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G = D152C8order 240 = 24·3·5

The semidirect product of D15 and C8 acting via C8/C4=C2

metabelian, supersoluble, monomial, A-group, 2-hyperelementary

Aliases: D152C8, D30.4C4, C20.31D6, C12.31D10, C60.31C22, Dic15.4C4, C53(S3×C8), C3⋊C86D5, C31(C8×D5), C157(C2×C8), C52C86S3, C6.1(C4×D5), C10.8(C4×S3), C4.24(S3×D5), C30.24(C2×C4), (C4×D15).4C2, C2.1(D30.C2), (C5×C3⋊C8)⋊5C2, (C3×C52C8)⋊5C2, SmallGroup(240,9)

Series: Derived Chief Lower central Upper central

C1C15 — D152C8
C1C5C15C30C60C3×C52C8 — D152C8
C15 — D152C8
C1C4

Generators and relations for D152C8
 G = < a,b,c | a15=b2=c8=1, bab=a-1, cac-1=a11, cbc-1=a10b >

15C2
15C2
15C4
15C22
5S3
5S3
3D5
3D5
3C8
5C8
15C2×C4
5Dic3
5D6
3Dic5
3D10
15C2×C8
5C24
5C4×S3
3C40
3C4×D5
5S3×C8
3C8×D5

Smallest permutation representation of D152C8
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 19)(2 18)(3 17)(4 16)(5 30)(6 29)(7 28)(8 27)(9 26)(10 25)(11 24)(12 23)(13 22)(14 21)(15 20)(31 46)(32 60)(33 59)(34 58)(35 57)(36 56)(37 55)(38 54)(39 53)(40 52)(41 51)(42 50)(43 49)(44 48)(45 47)(61 79)(62 78)(63 77)(64 76)(65 90)(66 89)(67 88)(68 87)(69 86)(70 85)(71 84)(72 83)(73 82)(74 81)(75 80)(91 107)(92 106)(93 120)(94 119)(95 118)(96 117)(97 116)(98 115)(99 114)(100 113)(101 112)(102 111)(103 110)(104 109)(105 108)
(1 115 58 79 20 99 35 62)(2 111 59 90 21 95 36 73)(3 107 60 86 22 91 37 69)(4 118 46 82 23 102 38 65)(5 114 47 78 24 98 39 61)(6 110 48 89 25 94 40 72)(7 106 49 85 26 105 41 68)(8 117 50 81 27 101 42 64)(9 113 51 77 28 97 43 75)(10 109 52 88 29 93 44 71)(11 120 53 84 30 104 45 67)(12 116 54 80 16 100 31 63)(13 112 55 76 17 96 32 74)(14 108 56 87 18 92 33 70)(15 119 57 83 19 103 34 66)

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

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

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

D152C8 is a maximal subgroup of
D15⋊C16  D30.C8  S3×C8×D5  C40⋊D6  C40.34D6  C40.55D6  D60.5C4  D60.4C4  D154M4(2)  D15⋊D8  Dic10⋊D6  D20.10D6  D30.11D4  D15⋊SD16  D15⋊Q16  D20.16D6  D12.D10
D152C8 is a maximal quotient of
D152C16  D30.5C8  Dic154C8  D304C8  C60.14Q8

48 conjugacy classes

class 1 2A2B2C 3 4A4B4C4D5A5B 6 8A8B8C8D8E8F8G8H10A10B12A12B15A15B20A20B20C20D24A24B24C24D30A30B40A···40H60A60B60C60D
order122234444556888888881010121215152020202024242424303040···4060606060
size111515211151522233335555222244222210101010446···64444

48 irreducible representations

dim111111122222222444
type++++++++++
imageC1C2C2C2C4C4C8S3D5D6D10C4×S3C4×D5S3×C8C8×D5S3×D5D30.C2D152C8
kernelD152C8C5×C3⋊C8C3×C52C8C4×D15Dic15D30D15C52C8C3⋊C8C20C12C10C6C5C3C4C2C1
# reps111122812122448224

Matrix representation of D152C8 in GL4(𝔽241) generated by

1905100
19024000
0023949
001771
,
0100
1000
002192
0064239
,
1000
0100
00300
00233211
G:=sub<GL(4,GF(241))| [190,190,0,0,51,240,0,0,0,0,239,177,0,0,49,1],[0,1,0,0,1,0,0,0,0,0,2,64,0,0,192,239],[1,0,0,0,0,1,0,0,0,0,30,233,0,0,0,211] >;

D152C8 in GAP, Magma, Sage, TeX

D_{15}\rtimes_2C_8
% in TeX

G:=Group("D15:2C8");
// GroupNames label

G:=SmallGroup(240,9);
// by ID

G=gap.SmallGroup(240,9);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-3,-5,24,31,50,490,6917]);
// Polycyclic

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

Export

Subgroup lattice of D152C8 in TeX

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