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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 [×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 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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