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G = Q83Dic15order 480 = 25·3·5

2nd semidirect product of Q8 and Dic15 acting via Dic15/C30=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: Q83Dic15, D42Dic15, C60.213D4, C1521C4≀C2, (D4×C15)⋊8C4, (Q8×C15)⋊8C4, (C2×C30).3D4, C60.83(C2×C4), (C3×Q8)⋊2Dic5, (C5×Q8)⋊8Dic3, (C3×D4)⋊2Dic5, C4○D4.3D15, (C5×D4)⋊5Dic3, (C2×C20).76D6, (C2×C4).40D30, (C4×Dic15)⋊2C2, (C2×C12).77D10, C60.7C416C2, C12.9(C2×Dic5), C4.3(C2×Dic15), C55(Q83Dic3), C33(D42Dic5), C4.31(C157D4), (C2×C60).62C22, C20.30(C2×Dic3), C12.110(C5⋊D4), C20.110(C3⋊D4), C22.3(C157D4), C6.19(C23.D5), C30.107(C22⋊C4), C2.8(C30.38D4), C10.30(C6.D4), (C5×C4○D4).5S3, (C3×C4○D4).1D5, (C15×C4○D4).1C2, (C2×C6).8(C5⋊D4), (C2×C10).7(C3⋊D4), SmallGroup(480,197)

Series: Derived Chief Lower central Upper central

C1C60 — Q83Dic15
C1C5C15C30C60C2×C60C60.7C4 — Q83Dic15
C15C30C60 — Q83Dic15
C1C4C2×C4C4○D4

Generators and relations for Q83Dic15
 G = < a,b,c,d | a4=c30=1, b2=a2, d2=c15, bab-1=a-1, ac=ca, ad=da, cbc-1=a2b, dbd-1=a-1b, dcd-1=c-1 >

Subgroups: 356 in 88 conjugacy classes, 39 normal (all characteristic)
C1, C2, C2 [×2], C3, C4 [×2], C4 [×3], C22, C22, C5, C6, C6 [×2], C8, C2×C4, C2×C4 [×2], D4, D4, Q8, C10, C10 [×2], Dic3 [×2], C12 [×2], C12, C2×C6, C2×C6, C15, C42, M4(2), C4○D4, Dic5 [×2], C20 [×2], C20, C2×C10, C2×C10, C3⋊C8, C2×Dic3, C2×C12, C2×C12, C3×D4, C3×D4, C3×Q8, C30, C30 [×2], C4≀C2, C52C8, C2×Dic5, C2×C20, C2×C20, C5×D4, C5×D4, C5×Q8, C4.Dic3, C4×Dic3, C3×C4○D4, Dic15 [×2], C60 [×2], C60, C2×C30, C2×C30, C4.Dic5, C4×Dic5, C5×C4○D4, Q83Dic3, C153C8, C2×Dic15, C2×C60, C2×C60, D4×C15, D4×C15, Q8×C15, D42Dic5, C60.7C4, C4×Dic15, C15×C4○D4, Q83Dic15
Quotients: C1, C2 [×3], C4 [×2], C22, S3, C2×C4, D4 [×2], D5, Dic3 [×2], D6, C22⋊C4, Dic5 [×2], D10, C2×Dic3, C3⋊D4 [×2], D15, C4≀C2, C2×Dic5, C5⋊D4 [×2], C6.D4, Dic15 [×2], D30, C23.D5, Q83Dic3, C2×Dic15, C157D4 [×2], D42Dic5, C30.38D4, Q83Dic15

Smallest permutation representation of Q83Dic15
On 120 points
Generators in S120
(1 59 30 44)(2 60 16 45)(3 46 17 31)(4 47 18 32)(5 48 19 33)(6 49 20 34)(7 50 21 35)(8 51 22 36)(9 52 23 37)(10 53 24 38)(11 54 25 39)(12 55 26 40)(13 56 27 41)(14 57 28 42)(15 58 29 43)(61 114 76 99)(62 115 77 100)(63 116 78 101)(64 117 79 102)(65 118 80 103)(66 119 81 104)(67 120 82 105)(68 91 83 106)(69 92 84 107)(70 93 85 108)(71 94 86 109)(72 95 87 110)(73 96 88 111)(74 97 89 112)(75 98 90 113)
(1 112 30 97)(2 98 16 113)(3 114 17 99)(4 100 18 115)(5 116 19 101)(6 102 20 117)(7 118 21 103)(8 104 22 119)(9 120 23 105)(10 106 24 91)(11 92 25 107)(12 108 26 93)(13 94 27 109)(14 110 28 95)(15 96 29 111)(31 76 46 61)(32 62 47 77)(33 78 48 63)(34 64 49 79)(35 80 50 65)(36 66 51 81)(37 82 52 67)(38 68 53 83)(39 84 54 69)(40 70 55 85)(41 86 56 71)(42 72 57 87)(43 88 58 73)(44 74 59 89)(45 90 60 75)
(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 30)(2 29)(3 28)(4 27)(5 26)(6 25)(7 24)(8 23)(9 22)(10 21)(11 20)(12 19)(13 18)(14 17)(15 16)(31 57)(32 56)(33 55)(34 54)(35 53)(36 52)(37 51)(38 50)(39 49)(40 48)(41 47)(42 46)(43 60)(44 59)(45 58)(61 110 76 95)(62 109 77 94)(63 108 78 93)(64 107 79 92)(65 106 80 91)(66 105 81 120)(67 104 82 119)(68 103 83 118)(69 102 84 117)(70 101 85 116)(71 100 86 115)(72 99 87 114)(73 98 88 113)(74 97 89 112)(75 96 90 111)

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

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

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

84 conjugacy classes

class 1 2A2B2C 3 4A4B4C4D4E4F4G4H5A5B6A6B6C6D8A8B10A10B10C···10H12A12B12C12D12E15A15B15C15D20A20B20C20D20E···20J30A30B30C30D30E···30P60A···60H60I···60T
order122234444444455666688101010···101212121212151515152020202020···203030303030···3060···6060···60
size112421124303030302224446060224···422444222222224···422224···42···24···4

84 irreducible representations

dim111111222222222222222222222444
type+++++++++--+--++--
imageC1C2C2C2C4C4S3D4D4D5D6Dic3Dic3D10Dic5Dic5C3⋊D4C3⋊D4D15C4≀C2C5⋊D4C5⋊D4D30Dic15Dic15C157D4C157D4Q83Dic3D42Dic5Q83Dic15
kernelQ83Dic15C60.7C4C4×Dic15C15×C4○D4D4×C15Q8×C15C5×C4○D4C60C2×C30C3×C4○D4C2×C20C5×D4C5×Q8C2×C12C3×D4C3×Q8C20C2×C10C4○D4C15C12C2×C6C2×C4D4Q8C4C22C5C3C1
# reps111122111211122222444444488248

Matrix representation of Q83Dic15 in GL4(𝔽241) generated by

240000
024000
00640
000177
,
1654900
1927600
000177
001770
,
11014700
948400
0010
000240
,
5119000
24019000
002400
000177
G:=sub<GL(4,GF(241))| [240,0,0,0,0,240,0,0,0,0,64,0,0,0,0,177],[165,192,0,0,49,76,0,0,0,0,0,177,0,0,177,0],[110,94,0,0,147,84,0,0,0,0,1,0,0,0,0,240],[51,240,0,0,190,190,0,0,0,0,240,0,0,0,0,177] >;

Q83Dic15 in GAP, Magma, Sage, TeX

Q_8\rtimes_3{\rm Dic}_{15}
% in TeX

G:=Group("Q8:3Dic15");
// GroupNames label

G:=SmallGroup(480,197);
// by ID

G=gap.SmallGroup(480,197);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,28,141,100,675,346,80,2693,18822]);
// Polycyclic

G:=Group<a,b,c,d|a^4=c^30=1,b^2=a^2,d^2=c^15,b*a*b^-1=a^-1,a*c=c*a,a*d=d*a,c*b*c^-1=a^2*b,d*b*d^-1=a^-1*b,d*c*d^-1=c^-1>;
// generators/relations

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