86)(48 97 58 87)(49 98 59 88)(50 99 60 89)(101 116 111 106)(102 117 112 107)(103 118 113 108)(104 119 114 109)(105 120 115 110)
(1 84 74)(2 85 75)(3 86 76)(4 87 77)(5 88 78)(6 89 79)(7 90 80)(8 91 61)(9 92 62)(10 93 63)(11 94 64)(12 95 65)(13 96 66)(14 97 67)(15 98 68)(16 99 69)(17 100 70)(18 81 71)(19 82 72)(20 83 73)(21 120 45)(22 101 46)(23 102 47)(24 103 48)(25 104 49)(26 105 50)(27 106 51)(28 107 52)(29 108 53)(30 109 54)(31 110 55)(32 111 56)(33 112 57)(34 113 58)(35 114 59)(36 115 60)(37 116 41)(38 117 42)(39 118 43)(40 119 44)

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

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

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

47 conjugacy classes

 class 1 2A 2B 2C 2D 3A 3B 4A 4B 4C 4D 5A 5B 6A 6B 10A 10B 10C 10D 12A 12B 12C 12D 12E 12F 15A 15B 15C 15D 20A 20B 20C 20D 20E 20F 30A 30B 30C 30D 60A ··· 60H order 1 2 2 2 2 3 3 4 4 4 4 5 5 6 6 10 10 10 10 12 12 12 12 12 12 15 15 15 15 20 20 20 20 20 20 30 30 30 30 60 ··· 60 size 1 1 6 30 30 4 4 2 6 10 10 2 2 4 4 2 2 12 12 8 8 40 40 40 40 8 8 8 8 2 2 2 2 12 12 8 8 8 8 8 ··· 8

47 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 3 3 3 4 4 4 4 6 6 type + + + + + + + + + + + + image C1 C2 C2 C3 C6 C6 D5 D10 C3×D5 C6×D5 A4 C2×A4 C2×A4 Q8.A4 Q8.A4 Dic10.A4 Dic10.A4 D5×A4 C2×D5×A4 kernel Dic10.A4 Dic5.A4 C5×C4.A4 D4⋊8D10 Q8⋊2D5 C5×C4○D4 C4.A4 SL2(𝔽3) C4○D4 Q8 Dic10 Dic5 C20 C5 C5 C1 C1 C4 C2 # reps 1 2 1 2 4 2 2 2 4 4 1 2 1 1 2 4 8 2 2

Matrix representation of Dic10.A4 in GL4(𝔽61) generated by

 0 50 0 0 50 31 0 0 0 25 2 36 50 2 27 29
,
 45 0 34 36 59 0 32 16 45 18 17 18 30 44 16 60
,
 1 0 59 0 0 0 17 1 1 0 60 0 44 60 17 0
,
 39 8 0 0 8 22 0 0 39 4 29 4 0 54 3 32
,
 60 57 30 4 0 19 18 41 41 57 0 0 43 49 18 41
G:=sub<GL(4,GF(61))| [0,50,0,50,50,31,25,2,0,0,2,27,0,0,36,29],[45,59,45,30,0,0,18,44,34,32,17,16,36,16,18,60],[1,0,1,44,0,0,0,60,59,17,60,17,0,1,0,0],[39,8,39,0,8,22,4,54,0,0,29,3,0,0,4,32],[60,0,41,43,57,19,57,49,30,18,0,18,4,41,0,41] >;

Dic10.A4 in GAP, Magma, Sage, TeX

{\rm Dic}_{10}.A_4
% in TeX

G:=Group("Dic10.A4");
// GroupNames label

G:=SmallGroup(480,1041);
// by ID

G=gap.SmallGroup(480,1041);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,2,-5,-2,1680,3389,1688,269,584,123,795,382,8069]);
// Polycyclic

G:=Group<a,b,c,d,e|a^20=e^3=1,b^2=c^2=d^2=a^10,b*a*b^-1=a^-1,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,d*c*d^-1=a^10*c,e*c*e^-1=a^10*c*d,e*d*e^-1=c>;
// generators/relations

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