Copied to
clipboard

G = C14×He3order 378 = 2·33·7

Direct product of C14 and He3

direct product, metabelian, nilpotent (class 2), monomial, 3-elementary

Aliases: C14×He3, C322C42, C42.7C32, (C3×C6)⋊C21, (C3×C42)⋊1C3, (C3×C21)⋊12C6, C3.1(C3×C42), C6.1(C3×C21), C21.16(C3×C6), SmallGroup(378,45)

Series: Derived Chief Lower central Upper central

C1C3 — C14×He3
C1C3C21C3×C21C7×He3 — C14×He3
C1C3 — C14×He3
C1C42 — C14×He3

Generators and relations for C14×He3
 G = < a,b,c,d | a14=b3=c3=d3=1, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=bc-1, cd=dc >

3C3
3C3
3C3
3C3
3C6
3C6
3C6
3C6
3C21
3C21
3C21
3C21
3C42
3C42
3C42
3C42

Smallest permutation representation of C14×He3
On 126 points
Generators in S126
(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)(113 114 115 116 117 118 119 120 121 122 123 124 125 126)
(15 60 117)(16 61 118)(17 62 119)(18 63 120)(19 64 121)(20 65 122)(21 66 123)(22 67 124)(23 68 125)(24 69 126)(25 70 113)(26 57 114)(27 58 115)(28 59 116)(29 86 79)(30 87 80)(31 88 81)(32 89 82)(33 90 83)(34 91 84)(35 92 71)(36 93 72)(37 94 73)(38 95 74)(39 96 75)(40 97 76)(41 98 77)(42 85 78)
(1 105 52)(2 106 53)(3 107 54)(4 108 55)(5 109 56)(6 110 43)(7 111 44)(8 112 45)(9 99 46)(10 100 47)(11 101 48)(12 102 49)(13 103 50)(14 104 51)(15 60 117)(16 61 118)(17 62 119)(18 63 120)(19 64 121)(20 65 122)(21 66 123)(22 67 124)(23 68 125)(24 69 126)(25 70 113)(26 57 114)(27 58 115)(28 59 116)(29 79 86)(30 80 87)(31 81 88)(32 82 89)(33 83 90)(34 84 91)(35 71 92)(36 72 93)(37 73 94)(38 74 95)(39 75 96)(40 76 97)(41 77 98)(42 78 85)
(1 96 116)(2 97 117)(3 98 118)(4 85 119)(5 86 120)(6 87 121)(7 88 122)(8 89 123)(9 90 124)(10 91 125)(11 92 126)(12 93 113)(13 94 114)(14 95 115)(15 106 40)(16 107 41)(17 108 42)(18 109 29)(19 110 30)(20 111 31)(21 112 32)(22 99 33)(23 100 34)(24 101 35)(25 102 36)(26 103 37)(27 104 38)(28 105 39)(43 80 64)(44 81 65)(45 82 66)(46 83 67)(47 84 68)(48 71 69)(49 72 70)(50 73 57)(51 74 58)(52 75 59)(53 76 60)(54 77 61)(55 78 62)(56 79 63)

G:=sub<Sym(126)| (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)(113,114,115,116,117,118,119,120,121,122,123,124,125,126), (15,60,117)(16,61,118)(17,62,119)(18,63,120)(19,64,121)(20,65,122)(21,66,123)(22,67,124)(23,68,125)(24,69,126)(25,70,113)(26,57,114)(27,58,115)(28,59,116)(29,86,79)(30,87,80)(31,88,81)(32,89,82)(33,90,83)(34,91,84)(35,92,71)(36,93,72)(37,94,73)(38,95,74)(39,96,75)(40,97,76)(41,98,77)(42,85,78), (1,105,52)(2,106,53)(3,107,54)(4,108,55)(5,109,56)(6,110,43)(7,111,44)(8,112,45)(9,99,46)(10,100,47)(11,101,48)(12,102,49)(13,103,50)(14,104,51)(15,60,117)(16,61,118)(17,62,119)(18,63,120)(19,64,121)(20,65,122)(21,66,123)(22,67,124)(23,68,125)(24,69,126)(25,70,113)(26,57,114)(27,58,115)(28,59,116)(29,79,86)(30,80,87)(31,81,88)(32,82,89)(33,83,90)(34,84,91)(35,71,92)(36,72,93)(37,73,94)(38,74,95)(39,75,96)(40,76,97)(41,77,98)(42,78,85), (1,96,116)(2,97,117)(3,98,118)(4,85,119)(5,86,120)(6,87,121)(7,88,122)(8,89,123)(9,90,124)(10,91,125)(11,92,126)(12,93,113)(13,94,114)(14,95,115)(15,106,40)(16,107,41)(17,108,42)(18,109,29)(19,110,30)(20,111,31)(21,112,32)(22,99,33)(23,100,34)(24,101,35)(25,102,36)(26,103,37)(27,104,38)(28,105,39)(43,80,64)(44,81,65)(45,82,66)(46,83,67)(47,84,68)(48,71,69)(49,72,70)(50,73,57)(51,74,58)(52,75,59)(53,76,60)(54,77,61)(55,78,62)(56,79,63)>;

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)(113,114,115,116,117,118,119,120,121,122,123,124,125,126), (15,60,117)(16,61,118)(17,62,119)(18,63,120)(19,64,121)(20,65,122)(21,66,123)(22,67,124)(23,68,125)(24,69,126)(25,70,113)(26,57,114)(27,58,115)(28,59,116)(29,86,79)(30,87,80)(31,88,81)(32,89,82)(33,90,83)(34,91,84)(35,92,71)(36,93,72)(37,94,73)(38,95,74)(39,96,75)(40,97,76)(41,98,77)(42,85,78), (1,105,52)(2,106,53)(3,107,54)(4,108,55)(5,109,56)(6,110,43)(7,111,44)(8,112,45)(9,99,46)(10,100,47)(11,101,48)(12,102,49)(13,103,50)(14,104,51)(15,60,117)(16,61,118)(17,62,119)(18,63,120)(19,64,121)(20,65,122)(21,66,123)(22,67,124)(23,68,125)(24,69,126)(25,70,113)(26,57,114)(27,58,115)(28,59,116)(29,79,86)(30,80,87)(31,81,88)(32,82,89)(33,83,90)(34,84,91)(35,71,92)(36,72,93)(37,73,94)(38,74,95)(39,75,96)(40,76,97)(41,77,98)(42,78,85), (1,96,116)(2,97,117)(3,98,118)(4,85,119)(5,86,120)(6,87,121)(7,88,122)(8,89,123)(9,90,124)(10,91,125)(11,92,126)(12,93,113)(13,94,114)(14,95,115)(15,106,40)(16,107,41)(17,108,42)(18,109,29)(19,110,30)(20,111,31)(21,112,32)(22,99,33)(23,100,34)(24,101,35)(25,102,36)(26,103,37)(27,104,38)(28,105,39)(43,80,64)(44,81,65)(45,82,66)(46,83,67)(47,84,68)(48,71,69)(49,72,70)(50,73,57)(51,74,58)(52,75,59)(53,76,60)(54,77,61)(55,78,62)(56,79,63) );

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),(113,114,115,116,117,118,119,120,121,122,123,124,125,126)], [(15,60,117),(16,61,118),(17,62,119),(18,63,120),(19,64,121),(20,65,122),(21,66,123),(22,67,124),(23,68,125),(24,69,126),(25,70,113),(26,57,114),(27,58,115),(28,59,116),(29,86,79),(30,87,80),(31,88,81),(32,89,82),(33,90,83),(34,91,84),(35,92,71),(36,93,72),(37,94,73),(38,95,74),(39,96,75),(40,97,76),(41,98,77),(42,85,78)], [(1,105,52),(2,106,53),(3,107,54),(4,108,55),(5,109,56),(6,110,43),(7,111,44),(8,112,45),(9,99,46),(10,100,47),(11,101,48),(12,102,49),(13,103,50),(14,104,51),(15,60,117),(16,61,118),(17,62,119),(18,63,120),(19,64,121),(20,65,122),(21,66,123),(22,67,124),(23,68,125),(24,69,126),(25,70,113),(26,57,114),(27,58,115),(28,59,116),(29,79,86),(30,80,87),(31,81,88),(32,82,89),(33,83,90),(34,84,91),(35,71,92),(36,72,93),(37,73,94),(38,74,95),(39,75,96),(40,76,97),(41,77,98),(42,78,85)], [(1,96,116),(2,97,117),(3,98,118),(4,85,119),(5,86,120),(6,87,121),(7,88,122),(8,89,123),(9,90,124),(10,91,125),(11,92,126),(12,93,113),(13,94,114),(14,95,115),(15,106,40),(16,107,41),(17,108,42),(18,109,29),(19,110,30),(20,111,31),(21,112,32),(22,99,33),(23,100,34),(24,101,35),(25,102,36),(26,103,37),(27,104,38),(28,105,39),(43,80,64),(44,81,65),(45,82,66),(46,83,67),(47,84,68),(48,71,69),(49,72,70),(50,73,57),(51,74,58),(52,75,59),(53,76,60),(54,77,61),(55,78,62),(56,79,63)]])

154 conjugacy classes

class 1  2 3A3B3C···3J6A6B6C···6J7A···7F14A···14F21A···21L21M···21BH42A···42L42M···42BH
order12333···3666···67···714···1421···2121···2142···4242···42
size11113···3113···31···11···11···13···31···13···3

154 irreducible representations

dim111111113333
type++
imageC1C2C3C6C7C14C21C42He3C2×He3C7×He3C14×He3
kernelC14×He3C7×He3C3×C42C3×C21C2×He3He3C3×C6C32C14C7C2C1
# reps1188664848221212

Matrix representation of C14×He3 in GL4(𝔽43) generated by

42000
01100
00110
00011
,
6000
0100
00360
0006
,
1000
03600
00360
00036
,
36000
0010
0001
0100
G:=sub<GL(4,GF(43))| [42,0,0,0,0,11,0,0,0,0,11,0,0,0,0,11],[6,0,0,0,0,1,0,0,0,0,36,0,0,0,0,6],[1,0,0,0,0,36,0,0,0,0,36,0,0,0,0,36],[36,0,0,0,0,0,0,1,0,1,0,0,0,0,1,0] >;

C14×He3 in GAP, Magma, Sage, TeX

C_{14}\times {\rm He}_3
% in TeX

G:=Group("C14xHe3");
// GroupNames label

G:=SmallGroup(378,45);
// by ID

G=gap.SmallGroup(378,45);
# by ID

G:=PCGroup([5,-2,-3,-3,-7,-3,997]);
// Polycyclic

G:=Group<a,b,c,d|a^14=b^3=c^3=d^3=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b*c^-1,c*d=d*c>;
// generators/relations

Export

Subgroup lattice of C14×He3 in TeX

׿
×
𝔽