direct product, metabelian, supersoluble, monomial, 3-hyperelementary
Aliases: C2×C7⋊He3, C14⋊He3, C42.6C32, C7⋊2(C2×He3), (C3×C42)⋊2C3, (C3×C21)⋊13C6, C21.12(C3×C6), (C6×C7⋊C3)⋊C3, (C3×C6)⋊(C7⋊C3), (C3×C7⋊C3)⋊2C6, C6.6(C3×C7⋊C3), C3.6(C6×C7⋊C3), C32⋊2(C2×C7⋊C3), SmallGroup(378,28)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×C7⋊He3
G = < a,b,c,d,e | a2=b7=c3=d3=e3=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, ebe-1=b4, cd=dc, ece-1=cd-1, de=ed >
Smallest permutation representation of C2×C7⋊He3
►On 126 points
Generators in S126
(1 64)(2 65)(3 66)(4 67)(5 68)(6 69)(7 70)(8 71)(9 72)(10 73)(11 74)(12 75)(13 76)(14 77)(15 78)(16 79)(17 80)(18 81)(19 82)(20 83)(21 84)(22 85)(23 86)(24 87)(25 88)(26 89)(27 90)(28 91)(29 92)(30 93)(31 94)(32 95)(33 96)(34 97)(35 98)(36 99)(37 100)(38 101)(39 102)(40 103)(41 104)(42 105)(43 106)(44 107)(45 108)(46 109)(47 110)(48 111)(49 112)(50 113)(51 114)(52 115)(53 116)(54 117)(55 118)(56 119)(57 120)(58 121)(59 122)(60 123)(61 124)(62 125)(63 126)
(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)
(1 43 22)(2 44 23)(3 45 24)(4 46 25)(5 47 26)(6 48 27)(7 49 28)(8 50 29)(9 51 30)(10 52 31)(11 53 32)(12 54 33)(13 55 34)(14 56 35)(15 57 36)(16 58 37)(17 59 38)(18 60 39)(19 61 40)(20 62 41)(21 63 42)(64 106 85)(65 107 86)(66 108 87)(67 109 88)(68 110 89)(69 111 90)(70 112 91)(71 113 92)(72 114 93)(73 115 94)(74 116 95)(75 117 96)(76 118 97)(77 119 98)(78 120 99)(79 121 100)(80 122 101)(81 123 102)(82 124 103)(83 125 104)(84 126 105)
(1 15 8)(2 16 9)(3 17 10)(4 18 11)(5 19 12)(6 20 13)(7 21 14)(22 36 29)(23 37 30)(24 38 31)(25 39 32)(26 40 33)(27 41 34)(28 42 35)(43 57 50)(44 58 51)(45 59 52)(46 60 53)(47 61 54)(48 62 55)(49 63 56)(64 78 71)(65 79 72)(66 80 73)(67 81 74)(68 82 75)(69 83 76)(70 84 77)(85 99 92)(86 100 93)(87 101 94)(88 102 95)(89 103 96)(90 104 97)(91 105 98)(106 120 113)(107 121 114)(108 122 115)(109 123 116)(110 124 117)(111 125 118)(112 126 119)
(2 3 5)(4 7 6)(9 10 12)(11 14 13)(16 17 19)(18 21 20)(22 29 36)(23 31 40)(24 33 37)(25 35 41)(26 30 38)(27 32 42)(28 34 39)(43 57 50)(44 59 54)(45 61 51)(46 63 55)(47 58 52)(48 60 56)(49 62 53)(65 66 68)(67 70 69)(72 73 75)(74 77 76)(79 80 82)(81 84 83)(85 92 99)(86 94 103)(87 96 100)(88 98 104)(89 93 101)(90 95 105)(91 97 102)(106 120 113)(107 122 117)(108 124 114)(109 126 118)(110 121 115)(111 123 119)(112 125 116)
G:=sub<Sym(126)| (1,64)(2,65)(3,66)(4,67)(5,68)(6,69)(7,70)(8,71)(9,72)(10,73)(11,74)(12,75)(13,76)(14,77)(15,78)(16,79)(17,80)(18,81)(19,82)(20,83)(21,84)(22,85)(23,86)(24,87)(25,88)(26,89)(27,90)(28,91)(29,92)(30,93)(31,94)(32,95)(33,96)(34,97)(35,98)(36,99)(37,100)(38,101)(39,102)(40,103)(41,104)(42,105)(43,106)(44,107)(45,108)(46,109)(47,110)(48,111)(49,112)(50,113)(51,114)(52,115)(53,116)(54,117)(55,118)(56,119)(57,120)(58,121)(59,122)(60,123)(61,124)(62,125)(63,126), (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), (1,43,22)(2,44,23)(3,45,24)(4,46,25)(5,47,26)(6,48,27)(7,49,28)(8,50,29)(9,51,30)(10,52,31)(11,53,32)(12,54,33)(13,55,34)(14,56,35)(15,57,36)(16,58,37)(17,59,38)(18,60,39)(19,61,40)(20,62,41)(21,63,42)(64,106,85)(65,107,86)(66,108,87)(67,109,88)(68,110,89)(69,111,90)(70,112,91)(71,113,92)(72,114,93)(73,115,94)(74,116,95)(75,117,96)(76,118,97)(77,119,98)(78,120,99)(79,121,100)(80,122,101)(81,123,102)(82,124,103)(83,125,104)(84,126,105), (1,15,8)(2,16,9)(3,17,10)(4,18,11)(5,19,12)(6,20,13)(7,21,14)(22,36,29)(23,37,30)(24,38,31)(25,39,32)(26,40,33)(27,41,34)(28,42,35)(43,57,50)(44,58,51)(45,59,52)(46,60,53)(47,61,54)(48,62,55)(49,63,56)(64,78,71)(65,79,72)(66,80,73)(67,81,74)(68,82,75)(69,83,76)(70,84,77)(85,99,92)(86,100,93)(87,101,94)(88,102,95)(89,103,96)(90,104,97)(91,105,98)(106,120,113)(107,121,114)(108,122,115)(109,123,116)(110,124,117)(111,125,118)(112,126,119), (2,3,5)(4,7,6)(9,10,12)(11,14,13)(16,17,19)(18,21,20)(22,29,36)(23,31,40)(24,33,37)(25,35,41)(26,30,38)(27,32,42)(28,34,39)(43,57,50)(44,59,54)(45,61,51)(46,63,55)(47,58,52)(48,60,56)(49,62,53)(65,66,68)(67,70,69)(72,73,75)(74,77,76)(79,80,82)(81,84,83)(85,92,99)(86,94,103)(87,96,100)(88,98,104)(89,93,101)(90,95,105)(91,97,102)(106,120,113)(107,122,117)(108,124,114)(109,126,118)(110,121,115)(111,123,119)(112,125,116)>;
G:=Group( (1,64)(2,65)(3,66)(4,67)(5,68)(6,69)(7,70)(8,71)(9,72)(10,73)(11,74)(12,75)(13,76)(14,77)(15,78)(16,79)(17,80)(18,81)(19,82)(20,83)(21,84)(22,85)(23,86)(24,87)(25,88)(26,89)(27,90)(28,91)(29,92)(30,93)(31,94)(32,95)(33,96)(34,97)(35,98)(36,99)(37,100)(38,101)(39,102)(40,103)(41,104)(42,105)(43,106)(44,107)(45,108)(46,109)(47,110)(48,111)(49,112)(50,113)(51,114)(52,115)(53,116)(54,117)(55,118)(56,119)(57,120)(58,121)(59,122)(60,123)(61,124)(62,125)(63,126), (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), (1,43,22)(2,44,23)(3,45,24)(4,46,25)(5,47,26)(6,48,27)(7,49,28)(8,50,29)(9,51,30)(10,52,31)(11,53,32)(12,54,33)(13,55,34)(14,56,35)(15,57,36)(16,58,37)(17,59,38)(18,60,39)(19,61,40)(20,62,41)(21,63,42)(64,106,85)(65,107,86)(66,108,87)(67,109,88)(68,110,89)(69,111,90)(70,112,91)(71,113,92)(72,114,93)(73,115,94)(74,116,95)(75,117,96)(76,118,97)(77,119,98)(78,120,99)(79,121,100)(80,122,101)(81,123,102)(82,124,103)(83,125,104)(84,126,105), (1,15,8)(2,16,9)(3,17,10)(4,18,11)(5,19,12)(6,20,13)(7,21,14)(22,36,29)(23,37,30)(24,38,31)(25,39,32)(26,40,33)(27,41,34)(28,42,35)(43,57,50)(44,58,51)(45,59,52)(46,60,53)(47,61,54)(48,62,55)(49,63,56)(64,78,71)(65,79,72)(66,80,73)(67,81,74)(68,82,75)(69,83,76)(70,84,77)(85,99,92)(86,100,93)(87,101,94)(88,102,95)(89,103,96)(90,104,97)(91,105,98)(106,120,113)(107,121,114)(108,122,115)(109,123,116)(110,124,117)(111,125,118)(112,126,119), (2,3,5)(4,7,6)(9,10,12)(11,14,13)(16,17,19)(18,21,20)(22,29,36)(23,31,40)(24,33,37)(25,35,41)(26,30,38)(27,32,42)(28,34,39)(43,57,50)(44,59,54)(45,61,51)(46,63,55)(47,58,52)(48,60,56)(49,62,53)(65,66,68)(67,70,69)(72,73,75)(74,77,76)(79,80,82)(81,84,83)(85,92,99)(86,94,103)(87,96,100)(88,98,104)(89,93,101)(90,95,105)(91,97,102)(106,120,113)(107,122,117)(108,124,114)(109,126,118)(110,121,115)(111,123,119)(112,125,116) );
G=PermutationGroup([[(1,64),(2,65),(3,66),(4,67),(5,68),(6,69),(7,70),(8,71),(9,72),(10,73),(11,74),(12,75),(13,76),(14,77),(15,78),(16,79),(17,80),(18,81),(19,82),(20,83),(21,84),(22,85),(23,86),(24,87),(25,88),(26,89),(27,90),(28,91),(29,92),(30,93),(31,94),(32,95),(33,96),(34,97),(35,98),(36,99),(37,100),(38,101),(39,102),(40,103),(41,104),(42,105),(43,106),(44,107),(45,108),(46,109),(47,110),(48,111),(49,112),(50,113),(51,114),(52,115),(53,116),(54,117),(55,118),(56,119),(57,120),(58,121),(59,122),(60,123),(61,124),(62,125),(63,126)], [(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)], [(1,43,22),(2,44,23),(3,45,24),(4,46,25),(5,47,26),(6,48,27),(7,49,28),(8,50,29),(9,51,30),(10,52,31),(11,53,32),(12,54,33),(13,55,34),(14,56,35),(15,57,36),(16,58,37),(17,59,38),(18,60,39),(19,61,40),(20,62,41),(21,63,42),(64,106,85),(65,107,86),(66,108,87),(67,109,88),(68,110,89),(69,111,90),(70,112,91),(71,113,92),(72,114,93),(73,115,94),(74,116,95),(75,117,96),(76,118,97),(77,119,98),(78,120,99),(79,121,100),(80,122,101),(81,123,102),(82,124,103),(83,125,104),(84,126,105)], [(1,15,8),(2,16,9),(3,17,10),(4,18,11),(5,19,12),(6,20,13),(7,21,14),(22,36,29),(23,37,30),(24,38,31),(25,39,32),(26,40,33),(27,41,34),(28,42,35),(43,57,50),(44,58,51),(45,59,52),(46,60,53),(47,61,54),(48,62,55),(49,63,56),(64,78,71),(65,79,72),(66,80,73),(67,81,74),(68,82,75),(69,83,76),(70,84,77),(85,99,92),(86,100,93),(87,101,94),(88,102,95),(89,103,96),(90,104,97),(91,105,98),(106,120,113),(107,121,114),(108,122,115),(109,123,116),(110,124,117),(111,125,118),(112,126,119)], [(2,3,5),(4,7,6),(9,10,12),(11,14,13),(16,17,19),(18,21,20),(22,29,36),(23,31,40),(24,33,37),(25,35,41),(26,30,38),(27,32,42),(28,34,39),(43,57,50),(44,59,54),(45,61,51),(46,63,55),(47,58,52),(48,60,56),(49,62,53),(65,66,68),(67,70,69),(72,73,75),(74,77,76),(79,80,82),(81,84,83),(85,92,99),(86,94,103),(87,96,100),(88,98,104),(89,93,101),(90,95,105),(91,97,102),(106,120,113),(107,122,117),(108,124,114),(109,126,118),(110,121,115),(111,123,119),(112,125,116)]])
58 conjugacy classes
| class | 1 | 2 | 3A | 3B | 3C | 3D | 3E | ··· | 3J | 6A | 6B | 6C | 6D | 6E | ··· | 6J | 7A | 7B | 14A | 14B | 21A | ··· | 21P | 42A | ··· | 42P |
| order | 1 | 2 | 3 | 3 | 3 | 3 | 3 | ··· | 3 | 6 | 6 | 6 | 6 | 6 | ··· | 6 | 7 | 7 | 14 | 14 | 21 | ··· | 21 | 42 | ··· | 42 |
| size | 1 | 1 | 1 | 1 | 3 | 3 | 21 | ··· | 21 | 1 | 1 | 3 | 3 | 21 | ··· | 21 | 3 | 3 | 3 | 3 | 3 | ··· | 3 | 3 | ··· | 3 |
58 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 |
| type | + | + | ||||||||||||
| image | C1 | C2 | C3 | C3 | C6 | C6 | C7⋊C3 | He3 | C2×C7⋊C3 | C2×He3 | C3×C7⋊C3 | C6×C7⋊C3 | C7⋊He3 | C2×C7⋊He3 |
| kernel | C2×C7⋊He3 | C7⋊He3 | C6×C7⋊C3 | C3×C42 | C3×C7⋊C3 | C3×C21 | C3×C6 | C14 | C32 | C7 | C6 | C3 | C2 | C1 |
| # reps | 1 | 1 | 6 | 2 | 6 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 12 | 12 |
Matrix representation of C2×C7⋊He3 ►in GL3(𝔽43) generated by
| 42 | 0 | 0 |
| 0 | 42 | 0 |
| 0 | 0 | 42 |
| 18 | 19 | 1 |
| 1 | 0 | 0 |
| 0 | 1 | 0 |
| 41 | 39 | 15 |
| 15 | 29 | 12 |
| 12 | 14 | 16 |
| 36 | 0 | 0 |
| 0 | 36 | 0 |
| 0 | 0 | 36 |
| 1 | 0 | 0 |
| 24 | 42 | 42 |
| 0 | 1 | 0 |
G:=sub<GL(3,GF(43))| [42,0,0,0,42,0,0,0,42],[18,1,0,19,0,1,1,0,0],[41,15,12,39,29,14,15,12,16],[36,0,0,0,36,0,0,0,36],[1,24,0,0,42,1,0,42,0] >;
C2×C7⋊He3 in GAP, Magma, Sage, TeX
C_2\times C_7\rtimes {\rm He}_3 % in TeX
G:=Group("C2xC7:He3"); // GroupNames label
G:=SmallGroup(378,28);
// by ID
G=gap.SmallGroup(378,28);
# by ID
G:=PCGroup([5,-2,-3,-3,-3,-7,187,1359]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^7=c^3=d^3=e^3=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,e*b*e^-1=b^4,c*d=d*c,e*c*e^-1=c*d^-1,d*e=e*d>;
// generators/relations
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