direct product, metacyclic, supersoluble, monomial, A-group
Aliases: C42×C7⋊C3, C28⋊4C12, C7⋊2(C4×C12), (C4×C28)⋊3C3, (C2×C28).10C6, C14.10(C2×C12), (C2×C14).14(C2×C6), C22.2(C22×C7⋊C3), (C22×C7⋊C3).13C22, C2.1(C2×C4×C7⋊C3), (C2×C4×C7⋊C3).10C2, (C2×C4).4(C2×C7⋊C3), (C2×C7⋊C3).10(C2×C4), SmallGroup(336,48)
Series: Derived ►Chief ►Lower central ►Upper central
| C7 — C42×C7⋊C3 |
Generators and relations for C42×C7⋊C3
G = < a,b,c,d | a4=b4=c7=d3=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c4 >
Subgroups: 150 in 60 conjugacy classes, 45 normal (9 characteristic)
C1, C2, C3, C4, C22, C6, C7, C2×C4, C12, C2×C6, C14, C42, C7⋊C3, C2×C12, C28, C2×C14, C2×C7⋊C3, C4×C12, C2×C28, C4×C7⋊C3, C22×C7⋊C3, C4×C28, C2×C4×C7⋊C3, C42×C7⋊C3
Quotients: C1, C2, C3, C4, C22, C6, C2×C4, C12, C2×C6, C42, C7⋊C3, C2×C12, C2×C7⋊C3, C4×C12, C4×C7⋊C3, C22×C7⋊C3, C2×C4×C7⋊C3, C42×C7⋊C3
►On 112 points
Generators in S112
(1 99 22 92)(2 100 23 93)(3 101 24 94)(4 102 25 95)(5 103 26 96)(6 104 27 97)(7 105 28 98)(8 106 15 85)(9 107 16 86)(10 108 17 87)(11 109 18 88)(12 110 19 89)(13 111 20 90)(14 112 21 91)(29 57 50 78)(30 58 51 79)(31 59 52 80)(32 60 53 81)(33 61 54 82)(34 62 55 83)(35 63 56 84)(36 64 43 71)(37 65 44 72)(38 66 45 73)(39 67 46 74)(40 68 47 75)(41 69 48 76)(42 70 49 77)
(1 43 15 29)(2 44 16 30)(3 45 17 31)(4 46 18 32)(5 47 19 33)(6 48 20 34)(7 49 21 35)(8 50 22 36)(9 51 23 37)(10 52 24 38)(11 53 25 39)(12 54 26 40)(13 55 27 41)(14 56 28 42)(57 99 71 85)(58 100 72 86)(59 101 73 87)(60 102 74 88)(61 103 75 89)(62 104 76 90)(63 105 77 91)(64 106 78 92)(65 107 79 93)(66 108 80 94)(67 109 81 95)(68 110 82 96)(69 111 83 97)(70 112 84 98)
(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)
(2 3 5)(4 7 6)(9 10 12)(11 14 13)(16 17 19)(18 21 20)(23 24 26)(25 28 27)(30 31 33)(32 35 34)(37 38 40)(39 42 41)(44 45 47)(46 49 48)(51 52 54)(53 56 55)(58 59 61)(60 63 62)(65 66 68)(67 70 69)(72 73 75)(74 77 76)(79 80 82)(81 84 83)(86 87 89)(88 91 90)(93 94 96)(95 98 97)(100 101 103)(102 105 104)(107 108 110)(109 112 111)
G:=sub<Sym(112)| (1,99,22,92)(2,100,23,93)(3,101,24,94)(4,102,25,95)(5,103,26,96)(6,104,27,97)(7,105,28,98)(8,106,15,85)(9,107,16,86)(10,108,17,87)(11,109,18,88)(12,110,19,89)(13,111,20,90)(14,112,21,91)(29,57,50,78)(30,58,51,79)(31,59,52,80)(32,60,53,81)(33,61,54,82)(34,62,55,83)(35,63,56,84)(36,64,43,71)(37,65,44,72)(38,66,45,73)(39,67,46,74)(40,68,47,75)(41,69,48,76)(42,70,49,77), (1,43,15,29)(2,44,16,30)(3,45,17,31)(4,46,18,32)(5,47,19,33)(6,48,20,34)(7,49,21,35)(8,50,22,36)(9,51,23,37)(10,52,24,38)(11,53,25,39)(12,54,26,40)(13,55,27,41)(14,56,28,42)(57,99,71,85)(58,100,72,86)(59,101,73,87)(60,102,74,88)(61,103,75,89)(62,104,76,90)(63,105,77,91)(64,106,78,92)(65,107,79,93)(66,108,80,94)(67,109,81,95)(68,110,82,96)(69,111,83,97)(70,112,84,98), (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), (2,3,5)(4,7,6)(9,10,12)(11,14,13)(16,17,19)(18,21,20)(23,24,26)(25,28,27)(30,31,33)(32,35,34)(37,38,40)(39,42,41)(44,45,47)(46,49,48)(51,52,54)(53,56,55)(58,59,61)(60,63,62)(65,66,68)(67,70,69)(72,73,75)(74,77,76)(79,80,82)(81,84,83)(86,87,89)(88,91,90)(93,94,96)(95,98,97)(100,101,103)(102,105,104)(107,108,110)(109,112,111)>;
G:=Group( (1,99,22,92)(2,100,23,93)(3,101,24,94)(4,102,25,95)(5,103,26,96)(6,104,27,97)(7,105,28,98)(8,106,15,85)(9,107,16,86)(10,108,17,87)(11,109,18,88)(12,110,19,89)(13,111,20,90)(14,112,21,91)(29,57,50,78)(30,58,51,79)(31,59,52,80)(32,60,53,81)(33,61,54,82)(34,62,55,83)(35,63,56,84)(36,64,43,71)(37,65,44,72)(38,66,45,73)(39,67,46,74)(40,68,47,75)(41,69,48,76)(42,70,49,77), (1,43,15,29)(2,44,16,30)(3,45,17,31)(4,46,18,32)(5,47,19,33)(6,48,20,34)(7,49,21,35)(8,50,22,36)(9,51,23,37)(10,52,24,38)(11,53,25,39)(12,54,26,40)(13,55,27,41)(14,56,28,42)(57,99,71,85)(58,100,72,86)(59,101,73,87)(60,102,74,88)(61,103,75,89)(62,104,76,90)(63,105,77,91)(64,106,78,92)(65,107,79,93)(66,108,80,94)(67,109,81,95)(68,110,82,96)(69,111,83,97)(70,112,84,98), (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), (2,3,5)(4,7,6)(9,10,12)(11,14,13)(16,17,19)(18,21,20)(23,24,26)(25,28,27)(30,31,33)(32,35,34)(37,38,40)(39,42,41)(44,45,47)(46,49,48)(51,52,54)(53,56,55)(58,59,61)(60,63,62)(65,66,68)(67,70,69)(72,73,75)(74,77,76)(79,80,82)(81,84,83)(86,87,89)(88,91,90)(93,94,96)(95,98,97)(100,101,103)(102,105,104)(107,108,110)(109,112,111) );
G=PermutationGroup([[(1,99,22,92),(2,100,23,93),(3,101,24,94),(4,102,25,95),(5,103,26,96),(6,104,27,97),(7,105,28,98),(8,106,15,85),(9,107,16,86),(10,108,17,87),(11,109,18,88),(12,110,19,89),(13,111,20,90),(14,112,21,91),(29,57,50,78),(30,58,51,79),(31,59,52,80),(32,60,53,81),(33,61,54,82),(34,62,55,83),(35,63,56,84),(36,64,43,71),(37,65,44,72),(38,66,45,73),(39,67,46,74),(40,68,47,75),(41,69,48,76),(42,70,49,77)], [(1,43,15,29),(2,44,16,30),(3,45,17,31),(4,46,18,32),(5,47,19,33),(6,48,20,34),(7,49,21,35),(8,50,22,36),(9,51,23,37),(10,52,24,38),(11,53,25,39),(12,54,26,40),(13,55,27,41),(14,56,28,42),(57,99,71,85),(58,100,72,86),(59,101,73,87),(60,102,74,88),(61,103,75,89),(62,104,76,90),(63,105,77,91),(64,106,78,92),(65,107,79,93),(66,108,80,94),(67,109,81,95),(68,110,82,96),(69,111,83,97),(70,112,84,98)], [(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)], [(2,3,5),(4,7,6),(9,10,12),(11,14,13),(16,17,19),(18,21,20),(23,24,26),(25,28,27),(30,31,33),(32,35,34),(37,38,40),(39,42,41),(44,45,47),(46,49,48),(51,52,54),(53,56,55),(58,59,61),(60,63,62),(65,66,68),(67,70,69),(72,73,75),(74,77,76),(79,80,82),(81,84,83),(86,87,89),(88,91,90),(93,94,96),(95,98,97),(100,101,103),(102,105,104),(107,108,110),(109,112,111)]])
80 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 4A | ··· | 4L | 6A | ··· | 6F | 7A | 7B | 12A | ··· | 12X | 14A | ··· | 14F | 28A | ··· | 28X |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 4 | ··· | 4 | 6 | ··· | 6 | 7 | 7 | 12 | ··· | 12 | 14 | ··· | 14 | 28 | ··· | 28 |
| size | 1 | 1 | 1 | 1 | 7 | 7 | 1 | ··· | 1 | 7 | ··· | 7 | 3 | 3 | 7 | ··· | 7 | 3 | ··· | 3 | 3 | ··· | 3 |
80 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 3 | 3 | 3 |
| type | + | + | |||||||
| image | C1 | C2 | C3 | C4 | C6 | C12 | C7⋊C3 | C2×C7⋊C3 | C4×C7⋊C3 |
| kernel | C42×C7⋊C3 | C2×C4×C7⋊C3 | C4×C28 | C4×C7⋊C3 | C2×C28 | C28 | C42 | C2×C4 | C4 |
| # reps | 1 | 3 | 2 | 12 | 6 | 24 | 2 | 6 | 24 |
Matrix representation of C42×C7⋊C3 ►in GL5(𝔽337)
| 148 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 336 | 0 | 0 | 0 | 0 |
| 0 | 148 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 124 | 125 | 1 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 208 | 0 | 0 | 0 | 0 |
| 0 | 208 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 212 | 336 | 336 |
| 0 | 0 | 0 | 1 | 0 |
G:=sub<GL(5,GF(337))| [148,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[336,0,0,0,0,0,148,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,124,1,0,0,0,125,0,1,0,0,1,0,0],[208,0,0,0,0,0,208,0,0,0,0,0,1,212,0,0,0,0,336,1,0,0,0,336,0] >;
C42×C7⋊C3 in GAP, Magma, Sage, TeX
C_4^2\times C_7\rtimes C_3
% in TeX
G:=Group("C4^2xC7:C3"); // GroupNames label
G:=SmallGroup(336,48);
// by ID
G=gap.SmallGroup(336,48);
# by ID
G:=PCGroup([6,-2,-2,-3,-2,-2,-7,144,79,881]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^7=d^3=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^4>;
// generators/relations