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G = D30.8D4order 480 = 25·3·5

8th non-split extension by D30 of D4 acting via D4/C2=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D30.8D4, D20.6D6, D12.6D10, C60.6C23, Dic15.40D4, Dic301C22, C3⋊C86D10, D4⋊D53S3, D4⋊S33D5, D46(S3×D5), (C5×D4)⋊4D6, C52C86D6, (C3×D4)⋊4D10, C53(D8⋊S3), C6.68(D4×D5), C20⋊D62C2, C33(D8⋊D5), C10.69(S3×D4), D42D151C2, C1515(C8⋊C22), D12.D51C2, C6.D201C2, C30.168(C2×D4), (D4×C15)⋊6C22, C20.6(C22×S3), C12.6(C22×D5), D30.5C41C2, (C5×D12).2C22, (C4×D15).2C22, (C3×D20).2C22, C2.21(D10⋊D6), C4.6(C2×S3×D5), (C3×D4⋊D5)⋊4C2, (C5×D4⋊S3)⋊4C2, (C5×C3⋊C8)⋊4C22, (C3×C52C8)⋊4C22, SmallGroup(480,558)

Series: Derived Chief Lower central Upper central

C1C60 — D30.8D4
C1C5C15C30C60C3×D20C20⋊D6 — D30.8D4
C15C30C60 — D30.8D4
C1C2C4D4

Generators and relations for D30.8D4
 G = < a,b,c,d | a20=b2=c6=1, d2=a5, bab=a-1, cac-1=a11, ad=da, cbc-1=dbd-1=a15b, dcd-1=a5c-1 >

Subgroups: 940 in 136 conjugacy classes, 38 normal (all characteristic)
C1, C2, C2 [×4], C3, C4, C4 [×2], C22 [×6], C5, S3 [×2], C6, C6 [×2], C8 [×2], C2×C4 [×2], D4, D4 [×4], Q8, C23, D5 [×2], C10, C10 [×2], Dic3 [×2], C12, D6 [×4], C2×C6 [×2], C15, M4(2), D8 [×2], SD16 [×2], C2×D4, C4○D4, Dic5 [×2], C20, D10 [×4], C2×C10 [×2], C3⋊C8, C24, Dic6, C4×S3, D12, C2×Dic3, C3⋊D4 [×2], C3×D4, C3×D4, C22×S3, C5×S3, C3×D5, D15, C30, C30, C8⋊C22, C52C8, C40, Dic10, C4×D5, D20, C2×Dic5, C5⋊D4 [×2], C5×D4, C5×D4, C22×D5, C8⋊S3, C24⋊C2, D4⋊S3, D4.S3, C3×D8, S3×D4, D42S3, Dic15, Dic15, C60, S3×D5 [×2], C6×D5, S3×C10, D30, C2×C30, C8⋊D5, C40⋊C2, D4⋊D5, D4.D5, C5×D8, D4×D5, D42D5, D8⋊S3, C5×C3⋊C8, C3×C52C8, C15⋊D4, C3×D20, C5×D12, Dic30, C4×D15, C2×Dic15, C157D4, D4×C15, C2×S3×D5, D8⋊D5, D30.5C4, C6.D20, D12.D5, C3×D4⋊D5, C5×D4⋊S3, C20⋊D6, D42D15, D30.8D4
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], C23, D5, D6 [×3], C2×D4, D10 [×3], C22×S3, C8⋊C22, C22×D5, S3×D4, S3×D5, D4×D5, D8⋊S3, C2×S3×D5, D8⋊D5, D10⋊D6, D30.8D4

Smallest permutation representation of D30.8D4
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 20)(2 19)(3 18)(4 17)(5 16)(6 15)(7 14)(8 13)(9 12)(10 11)(21 39)(22 38)(23 37)(24 36)(25 35)(26 34)(27 33)(28 32)(29 31)(41 52)(42 51)(43 50)(44 49)(45 48)(46 47)(53 60)(54 59)(55 58)(56 57)(61 64)(62 63)(65 80)(66 79)(67 78)(68 77)(69 76)(70 75)(71 74)(72 73)(81 93)(82 92)(83 91)(84 90)(85 89)(86 88)(94 100)(95 99)(96 98)(101 117)(102 116)(103 115)(104 114)(105 113)(106 112)(107 111)(108 110)(118 120)
(1 95 73 28 47 117)(2 86 74 39 48 108)(3 97 75 30 49 119)(4 88 76 21 50 110)(5 99 77 32 51 101)(6 90 78 23 52 112)(7 81 79 34 53 103)(8 92 80 25 54 114)(9 83 61 36 55 105)(10 94 62 27 56 116)(11 85 63 38 57 107)(12 96 64 29 58 118)(13 87 65 40 59 109)(14 98 66 31 60 120)(15 89 67 22 41 111)(16 100 68 33 42 102)(17 91 69 24 43 113)(18 82 70 35 44 104)(19 93 71 26 45 115)(20 84 72 37 46 106)
(1 117 6 102 11 107 16 112)(2 118 7 103 12 108 17 113)(3 119 8 104 13 109 18 114)(4 120 9 105 14 110 19 115)(5 101 10 106 15 111 20 116)(21 71 26 76 31 61 36 66)(22 72 27 77 32 62 37 67)(23 73 28 78 33 63 38 68)(24 74 29 79 34 64 39 69)(25 75 30 80 35 65 40 70)(41 89 46 94 51 99 56 84)(42 90 47 95 52 100 57 85)(43 91 48 96 53 81 58 86)(44 92 49 97 54 82 59 87)(45 93 50 98 55 83 60 88)

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

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

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

42 conjugacy classes

class 1 2A2B2C2D2E 3 4A4B4C5A5B6A6B6C8A8B10A10B10C10D10E10F 12 15A15B20A20B24A24B30A30B30C30D30E30F40A40B40C40D60A60B
order1222223444556668810101010101012151520202424303030303030404040406060
size1141220302230602228401220228824244444420204488881212121288

42 irreducible representations

dim111111112222222222444444448
type++++++++++++++++++++++++-
imageC1C2C2C2C2C2C2C2S3D4D4D5D6D6D6D10D10D10C8⋊C22S3×D4S3×D5D4×D5D8⋊S3C2×S3×D5D8⋊D5D10⋊D6D30.8D4
kernelD30.8D4D30.5C4C6.D20D12.D5C3×D4⋊D5C5×D4⋊S3C20⋊D6D42D15D4⋊D5Dic15D30D4⋊S3C52C8D20C5×D4C3⋊C8D12C3×D4C15C10D4C6C5C4C3C2C1
# reps111111111112111222112222442

Matrix representation of D30.8D4 in GL6(𝔽241)

18910000
24000000
000010
000001
00240000
00024000
,
11890000
02400000
000010
000001
001000
000100
,
24000000
02400000
001944719447
00194147194147
001944747194
001941474794
,
24000000
02400000
004719419447
001471949447
004719447194
00147194147194

G:=sub<GL(6,GF(241))| [189,240,0,0,0,0,1,0,0,0,0,0,0,0,0,0,240,0,0,0,0,0,0,240,0,0,1,0,0,0,0,0,0,1,0,0],[1,0,0,0,0,0,189,240,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,1,0,0],[240,0,0,0,0,0,0,240,0,0,0,0,0,0,194,194,194,194,0,0,47,147,47,147,0,0,194,194,47,47,0,0,47,147,194,94],[240,0,0,0,0,0,0,240,0,0,0,0,0,0,47,147,47,147,0,0,194,194,194,194,0,0,194,94,47,147,0,0,47,47,194,194] >;

D30.8D4 in GAP, Magma, Sage, TeX

D_{30}._8D_4
% in TeX

G:=Group("D30.8D4");
// GroupNames label

G:=SmallGroup(480,558);
// by ID

G=gap.SmallGroup(480,558);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,254,303,675,346,185,80,1356,18822]);
// Polycyclic

G:=Group<a,b,c,d|a^20=b^2=c^6=1,d^2=a^5,b*a*b=a^-1,c*a*c^-1=a^11,a*d=d*a,c*b*c^-1=d*b*d^-1=a^15*b,d*c*d^-1=a^5*c^-1>;
// generators/relations

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