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G = Q8⋊3Dic15order 480 = 25·3·5

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C60 — Q8⋊3Dic15
 Chief series C1 — C5 — C15 — C30 — C60 — C2×C60 — C60.7C4 — Q8⋊3Dic15
 Lower central C15 — C30 — C60 — Q8⋊3Dic15
 Upper central C1 — C4 — C2×C4 — C4○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, C3, C4, C4, C22, C22, C5, C6, C6, C8, C2×C4, C2×C4, D4, D4, Q8, C10, C10, Dic3, C12, C12, C2×C6, C2×C6, C15, C42, M4(2), C4○D4, Dic5, C20, C20, C2×C10, C2×C10, C3⋊C8, C2×Dic3, C2×C12, C2×C12, C3×D4, C3×D4, C3×Q8, C30, C30, C4≀C2, C52C8, C2×Dic5, C2×C20, C2×C20, C5×D4, C5×D4, C5×Q8, C4.Dic3, C4×Dic3, C3×C4○D4, Dic15, C60, 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, C4, C22, S3, C2×C4, D4, D5, Dic3, D6, C22⋊C4, Dic5, D10, C2×Dic3, C3⋊D4, D15, C4≀C2, C2×Dic5, C5⋊D4, C6.D4, Dic15, D30, C23.D5, Q83Dic3, C2×Dic15, C157D4, D42Dic5, C30.38D4, Q83Dic15

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

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

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

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

84 conjugacy classes

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

84 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 4 4 4 type + + + + + + + + + - - + - - + + - - image C1 C2 C2 C2 C4 C4 S3 D4 D4 D5 D6 Dic3 Dic3 D10 Dic5 Dic5 C3⋊D4 C3⋊D4 D15 C4≀C2 C5⋊D4 C5⋊D4 D30 Dic15 Dic15 C15⋊7D4 C15⋊7D4 Q8⋊3Dic3 D4⋊2Dic5 Q8⋊3Dic15 kernel Q8⋊3Dic15 C60.7C4 C4×Dic15 C15×C4○D4 D4×C15 Q8×C15 C5×C4○D4 C60 C2×C30 C3×C4○D4 C2×C20 C5×D4 C5×Q8 C2×C12 C3×D4 C3×Q8 C20 C2×C10 C4○D4 C15 C12 C2×C6 C2×C4 D4 Q8 C4 C22 C5 C3 C1 # reps 1 1 1 1 2 2 1 1 1 2 1 1 1 2 2 2 2 2 4 4 4 4 4 4 4 8 8 2 4 8

Matrix representation of Q83Dic15 in GL4(𝔽241) generated by

 240 0 0 0 0 240 0 0 0 0 64 0 0 0 0 177
,
 165 49 0 0 192 76 0 0 0 0 0 177 0 0 177 0
,
 110 147 0 0 94 84 0 0 0 0 1 0 0 0 0 240
,
 51 190 0 0 240 190 0 0 0 0 240 0 0 0 0 177
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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