Copied to
clipboard

G = D154M4(2)  order 480 = 25·3·5

The semidirect product of D15 and M4(2) acting via M4(2)/C2×C4=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D154M4(2), C60.185C23, C3⋊C822D10, C52C822D6, C57(S3×M4(2)), C33(D5×M4(2)), C20.54(C4×S3), (C2×C20).82D6, C12.22(C4×D5), C4.Dic56S3, C4.Dic36D5, C60.131(C2×C4), D30.37(C2×C4), (C4×D15).10C4, (C2×C12).83D10, C1517(C2×M4(2)), D152C813C2, D30.5C414C2, C30.107(C22×C4), (C2×C60).215C22, C20.182(C22×S3), C4.14(D30.C2), (C2×Dic15).22C4, Dic15.45(C2×C4), (C22×D15).13C4, (C4×D15).65C22, C12.182(C22×D5), C22.6(D30.C2), C6.41(C2×C4×D5), C10.74(S3×C2×C4), C4.155(C2×S3×D5), (C5×C3⋊C8)⋊22C22, (C2×C4×D15).18C2, (C2×C6).10(C4×D5), (C2×C10).33(C4×S3), (C2×C4).195(S3×D5), C2.6(C2×D30.C2), (C2×C30).104(C2×C4), (C3×C52C8)⋊22C22, (C5×C4.Dic3)⋊12C2, (C3×C4.Dic5)⋊12C2, SmallGroup(480,368)

Series: Derived Chief Lower central Upper central

C1C30 — D154M4(2)
C1C5C15C30C60C3×C52C8D152C8 — D154M4(2)
C15C30 — D154M4(2)
C1C4C2×C4

Generators and relations for D154M4(2)
 G = < a,b,c,d | a15=b2=c8=d2=1, bab=a-1, cac-1=a11, ad=da, cbc-1=a10b, bd=db, dcd=c5 >

Subgroups: 668 in 136 conjugacy classes, 54 normal (40 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C5, S3, C6, C6, C8, C2×C4, C2×C4, C23, D5, C10, C10, Dic3, C12, D6, C2×C6, C15, C2×C8, 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), C5×C3⋊C8, C3×C52C8, C4×D15, C2×Dic15, C2×C60, C22×D15, D5×M4(2), D152C8, D30.5C4, C3×C4.Dic5, C5×C4.Dic3, C2×C4×D15, D154M4(2)
Quotients: C1, C2, C4, C22, S3, C2×C4, C23, D5, D6, M4(2), C22×C4, D10, C4×S3, C22×S3, C2×M4(2), C4×D5, C22×D5, S3×C2×C4, S3×D5, C2×C4×D5, S3×M4(2), D30.C2, C2×S3×D5, D5×M4(2), C2×D30.C2, D154M4(2)

Smallest permutation representation of D154M4(2)
On 120 points
Generators in S120
(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 23)(17 22)(18 21)(19 20)(24 30)(25 29)(26 28)(31 32)(33 45)(34 44)(35 43)(36 42)(37 41)(38 40)(46 54)(47 53)(48 52)(49 51)(55 60)(56 59)(57 58)(61 72)(62 71)(63 70)(64 69)(65 68)(66 67)(73 75)(76 83)(77 82)(78 81)(79 80)(84 90)(85 89)(86 88)(91 100)(92 99)(93 98)(94 97)(95 96)(101 105)(102 104)(106 117)(107 116)(108 115)(109 114)(110 113)(111 112)(118 120)
(1 112 32 67 20 96 58 80)(2 108 33 63 21 92 59 76)(3 119 34 74 22 103 60 87)(4 115 35 70 23 99 46 83)(5 111 36 66 24 95 47 79)(6 107 37 62 25 91 48 90)(7 118 38 73 26 102 49 86)(8 114 39 69 27 98 50 82)(9 110 40 65 28 94 51 78)(10 106 41 61 29 105 52 89)(11 117 42 72 30 101 53 85)(12 113 43 68 16 97 54 81)(13 109 44 64 17 93 55 77)(14 120 45 75 18 104 56 88)(15 116 31 71 19 100 57 84)
(1 20)(2 21)(3 22)(4 23)(5 24)(6 25)(7 26)(8 27)(9 28)(10 29)(11 30)(12 16)(13 17)(14 18)(15 19)(31 57)(32 58)(33 59)(34 60)(35 46)(36 47)(37 48)(38 49)(39 50)(40 51)(41 52)(42 53)(43 54)(44 55)(45 56)

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

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

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

66 conjugacy classes

class 1 2A2B2C2D2E 3 4A4B4C4D4E4F5A5B6A6B8A8B8C8D8E8F8G8H10A10B10C10D12A12B12C15A15B20A20B20C20D20E20F24A24B24C24D30A···30F40A···40H60A···60H
order12222234444445566888888881010101012121215152020202020202424242430···3040···4060···60
size11215153021121515302224666610101010224422444222244202020204···412···124···4

66 irreducible representations

dim111111111222222222224444444
type++++++++++++++++
imageC1C2C2C2C2C2C4C4C4S3D5D6D6M4(2)D10D10C4×S3C4×S3C4×D5C4×D5S3×D5S3×M4(2)D30.C2C2×S3×D5D30.C2D5×M4(2)D154M4(2)
kernelD154M4(2)D152C8D30.5C4C3×C4.Dic5C5×C4.Dic3C2×C4×D15C4×D15C2×Dic15C22×D15C4.Dic5C4.Dic3C52C8C2×C20D15C3⋊C8C2×C12C20C2×C10C12C2×C6C2×C4C5C4C4C22C3C1
# reps122111422122144222442222248

Matrix representation of D154M4(2) in GL6(𝔽241)

0510000
1891890000
00240100
00240000
000010
000001
,
178840000
79630000
00240000
00240100
000010
000001
,
6400000
0640000
00017700
00177000
00000240
0000640
,
24000000
02400000
001000
000100
00002400
000001

G:=sub<GL(6,GF(241))| [0,189,0,0,0,0,51,189,0,0,0,0,0,0,240,240,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[178,79,0,0,0,0,84,63,0,0,0,0,0,0,240,240,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[64,0,0,0,0,0,0,64,0,0,0,0,0,0,0,177,0,0,0,0,177,0,0,0,0,0,0,0,0,64,0,0,0,0,240,0],[240,0,0,0,0,0,0,240,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,240,0,0,0,0,0,0,1] >;

D154M4(2) in GAP, Magma, Sage, TeX

D_{15}\rtimes_4M_4(2)
% in TeX

G:=Group("D15:4M4(2)");
// GroupNames label

G:=SmallGroup(480,368);
// by ID

G=gap.SmallGroup(480,368);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,56,64,422,219,80,1356,18822]);
// Polycyclic

G:=Group<a,b,c,d|a^15=b^2=c^8=d^2=1,b*a*b=a^-1,c*a*c^-1=a^11,a*d=d*a,c*b*c^-1=a^10*b,b*d=d*b,d*c*d=c^5>;
// generators/relations

׿
×
𝔽