Copied to
clipboard

## G = M4(2)×D15order 480 = 25·3·5

### Direct product of M4(2) and D15

Series: Derived Chief Lower central Upper central

 Derived series C1 — C30 — M4(2)×D15
 Chief series C1 — C5 — C15 — C30 — C60 — C4×D15 — C2×C4×D15 — M4(2)×D15
 Lower central C15 — C30 — M4(2)×D15
 Upper central C1 — C4 — M4(2)

Generators and relations for M4(2)×D15
G = < a,b,c,d | a8=b2=c15=d2=1, bab=a5, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >

Subgroups: 692 in 136 conjugacy classes, 57 normal (41 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C5, S3, C6, C6, C8, C8, C2×C4, C2×C4, C23, D5, C10, C10, Dic3, C12, D6, C2×C6, C15, C2×C8, M4(2), M4(2), C22×C4, Dic5, C20, D10, C2×C10, C3⋊C8, C24, C4×S3, C2×Dic3, C2×C12, C22×S3, D15, D15, C30, C30, C2×M4(2), C52C8, C40, C4×D5, C2×Dic5, C2×C20, C22×D5, S3×C8, C8⋊S3, C4.Dic3, C3×M4(2), S3×C2×C4, Dic15, C60, D30, D30, C2×C30, C8×D5, C8⋊D5, C4.Dic5, C5×M4(2), C2×C4×D5, S3×M4(2), C153C8, C120, C4×D15, C2×Dic15, C2×C60, C22×D15, D5×M4(2), C8×D15, C40⋊S3, C60.7C4, C15×M4(2), C2×C4×D15, M4(2)×D15
Quotients: C1, C2, C4, C22, S3, C2×C4, C23, D5, D6, M4(2), C22×C4, D10, C4×S3, C22×S3, D15, C2×M4(2), C4×D5, C22×D5, S3×C2×C4, D30, C2×C4×D5, S3×M4(2), C4×D15, C22×D15, D5×M4(2), C2×C4×D15, M4(2)×D15

Smallest permutation representation of M4(2)×D15
On 120 points
Generators in S120
(1 91 35 76 16 106 47 62)(2 92 36 77 17 107 48 63)(3 93 37 78 18 108 49 64)(4 94 38 79 19 109 50 65)(5 95 39 80 20 110 51 66)(6 96 40 81 21 111 52 67)(7 97 41 82 22 112 53 68)(8 98 42 83 23 113 54 69)(9 99 43 84 24 114 55 70)(10 100 44 85 25 115 56 71)(11 101 45 86 26 116 57 72)(12 102 31 87 27 117 58 73)(13 103 32 88 28 118 59 74)(14 104 33 89 29 119 60 75)(15 105 34 90 30 120 46 61)
(61 90)(62 76)(63 77)(64 78)(65 79)(66 80)(67 81)(68 82)(69 83)(70 84)(71 85)(72 86)(73 87)(74 88)(75 89)(91 106)(92 107)(93 108)(94 109)(95 110)(96 111)(97 112)(98 113)(99 114)(100 115)(101 116)(102 117)(103 118)(104 119)(105 120)
(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 15)(2 14)(3 13)(4 12)(5 11)(6 10)(7 9)(16 30)(17 29)(18 28)(19 27)(20 26)(21 25)(22 24)(31 38)(32 37)(33 36)(34 35)(39 45)(40 44)(41 43)(46 47)(48 60)(49 59)(50 58)(51 57)(52 56)(53 55)(61 62)(63 75)(64 74)(65 73)(66 72)(67 71)(68 70)(76 90)(77 89)(78 88)(79 87)(80 86)(81 85)(82 84)(91 105)(92 104)(93 103)(94 102)(95 101)(96 100)(97 99)(106 120)(107 119)(108 118)(109 117)(110 116)(111 115)(112 114)

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

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

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

90 conjugacy classes

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

90 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 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 C2 C2 C4 C4 C4 S3 D5 D6 D6 M4(2) D10 D10 C4×S3 C4×S3 D15 C4×D5 C4×D5 D30 D30 C4×D15 C4×D15 S3×M4(2) D5×M4(2) M4(2)×D15 kernel M4(2)×D15 C8×D15 C40⋊S3 C60.7C4 C15×M4(2) C2×C4×D15 C4×D15 C2×Dic15 C22×D15 C5×M4(2) C3×M4(2) C40 C2×C20 D15 C24 C2×C12 C20 C2×C10 M4(2) C12 C2×C6 C8 C2×C4 C4 C22 C5 C3 C1 # reps 1 2 2 1 1 1 4 2 2 1 2 2 1 4 4 2 2 2 4 4 4 8 4 8 8 2 4 8

Matrix representation of M4(2)×D15 in GL4(𝔽241) generated by

 177 0 0 0 0 177 0 0 0 0 240 239 0 0 153 1
,
 1 0 0 0 0 1 0 0 0 0 1 0 0 0 240 240
,
 68 80 0 0 98 211 0 0 0 0 1 0 0 0 0 1
,
 240 1 0 0 0 1 0 0 0 0 1 0 0 0 0 1
G:=sub<GL(4,GF(241))| [177,0,0,0,0,177,0,0,0,0,240,153,0,0,239,1],[1,0,0,0,0,1,0,0,0,0,1,240,0,0,0,240],[68,98,0,0,80,211,0,0,0,0,1,0,0,0,0,1],[240,0,0,0,1,1,0,0,0,0,1,0,0,0,0,1] >;

M4(2)×D15 in GAP, Magma, Sage, TeX

M_4(2)\times D_{15}
% in TeX

G:=Group("M4(2)xD15");
// GroupNames label

G:=SmallGroup(480,871);
// by ID

G=gap.SmallGroup(480,871);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,219,58,80,2693,18822]);
// Polycyclic

G:=Group<a,b,c,d|a^8=b^2=c^15=d^2=1,b*a*b=a^5,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations

׿
×
𝔽