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G = Dic10.A4order 480 = 25·3·5

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

non-abelian, soluble

Aliases: Dic10.A4, SL2(𝔽3)⋊6D10, C5⋊(Q8.A4), D48D10⋊C3, C4.A43D5, C4.2(D5×A4), C20.2(C2×A4), Q8.3(C6×D5), Dic5.A45C2, C10.7(C22×A4), Q82D5.1C6, Dic5.3(C2×A4), (C5×SL2(𝔽3))⋊7C22, C2.8(C2×D5×A4), C4○D4.(C3×D5), (C5×C4.A4)⋊3C2, (C5×C4○D4).2C6, (C5×Q8).3(C2×C6), SmallGroup(480,1041)

Series: Derived Chief Lower central Upper central

C1C2C5×Q8 — Dic10.A4
C1C2C10C5×Q8C5×SL2(𝔽3)Dic5.A4 — Dic10.A4
C5×Q8 — Dic10.A4
C1C2C4

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 2A2B2C2D3A3B4A4B4C4D5A5B6A6B10A10B10C10D12A12B12C12D12E12F15A15B15C15D20A20B20C20D20E20F30A30B30C30D60A···60H
order12222334444556610101010121212121212151515152020202020203030303060···60
size1163030442610102244221212884040404088882222121288888···8

47 irreducible representations

dim1111112222333444466
type++++++++++++
imageC1C2C2C3C6C6D5D10C3×D5C6×D5A4C2×A4C2×A4Q8.A4Q8.A4Dic10.A4Dic10.A4D5×A4C2×D5×A4
kernelDic10.A4Dic5.A4C5×C4.A4D48D10Q82D5C5×C4○D4C4.A4SL2(𝔽3)C4○D4Q8Dic10Dic5C20C5C5C1C1C4C2
# reps1212422244121124822

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

05000
503100
025236
5022729
,
4503436
5903216
45181718
30441660
,
10590
00171
10600
4460170
,
39800
82200
394294
054332
,
6057304
0191841
415700
43491841
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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