direct product, metabelian, supersoluble, monomial, 3-hyperelementary
Aliases: C2×C21.C32, C42.5C32, C14⋊33- 1+2, C7⋊C9⋊6C6, (C3×C42).3C3, (C3×C21).8C6, C21.11(C3×C6), C7⋊6(C2×3- 1+2), (C2×C7⋊C9)⋊3C3, C6.5(C3×C7⋊C3), C3.5(C6×C7⋊C3), (C3×C6).(C7⋊C3), C32.(C2×C7⋊C3), SmallGroup(378,27)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×C21.C32
G = < a,b,c,d | a2=b21=d3=1, c3=b7, ab=ba, ac=ca, ad=da, cbc-1=b4, bd=db, dcd-1=b7c >
Smallest permutation representation of C2×C21.C32
►On 126 points
Generators in S126
(1 74)(2 75)(3 76)(4 77)(5 78)(6 79)(7 80)(8 81)(9 82)(10 83)(11 84)(12 64)(13 65)(14 66)(15 67)(16 68)(17 69)(18 70)(19 71)(20 72)(21 73)(22 96)(23 97)(24 98)(25 99)(26 100)(27 101)(28 102)(29 103)(30 104)(31 105)(32 85)(33 86)(34 87)(35 88)(36 89)(37 90)(38 91)(39 92)(40 93)(41 94)(42 95)(43 110)(44 111)(45 112)(46 113)(47 114)(48 115)(49 116)(50 117)(51 118)(52 119)(53 120)(54 121)(55 122)(56 123)(57 124)(58 125)(59 126)(60 106)(61 107)(62 108)(63 109)
(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)(106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126)
(1 62 26 8 48 33 15 55 40)(2 57 30 9 43 37 16 50 23)(3 52 34 10 59 41 17 45 27)(4 47 38 11 54 24 18 61 31)(5 63 42 12 49 28 19 56 35)(6 58 25 13 44 32 20 51 39)(7 53 29 14 60 36 21 46 22)(64 116 102 71 123 88 78 109 95)(65 111 85 72 118 92 79 125 99)(66 106 89 73 113 96 80 120 103)(67 122 93 74 108 100 81 115 86)(68 117 97 75 124 104 82 110 90)(69 112 101 76 119 87 83 126 94)(70 107 105 77 114 91 84 121 98)
(22 29 36)(23 30 37)(24 31 38)(25 32 39)(26 33 40)(27 34 41)(28 35 42)(43 57 50)(44 58 51)(45 59 52)(46 60 53)(47 61 54)(48 62 55)(49 63 56)(85 92 99)(86 93 100)(87 94 101)(88 95 102)(89 96 103)(90 97 104)(91 98 105)(106 120 113)(107 121 114)(108 122 115)(109 123 116)(110 124 117)(111 125 118)(112 126 119)
G:=sub<Sym(126)| (1,74)(2,75)(3,76)(4,77)(5,78)(6,79)(7,80)(8,81)(9,82)(10,83)(11,84)(12,64)(13,65)(14,66)(15,67)(16,68)(17,69)(18,70)(19,71)(20,72)(21,73)(22,96)(23,97)(24,98)(25,99)(26,100)(27,101)(28,102)(29,103)(30,104)(31,105)(32,85)(33,86)(34,87)(35,88)(36,89)(37,90)(38,91)(39,92)(40,93)(41,94)(42,95)(43,110)(44,111)(45,112)(46,113)(47,114)(48,115)(49,116)(50,117)(51,118)(52,119)(53,120)(54,121)(55,122)(56,123)(57,124)(58,125)(59,126)(60,106)(61,107)(62,108)(63,109), (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)(106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126), (1,62,26,8,48,33,15,55,40)(2,57,30,9,43,37,16,50,23)(3,52,34,10,59,41,17,45,27)(4,47,38,11,54,24,18,61,31)(5,63,42,12,49,28,19,56,35)(6,58,25,13,44,32,20,51,39)(7,53,29,14,60,36,21,46,22)(64,116,102,71,123,88,78,109,95)(65,111,85,72,118,92,79,125,99)(66,106,89,73,113,96,80,120,103)(67,122,93,74,108,100,81,115,86)(68,117,97,75,124,104,82,110,90)(69,112,101,76,119,87,83,126,94)(70,107,105,77,114,91,84,121,98), (22,29,36)(23,30,37)(24,31,38)(25,32,39)(26,33,40)(27,34,41)(28,35,42)(43,57,50)(44,58,51)(45,59,52)(46,60,53)(47,61,54)(48,62,55)(49,63,56)(85,92,99)(86,93,100)(87,94,101)(88,95,102)(89,96,103)(90,97,104)(91,98,105)(106,120,113)(107,121,114)(108,122,115)(109,123,116)(110,124,117)(111,125,118)(112,126,119)>;
G:=Group( (1,74)(2,75)(3,76)(4,77)(5,78)(6,79)(7,80)(8,81)(9,82)(10,83)(11,84)(12,64)(13,65)(14,66)(15,67)(16,68)(17,69)(18,70)(19,71)(20,72)(21,73)(22,96)(23,97)(24,98)(25,99)(26,100)(27,101)(28,102)(29,103)(30,104)(31,105)(32,85)(33,86)(34,87)(35,88)(36,89)(37,90)(38,91)(39,92)(40,93)(41,94)(42,95)(43,110)(44,111)(45,112)(46,113)(47,114)(48,115)(49,116)(50,117)(51,118)(52,119)(53,120)(54,121)(55,122)(56,123)(57,124)(58,125)(59,126)(60,106)(61,107)(62,108)(63,109), (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)(106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126), (1,62,26,8,48,33,15,55,40)(2,57,30,9,43,37,16,50,23)(3,52,34,10,59,41,17,45,27)(4,47,38,11,54,24,18,61,31)(5,63,42,12,49,28,19,56,35)(6,58,25,13,44,32,20,51,39)(7,53,29,14,60,36,21,46,22)(64,116,102,71,123,88,78,109,95)(65,111,85,72,118,92,79,125,99)(66,106,89,73,113,96,80,120,103)(67,122,93,74,108,100,81,115,86)(68,117,97,75,124,104,82,110,90)(69,112,101,76,119,87,83,126,94)(70,107,105,77,114,91,84,121,98), (22,29,36)(23,30,37)(24,31,38)(25,32,39)(26,33,40)(27,34,41)(28,35,42)(43,57,50)(44,58,51)(45,59,52)(46,60,53)(47,61,54)(48,62,55)(49,63,56)(85,92,99)(86,93,100)(87,94,101)(88,95,102)(89,96,103)(90,97,104)(91,98,105)(106,120,113)(107,121,114)(108,122,115)(109,123,116)(110,124,117)(111,125,118)(112,126,119) );
G=PermutationGroup([[(1,74),(2,75),(3,76),(4,77),(5,78),(6,79),(7,80),(8,81),(9,82),(10,83),(11,84),(12,64),(13,65),(14,66),(15,67),(16,68),(17,69),(18,70),(19,71),(20,72),(21,73),(22,96),(23,97),(24,98),(25,99),(26,100),(27,101),(28,102),(29,103),(30,104),(31,105),(32,85),(33,86),(34,87),(35,88),(36,89),(37,90),(38,91),(39,92),(40,93),(41,94),(42,95),(43,110),(44,111),(45,112),(46,113),(47,114),(48,115),(49,116),(50,117),(51,118),(52,119),(53,120),(54,121),(55,122),(56,123),(57,124),(58,125),(59,126),(60,106),(61,107),(62,108),(63,109)], [(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),(106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126)], [(1,62,26,8,48,33,15,55,40),(2,57,30,9,43,37,16,50,23),(3,52,34,10,59,41,17,45,27),(4,47,38,11,54,24,18,61,31),(5,63,42,12,49,28,19,56,35),(6,58,25,13,44,32,20,51,39),(7,53,29,14,60,36,21,46,22),(64,116,102,71,123,88,78,109,95),(65,111,85,72,118,92,79,125,99),(66,106,89,73,113,96,80,120,103),(67,122,93,74,108,100,81,115,86),(68,117,97,75,124,104,82,110,90),(69,112,101,76,119,87,83,126,94),(70,107,105,77,114,91,84,121,98)], [(22,29,36),(23,30,37),(24,31,38),(25,32,39),(26,33,40),(27,34,41),(28,35,42),(43,57,50),(44,58,51),(45,59,52),(46,60,53),(47,61,54),(48,62,55),(49,63,56),(85,92,99),(86,93,100),(87,94,101),(88,95,102),(89,96,103),(90,97,104),(91,98,105),(106,120,113),(107,121,114),(108,122,115),(109,123,116),(110,124,117),(111,125,118),(112,126,119)]])
58 conjugacy classes
| class | 1 | 2 | 3A | 3B | 3C | 3D | 6A | 6B | 6C | 6D | 7A | 7B | 9A | ··· | 9F | 14A | 14B | 18A | ··· | 18F | 21A | ··· | 21P | 42A | ··· | 42P |
| order | 1 | 2 | 3 | 3 | 3 | 3 | 6 | 6 | 6 | 6 | 7 | 7 | 9 | ··· | 9 | 14 | 14 | 18 | ··· | 18 | 21 | ··· | 21 | 42 | ··· | 42 |
| size | 1 | 1 | 1 | 1 | 3 | 3 | 1 | 1 | 3 | 3 | 3 | 3 | 21 | ··· | 21 | 3 | 3 | 21 | ··· | 21 | 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 | 3- 1+2 | C2×C7⋊C3 | C2×3- 1+2 | C3×C7⋊C3 | C6×C7⋊C3 | C21.C32 | C2×C21.C32 |
| kernel | C2×C21.C32 | C21.C32 | C2×C7⋊C9 | C3×C42 | C7⋊C9 | 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×C21.C32 ►in GL4(𝔽127) generated by
| 126 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 61 | 0 | 0 |
| 0 | 0 | 87 | 0 |
| 0 | 0 | 0 | 47 |
| 19 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 107 | 0 | 0 |
| 19 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 107 | 0 |
| 0 | 0 | 0 | 19 |
G:=sub<GL(4,GF(127))| [126,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,61,0,0,0,0,87,0,0,0,0,47],[19,0,0,0,0,0,0,107,0,1,0,0,0,0,1,0],[19,0,0,0,0,1,0,0,0,0,107,0,0,0,0,19] >;
C2×C21.C32 in GAP, Magma, Sage, TeX
C_2\times C_{21}.C_3^2 % in TeX
G:=Group("C2xC21.C3^2"); // GroupNames label
G:=SmallGroup(378,27);
// by ID
G=gap.SmallGroup(378,27);
# by ID
G:=PCGroup([5,-2,-3,-3,-3,-7,96,187,1359]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^21=d^3=1,c^3=b^7,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=b^4,b*d=d*b,d*c*d^-1=b^7*c>;
// generators/relations
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