Copied to
clipboard

## G = C5×Q8⋊3D6order 480 = 25·3·5

### Direct product of C5 and Q8⋊3D6

Series: Derived Chief Lower central Upper central

 Derived series C1 — C12 — C5×Q8⋊3D6
 Chief series C1 — C3 — C6 — C12 — C60 — S3×C20 — C5×S3×D4 — C5×Q8⋊3D6
 Lower central C3 — C6 — C12 — C5×Q8⋊3D6
 Upper central C1 — C10 — C20 — C5×SD16

Generators and relations for C5×Q83D6
G = < a,b,c,d,e | a5=b4=d6=e2=1, c2=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=dbd-1=ebe=b-1, dcd-1=b-1c, ece=bc, ede=d-1 >

Subgroups: 404 in 136 conjugacy classes, 54 normal (all characteristic)
C1, C2, C2 [×4], C3, C4, C4 [×2], C22 [×6], C5, S3 [×3], C6, C6, C8, C8, C2×C4 [×2], D4, D4 [×4], Q8, C23, C10, C10 [×4], Dic3, C12, C12, D6, D6 [×4], C2×C6, C15, M4(2), D8 [×2], SD16, SD16, C2×D4, C4○D4, C20, C20 [×2], C2×C10 [×6], C3⋊C8, C24, C4×S3, C4×S3, D12 [×2], D12, C3⋊D4, C3×D4, C3×Q8, C22×S3, C5×S3 [×3], C30, C30, C8⋊C22, C40, C40, C2×C20 [×2], C5×D4, C5×D4 [×4], C5×Q8, C22×C10, C8⋊S3, D24, D4⋊S3, Q82S3, C3×SD16, S3×D4, Q83S3, C5×Dic3, C60, C60, S3×C10, S3×C10 [×4], C2×C30, C5×M4(2), C5×D8 [×2], C5×SD16, C5×SD16, D4×C10, C5×C4○D4, Q83D6, C5×C3⋊C8, C120, S3×C20, S3×C20, C5×D12 [×2], C5×D12, C5×C3⋊D4, D4×C15, Q8×C15, S3×C2×C10, C5×C8⋊C22, C5×C8⋊S3, C5×D24, C5×D4⋊S3, C5×Q82S3, C15×SD16, C5×S3×D4, C5×Q83S3, C5×Q83D6
Quotients: C1, C2 [×7], C22 [×7], C5, S3, D4 [×2], C23, C10 [×7], D6 [×3], C2×D4, C2×C10 [×7], C22×S3, C5×S3, C8⋊C22, C5×D4 [×2], C22×C10, S3×D4, S3×C10 [×3], D4×C10, Q83D6, S3×C2×C10, C5×C8⋊C22, C5×S3×D4, C5×Q83D6

Smallest permutation representation of C5×Q83D6
On 120 points
Generators in S120
(1 19 10 6 23)(2 20 11 4 24)(3 21 12 5 22)(7 13 27 17 28)(8 14 25 18 29)(9 15 26 16 30)(31 66 87 104 69)(32 61 88 105 70)(33 62 89 106 71)(34 63 90 107 72)(35 64 85 108 67)(36 65 86 103 68)(37 50 118 94 98)(38 51 119 95 99)(39 52 120 96 100)(40 53 115 91 101)(41 54 116 92 102)(42 49 117 93 97)(43 83 111 73 55)(44 84 112 74 56)(45 79 113 75 57)(46 80 114 76 58)(47 81 109 77 59)(48 82 110 78 60)
(1 114 30 111)(2 112 28 109)(3 110 29 113)(4 44 27 47)(5 48 25 45)(6 46 26 43)(7 77 20 74)(8 75 21 78)(9 73 19 76)(10 58 15 55)(11 56 13 59)(12 60 14 57)(16 83 23 80)(17 81 24 84)(18 79 22 82)(31 34 101 98)(32 99 102 35)(33 36 97 100)(37 66 63 40)(38 41 64 61)(39 62 65 42)(49 52 89 86)(50 87 90 53)(51 54 85 88)(67 70 95 92)(68 93 96 71)(69 72 91 94)(103 117 120 106)(104 107 115 118)(105 119 116 108)
(1 106 30 117)(2 104 28 115)(3 108 29 119)(4 66 27 40)(5 64 25 38)(6 62 26 42)(7 91 20 69)(8 95 21 67)(9 93 19 71)(10 33 15 97)(11 31 13 101)(12 35 14 99)(16 49 23 89)(17 53 24 87)(18 51 22 85)(32 60 102 57)(34 56 98 59)(36 58 100 55)(37 47 63 44)(39 43 65 46)(41 45 61 48)(50 81 90 84)(52 83 86 80)(54 79 88 82)(68 76 96 73)(70 78 92 75)(72 74 94 77)(103 114 120 111)(105 110 116 113)(107 112 118 109)
(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 28)(2 30)(3 29)(4 26)(5 25)(6 27)(7 19)(8 21)(9 20)(10 13)(11 15)(12 14)(16 24)(17 23)(18 22)(31 36)(32 35)(33 34)(37 42)(38 41)(39 40)(43 47)(44 46)(49 50)(51 54)(52 53)(55 59)(56 58)(61 64)(62 63)(65 66)(67 70)(68 69)(71 72)(73 77)(74 76)(80 84)(81 83)(85 88)(86 87)(89 90)(91 96)(92 95)(93 94)(97 98)(99 102)(100 101)(103 104)(105 108)(106 107)(109 111)(112 114)(115 120)(116 119)(117 118)

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

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

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

90 conjugacy classes

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

90 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 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 C5 C10 C10 C10 C10 C10 C10 C10 S3 D4 D4 D6 D6 D6 C5×S3 C5×D4 C5×D4 S3×C10 S3×C10 S3×C10 C8⋊C22 S3×D4 Q8⋊3D6 C5×C8⋊C22 C5×S3×D4 C5×Q8⋊3D6 kernel C5×Q8⋊3D6 C5×C8⋊S3 C5×D24 C5×D4⋊S3 C5×Q8⋊2S3 C15×SD16 C5×S3×D4 C5×Q8⋊3S3 Q8⋊3D6 C8⋊S3 D24 D4⋊S3 Q8⋊2S3 C3×SD16 S3×D4 Q8⋊3S3 C5×SD16 C5×Dic3 S3×C10 C40 C5×D4 C5×Q8 SD16 Dic3 D6 C8 D4 Q8 C15 C10 C5 C3 C2 C1 # reps 1 1 1 1 1 1 1 1 4 4 4 4 4 4 4 4 1 1 1 1 1 1 4 4 4 4 4 4 1 1 2 4 4 8

Matrix representation of C5×Q83D6 in GL4(𝔽241) generated by

 205 0 0 0 0 205 0 0 0 0 205 0 0 0 0 205
,
 240 0 240 0 0 240 0 240 2 0 1 0 0 2 0 1
,
 0 0 116 9 0 0 232 125 232 18 0 0 223 9 0 0
,
 240 1 0 0 240 0 0 0 2 239 1 240 2 0 1 0
,
 0 1 0 0 1 0 0 0 0 239 0 240 239 0 240 0
G:=sub<GL(4,GF(241))| [205,0,0,0,0,205,0,0,0,0,205,0,0,0,0,205],[240,0,2,0,0,240,0,2,240,0,1,0,0,240,0,1],[0,0,232,223,0,0,18,9,116,232,0,0,9,125,0,0],[240,240,2,2,1,0,239,0,0,0,1,1,0,0,240,0],[0,1,0,239,1,0,239,0,0,0,0,240,0,0,240,0] >;

C5×Q83D6 in GAP, Magma, Sage, TeX

C_5\times Q_8\rtimes_3D_6
% in TeX

G:=Group("C5xQ8:3D6");
// GroupNames label

G:=SmallGroup(480,793);
// by ID

G=gap.SmallGroup(480,793);
# by ID

G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-3,1766,471,436,2111,1068,102,15686]);
// Polycyclic

G:=Group<a,b,c,d,e|a^5=b^4=d^6=e^2=1,c^2=b^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c^-1=d*b*d^-1=e*b*e=b^-1,d*c*d^-1=b^-1*c,e*c*e=b*c,e*d*e=d^-1>;
// generators/relations

׿
×
𝔽