Copied to
clipboard

G = D10.4D12order 480 = 25·3·5

4th non-split extension by D10 of D12 acting via D12/C6=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D10.4D12, (C2×Dic3)⋊F5, C22⋊F52S3, C152(C23⋊C4), (C6×D5).29D4, C22.4(S3×F5), (C10×Dic3)⋊3C4, (C22×D15)⋊3C4, C2.13(D6⋊F5), C10.13(D6⋊C4), D10.4(C3⋊D4), C31(D10.D4), (C22×D5).35D6, C51(C23.6D6), C6.13(C22⋊F5), D10.D62C2, C30.13(C22⋊C4), (C2×C6).2(C2×F5), (C2×C10).9(C4×S3), (C2×C30).7(C2×C4), (C3×C22⋊F5)⋊2C2, (D5×C2×C6).65C22, (C2×C3⋊D20).10C2, SmallGroup(480,249)

Series: Derived Chief Lower central Upper central

C1C2×C30 — D10.4D12
C1C5C15C30C6×D5D5×C2×C6C3×C22⋊F5 — D10.4D12
C15C30C2×C30 — D10.4D12
C1C2C22

Generators and relations for D10.4D12
 G = < a,b,c,d | a10=b2=c12=1, d2=a4b, bab=a-1, cac-1=dad-1=a3, cbc-1=a7b, dbd-1=a2b, dcd-1=a-1bc-1 >

Subgroups: 836 in 104 conjugacy classes, 26 normal (all characteristic)
C1, C2, C2 [×4], C3, C4 [×3], C22, C22 [×5], C5, S3, C6, C6 [×3], C2×C4 [×3], D4 [×2], C23 [×2], D5 [×3], C10, C10, Dic3 [×2], C12, D6 [×2], C2×C6, C2×C6 [×3], C15, C22⋊C4 [×2], C2×D4, C20, F5 [×2], D10 [×2], D10 [×3], C2×C10, C2×Dic3, C2×Dic3, C3⋊D4 [×2], C2×C12, C22×S3, C22×C6, C3×D5 [×2], D15, C30, C30, C23⋊C4, D20 [×2], C2×C20, C2×F5 [×2], C22×D5, C22×D5, C6.D4, C3×C22⋊C4, C2×C3⋊D4, C5×Dic3, C3×F5, C3⋊F5, C6×D5 [×2], C6×D5, D30 [×2], C2×C30, C22⋊F5, C22⋊F5, C2×D20, C23.6D6, C3⋊D20 [×2], C10×Dic3, C6×F5, C2×C3⋊F5, D5×C2×C6, C22×D15, D10.D4, C3×C22⋊F5, D10.D6, C2×C3⋊D20, D10.4D12
Quotients: C1, C2 [×3], C4 [×2], C22, S3, C2×C4, D4 [×2], D6, C22⋊C4, F5, C4×S3, D12, C3⋊D4, C23⋊C4, C2×F5, D6⋊C4, C22⋊F5, C23.6D6, S3×F5, D10.D4, D6⋊F5, D10.4D12

Smallest permutation representation of D10.4D12
On 120 points
Generators in S120
(1 54 99 57 108 5 102 51 105 60)(2 58 103 49 100 6 106 55 97 52)(3 50 107 53 104 4 98 59 101 56)(7 14 23 30 33 10 27 36 17 20)(8 31 28 21 24 11 18 15 34 25)(9 22 19 26 29 12 35 32 13 16)(37 43 109 77 64 88 94 70 83 115)(38 78 95 116 110 89 84 44 65 71)(39 117 73 72 96 90 66 79 111 45)(40 61 67 46 74 91 112 118 85 80)(41 47 113 81 68 92 86 62 75 119)(42 82 87 120 114 93 76 48 69 63)
(1 33)(2 25)(3 29)(4 16)(5 20)(6 24)(7 108)(8 52)(9 104)(10 60)(11 100)(12 56)(13 98)(14 57)(15 103)(17 102)(18 49)(19 107)(21 106)(22 53)(23 99)(26 50)(27 105)(28 55)(30 54)(31 97)(32 59)(34 58)(35 101)(36 51)(37 61)(38 75)(39 48)(40 43)(41 65)(42 79)(44 47)(45 69)(46 83)(62 78)(63 111)(64 118)(66 82)(67 115)(68 110)(70 74)(71 119)(72 114)(73 93)(76 117)(77 85)(80 109)(81 89)(84 113)(86 95)(87 90)(88 112)(91 94)(92 116)(96 120)
(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 79 7 76)(2 75 11 84)(3 83 9 80)(4 77 12 74)(5 73 10 82)(6 81 8 78)(13 91 107 109)(14 120 60 39)(15 38 97 68)(16 67 50 88)(17 87 99 117)(18 116 52 47)(19 46 101 64)(20 63 54 96)(21 95 103 113)(22 112 56 43)(23 42 105 72)(24 71 58 92)(25 110 106 41)(26 40 59 70)(27 69 108 90)(28 89 49 119)(29 118 98 37)(30 48 51 66)(31 65 100 86)(32 85 53 115)(33 114 102 45)(34 44 55 62)(35 61 104 94)(36 93 57 111)

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

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

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

33 conjugacy classes

class 1 2A2B2C2D2E 3 4A4B4C4D4E 5 6A6B6C6D6E10A10B10C12A12B12C12D 15 20A20B20C20D30A30B30C
order122222344444566666101010121212121520202020303030
size1121010602122020606042410102044420202020812121212888

33 irreducible representations

dim111111222222444444888
type++++++++++++++++
imageC1C2C2C2C4C4S3D4D6D12C3⋊D4C4×S3F5C23⋊C4C2×F5C22⋊F5C23.6D6D10.D4S3×F5D6⋊F5D10.4D12
kernelD10.4D12C3×C22⋊F5D10.D6C2×C3⋊D20C10×Dic3C22×D15C22⋊F5C6×D5C22×D5D10D10C2×C10C2×Dic3C15C2×C6C6C5C3C22C2C1
# reps111122121222111224112

Matrix representation of D10.4D12 in GL8(𝔽61)

600000000
060000000
006000000
000600000
000000601
000000600
000010600
000001600
,
060000000
600000000
00010000
00100000
000006010
000060010
00000010
000000160
,
130000000
048000000
000140000
004700000
00000010
00001000
00000001
00000100
,
000140000
004700000
480000000
013000000
00001447540
0000747014
0000140477
00000544714

G:=sub<GL(8,GF(61))| [60,0,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,60,60,60,60,0,0,0,0,1,0,0,0],[0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,0,1,1,1,1,0,0,0,0,0,0,0,60],[13,0,0,0,0,0,0,0,0,48,0,0,0,0,0,0,0,0,0,47,0,0,0,0,0,0,14,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0],[0,0,48,0,0,0,0,0,0,0,0,13,0,0,0,0,0,47,0,0,0,0,0,0,14,0,0,0,0,0,0,0,0,0,0,0,14,7,14,0,0,0,0,0,47,47,0,54,0,0,0,0,54,0,47,47,0,0,0,0,0,14,7,14] >;

D10.4D12 in GAP, Magma, Sage, TeX

D_{10}._4D_{12}
% in TeX

G:=Group("D10.4D12");
// GroupNames label

G:=SmallGroup(480,249);
// by ID

G=gap.SmallGroup(480,249);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,28,141,120,219,675,1356,9414,4724]);
// Polycyclic

G:=Group<a,b,c,d|a^10=b^2=c^12=1,d^2=a^4*b,b*a*b=a^-1,c*a*c^-1=d*a*d^-1=a^3,c*b*c^-1=a^7*b,d*b*d^-1=a^2*b,d*c*d^-1=a^-1*b*c^-1>;
// generators/relations

׿
×
𝔽