Copied to
clipboard

## G = SD16×D15order 480 = 25·3·5

### Direct product of SD16 and D15

Series: Derived Chief Lower central Upper central

 Derived series C1 — C60 — SD16×D15
 Chief series C1 — C5 — C15 — C30 — C60 — C4×D15 — D4×D15 — SD16×D15
 Lower central C15 — C30 — C60 — SD16×D15
 Upper central C1 — C2 — C4 — SD16

Generators and relations for SD16×D15
G = < a,b,c,d | a8=b2=c15=d2=1, bab=a3, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >

Subgroups: 996 in 136 conjugacy classes, 43 normal (41 characteristic)
C1, C2, C2, C3, C4, C4, C22, C5, S3, C6, C6, C8, C8, C2×C4, D4, D4, Q8, Q8, C23, D5, C10, C10, Dic3, C12, C12, D6, C2×C6, C15, C2×C8, SD16, SD16, C2×D4, C2×Q8, Dic5, C20, C20, D10, C2×C10, C3⋊C8, C24, Dic6, C4×S3, D12, C3⋊D4, C3×D4, C3×Q8, C22×S3, D15, D15, C30, C30, C2×SD16, C52C8, C40, Dic10, C4×D5, D20, C5⋊D4, C5×D4, C5×Q8, C22×D5, S3×C8, C24⋊C2, D4.S3, Q82S3, C3×SD16, S3×D4, S3×Q8, Dic15, Dic15, C60, C60, D30, D30, C2×C30, C8×D5, C40⋊C2, D4.D5, Q8⋊D5, C5×SD16, D4×D5, Q8×D5, S3×SD16, C153C8, C120, Dic30, Dic30, C4×D15, C4×D15, D60, C157D4, D4×C15, Q8×C15, C22×D15, D5×SD16, C8×D15, C24⋊D5, D4.D15, Q82D15, C15×SD16, D4×D15, Q8×D15, SD16×D15
Quotients: C1, C2, C22, S3, D4, C23, D5, D6, SD16, C2×D4, D10, C22×S3, D15, C2×SD16, C22×D5, S3×D4, D30, D4×D5, S3×SD16, C22×D15, D5×SD16, D4×D15, SD16×D15

Smallest permutation representation of SD16×D15
On 120 points
Generators in S120
(1 110 31 84 16 99 50 65)(2 111 32 85 17 100 51 66)(3 112 33 86 18 101 52 67)(4 113 34 87 19 102 53 68)(5 114 35 88 20 103 54 69)(6 115 36 89 21 104 55 70)(7 116 37 90 22 105 56 71)(8 117 38 76 23 91 57 72)(9 118 39 77 24 92 58 73)(10 119 40 78 25 93 59 74)(11 120 41 79 26 94 60 75)(12 106 42 80 27 95 46 61)(13 107 43 81 28 96 47 62)(14 108 44 82 29 97 48 63)(15 109 45 83 30 98 49 64)
(31 50)(32 51)(33 52)(34 53)(35 54)(36 55)(37 56)(38 57)(39 58)(40 59)(41 60)(42 46)(43 47)(44 48)(45 49)(61 95)(62 96)(63 97)(64 98)(65 99)(66 100)(67 101)(68 102)(69 103)(70 104)(71 105)(72 91)(73 92)(74 93)(75 94)(76 117)(77 118)(78 119)(79 120)(80 106)(81 107)(82 108)(83 109)(84 110)(85 111)(86 112)(87 113)(88 114)(89 115)(90 116)
(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 49)(32 48)(33 47)(34 46)(35 60)(36 59)(37 58)(38 57)(39 56)(40 55)(41 54)(42 53)(43 52)(44 51)(45 50)(61 87)(62 86)(63 85)(64 84)(65 83)(66 82)(67 81)(68 80)(69 79)(70 78)(71 77)(72 76)(73 90)(74 89)(75 88)(91 117)(92 116)(93 115)(94 114)(95 113)(96 112)(97 111)(98 110)(99 109)(100 108)(101 107)(102 106)(103 120)(104 119)(105 118)

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

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

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

63 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3 4A 4B 4C 4D 5A 5B 6A 6B 8A 8B 8C 8D 10A 10B 10C 10D 12A 12B 15A 15B 15C 15D 20A 20B 20C 20D 24A 24B 30A 30B 30C 30D 30E 30F 30G 30H 40A 40B 40C 40D 60A 60B 60C 60D 60E 60F 60G 60H 120A ··· 120H order 1 2 2 2 2 2 3 4 4 4 4 5 5 6 6 8 8 8 8 10 10 10 10 12 12 15 15 15 15 20 20 20 20 24 24 30 30 30 30 30 30 30 30 40 40 40 40 60 60 60 60 60 60 60 60 120 ··· 120 size 1 1 4 15 15 60 2 2 4 30 60 2 2 2 8 2 2 30 30 2 2 8 8 4 8 2 2 2 2 4 4 8 8 4 4 2 2 2 2 8 8 8 8 4 4 4 4 4 4 4 4 8 8 8 8 4 ··· 4

63 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 type + + + + + + + + + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 S3 D4 D4 D5 D6 D6 D6 SD16 D10 D10 D10 D15 D30 D30 D30 S3×D4 D4×D5 S3×SD16 D5×SD16 D4×D15 SD16×D15 kernel SD16×D15 C8×D15 C24⋊D5 D4.D15 Q8⋊2D15 C15×SD16 D4×D15 Q8×D15 C5×SD16 Dic15 D30 C3×SD16 C40 C5×D4 C5×Q8 D15 C24 C3×D4 C3×Q8 SD16 C8 D4 Q8 C10 C6 C5 C3 C2 C1 # reps 1 1 1 1 1 1 1 1 1 1 1 2 1 1 1 4 2 2 2 4 4 4 4 1 2 2 4 4 8

Matrix representation of SD16×D15 in GL6(𝔽241)

 240 0 0 0 0 0 0 240 0 0 0 0 0 0 240 0 0 0 0 0 0 240 0 0 0 0 0 0 38 39 0 0 0 0 68 0
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 67 240
,
 190 51 0 0 0 0 190 240 0 0 0 0 0 0 1 192 0 0 0 0 64 239 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 0 1 0 0 0 0 1 0 0 0 0 0 0 0 1 192 0 0 0 0 0 240 0 0 0 0 0 0 240 0 0 0 0 0 0 240

G:=sub<GL(6,GF(241))| [240,0,0,0,0,0,0,240,0,0,0,0,0,0,240,0,0,0,0,0,0,240,0,0,0,0,0,0,38,68,0,0,0,0,39,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,67,0,0,0,0,0,240],[190,190,0,0,0,0,51,240,0,0,0,0,0,0,1,64,0,0,0,0,192,239,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,192,240,0,0,0,0,0,0,240,0,0,0,0,0,0,240] >;

SD16×D15 in GAP, Magma, Sage, TeX

{\rm SD}_{16}\times D_{15}
% in TeX

G:=Group("SD16xD15");
// GroupNames label

G:=SmallGroup(480,878);
// by ID

G=gap.SmallGroup(480,878);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,135,100,346,185,80,2693,18822]);
// Polycyclic

G:=Group<a,b,c,d|a^8=b^2=c^15=d^2=1,b*a*b=a^3,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations

׿
×
𝔽