direct product, metabelian, nilpotent (class 4), monomial, 2-elementary
Aliases: C7×M5(2)⋊C2, D8.2C28, C56.96D4, C28.62D8, M5(2)⋊6C14, C8.2(C2×C28), (C7×D8).4C4, C8.16(C7×D4), C4.11(C7×D8), C56.43(C2×C4), (C2×D8).5C14, C8.C4⋊2C14, (C14×D8).12C2, (C2×C28).280D4, (C7×M5(2))⋊14C2, (C2×C14).24SD16, C22.3(C7×SD16), C28.73(C22⋊C4), (C2×C56).266C22, C14.41(D4⋊C4), (C2×C4).11(C7×D4), C4.5(C7×C22⋊C4), (C2×C8).13(C2×C14), (C7×C8.C4)⋊11C2, C2.10(C7×D4⋊C4), SmallGroup(448,165)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C7×M5(2)⋊C2
G = < a,b,c,d | a7=b16=c2=d2=1, ab=ba, ac=ca, ad=da, cbc=b9, dbd=b3c, cd=dc >
Subgroups: 162 in 62 conjugacy classes, 30 normal (26 characteristic)
C1, C2, C2, C4, C22, C22, C7, C8, C8, C2×C4, D4, C23, C14, C14, C16, C2×C8, M4(2), D8, D8, C2×D4, C28, C2×C14, C2×C14, C8.C4, M5(2), C2×D8, C56, C56, C2×C28, C7×D4, C22×C14, M5(2)⋊C2, C112, C2×C56, C7×M4(2), C7×D8, C7×D8, D4×C14, C7×C8.C4, C7×M5(2), C14×D8, C7×M5(2)⋊C2
Quotients: C1, C2, C4, C22, C7, C2×C4, D4, C14, C22⋊C4, D8, SD16, C28, C2×C14, D4⋊C4, C2×C28, C7×D4, M5(2)⋊C2, C7×C22⋊C4, C7×D8, C7×SD16, C7×D4⋊C4, C7×M5(2)⋊C2
►On 112 points
Generators in S112
(1 50 19 75 110 47 87)(2 51 20 76 111 48 88)(3 52 21 77 112 33 89)(4 53 22 78 97 34 90)(5 54 23 79 98 35 91)(6 55 24 80 99 36 92)(7 56 25 65 100 37 93)(8 57 26 66 101 38 94)(9 58 27 67 102 39 95)(10 59 28 68 103 40 96)(11 60 29 69 104 41 81)(12 61 30 70 105 42 82)(13 62 31 71 106 43 83)(14 63 32 72 107 44 84)(15 64 17 73 108 45 85)(16 49 18 74 109 46 86)
(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 10)(4 12)(6 14)(8 16)(18 26)(20 28)(22 30)(24 32)(34 42)(36 44)(38 46)(40 48)(49 57)(51 59)(53 61)(55 63)(66 74)(68 76)(70 78)(72 80)(82 90)(84 92)(86 94)(88 96)(97 105)(99 107)(101 109)(103 111)
(2 12)(3 15)(4 10)(5 13)(6 8)(7 11)(14 16)(17 21)(18 32)(20 30)(22 28)(23 31)(24 26)(25 29)(33 45)(34 40)(35 43)(36 38)(37 41)(42 48)(44 46)(49 63)(51 61)(52 64)(53 59)(54 62)(55 57)(56 60)(65 69)(66 80)(68 78)(70 76)(71 79)(72 74)(73 77)(81 93)(82 88)(83 91)(84 86)(85 89)(90 96)(92 94)(97 103)(98 106)(99 101)(100 104)(105 111)(107 109)(108 112)
G:=sub<Sym(112)| (1,50,19,75,110,47,87)(2,51,20,76,111,48,88)(3,52,21,77,112,33,89)(4,53,22,78,97,34,90)(5,54,23,79,98,35,91)(6,55,24,80,99,36,92)(7,56,25,65,100,37,93)(8,57,26,66,101,38,94)(9,58,27,67,102,39,95)(10,59,28,68,103,40,96)(11,60,29,69,104,41,81)(12,61,30,70,105,42,82)(13,62,31,71,106,43,83)(14,63,32,72,107,44,84)(15,64,17,73,108,45,85)(16,49,18,74,109,46,86), (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), (2,10)(4,12)(6,14)(8,16)(18,26)(20,28)(22,30)(24,32)(34,42)(36,44)(38,46)(40,48)(49,57)(51,59)(53,61)(55,63)(66,74)(68,76)(70,78)(72,80)(82,90)(84,92)(86,94)(88,96)(97,105)(99,107)(101,109)(103,111), (2,12)(3,15)(4,10)(5,13)(6,8)(7,11)(14,16)(17,21)(18,32)(20,30)(22,28)(23,31)(24,26)(25,29)(33,45)(34,40)(35,43)(36,38)(37,41)(42,48)(44,46)(49,63)(51,61)(52,64)(53,59)(54,62)(55,57)(56,60)(65,69)(66,80)(68,78)(70,76)(71,79)(72,74)(73,77)(81,93)(82,88)(83,91)(84,86)(85,89)(90,96)(92,94)(97,103)(98,106)(99,101)(100,104)(105,111)(107,109)(108,112)>;
G:=Group( (1,50,19,75,110,47,87)(2,51,20,76,111,48,88)(3,52,21,77,112,33,89)(4,53,22,78,97,34,90)(5,54,23,79,98,35,91)(6,55,24,80,99,36,92)(7,56,25,65,100,37,93)(8,57,26,66,101,38,94)(9,58,27,67,102,39,95)(10,59,28,68,103,40,96)(11,60,29,69,104,41,81)(12,61,30,70,105,42,82)(13,62,31,71,106,43,83)(14,63,32,72,107,44,84)(15,64,17,73,108,45,85)(16,49,18,74,109,46,86), (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), (2,10)(4,12)(6,14)(8,16)(18,26)(20,28)(22,30)(24,32)(34,42)(36,44)(38,46)(40,48)(49,57)(51,59)(53,61)(55,63)(66,74)(68,76)(70,78)(72,80)(82,90)(84,92)(86,94)(88,96)(97,105)(99,107)(101,109)(103,111), (2,12)(3,15)(4,10)(5,13)(6,8)(7,11)(14,16)(17,21)(18,32)(20,30)(22,28)(23,31)(24,26)(25,29)(33,45)(34,40)(35,43)(36,38)(37,41)(42,48)(44,46)(49,63)(51,61)(52,64)(53,59)(54,62)(55,57)(56,60)(65,69)(66,80)(68,78)(70,76)(71,79)(72,74)(73,77)(81,93)(82,88)(83,91)(84,86)(85,89)(90,96)(92,94)(97,103)(98,106)(99,101)(100,104)(105,111)(107,109)(108,112) );
G=PermutationGroup([[(1,50,19,75,110,47,87),(2,51,20,76,111,48,88),(3,52,21,77,112,33,89),(4,53,22,78,97,34,90),(5,54,23,79,98,35,91),(6,55,24,80,99,36,92),(7,56,25,65,100,37,93),(8,57,26,66,101,38,94),(9,58,27,67,102,39,95),(10,59,28,68,103,40,96),(11,60,29,69,104,41,81),(12,61,30,70,105,42,82),(13,62,31,71,106,43,83),(14,63,32,72,107,44,84),(15,64,17,73,108,45,85),(16,49,18,74,109,46,86)], [(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)], [(2,10),(4,12),(6,14),(8,16),(18,26),(20,28),(22,30),(24,32),(34,42),(36,44),(38,46),(40,48),(49,57),(51,59),(53,61),(55,63),(66,74),(68,76),(70,78),(72,80),(82,90),(84,92),(86,94),(88,96),(97,105),(99,107),(101,109),(103,111)], [(2,12),(3,15),(4,10),(5,13),(6,8),(7,11),(14,16),(17,21),(18,32),(20,30),(22,28),(23,31),(24,26),(25,29),(33,45),(34,40),(35,43),(36,38),(37,41),(42,48),(44,46),(49,63),(51,61),(52,64),(53,59),(54,62),(55,57),(56,60),(65,69),(66,80),(68,78),(70,76),(71,79),(72,74),(73,77),(81,93),(82,88),(83,91),(84,86),(85,89),(90,96),(92,94),(97,103),(98,106),(99,101),(100,104),(105,111),(107,109),(108,112)]])
112 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 4A | 4B | 7A | ··· | 7F | 8A | 8B | 8C | 8D | 8E | 14A | ··· | 14F | 14G | ··· | 14L | 14M | ··· | 14X | 16A | 16B | 16C | 16D | 28A | ··· | 28L | 56A | ··· | 56L | 56M | ··· | 56R | 56S | ··· | 56AD | 112A | ··· | 112X |
| order | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 7 | ··· | 7 | 8 | 8 | 8 | 8 | 8 | 14 | ··· | 14 | 14 | ··· | 14 | 14 | ··· | 14 | 16 | 16 | 16 | 16 | 28 | ··· | 28 | 56 | ··· | 56 | 56 | ··· | 56 | 56 | ··· | 56 | 112 | ··· | 112 |
| size | 1 | 1 | 2 | 8 | 8 | 2 | 2 | 1 | ··· | 1 | 2 | 2 | 4 | 8 | 8 | 1 | ··· | 1 | 2 | ··· | 2 | 8 | ··· | 8 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | ··· | 8 | 4 | ··· | 4 |
112 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | ||||||||||||
| image | C1 | C2 | C2 | C2 | C4 | C7 | C14 | C14 | C14 | C28 | D4 | D4 | D8 | SD16 | C7×D4 | C7×D4 | C7×D8 | C7×SD16 | M5(2)⋊C2 | C7×M5(2)⋊C2 |
| kernel | C7×M5(2)⋊C2 | C7×C8.C4 | C7×M5(2) | C14×D8 | C7×D8 | M5(2)⋊C2 | C8.C4 | M5(2) | C2×D8 | D8 | C56 | C2×C28 | C28 | C2×C14 | C8 | C2×C4 | C4 | C22 | C7 | C1 |
| # reps | 1 | 1 | 1 | 1 | 4 | 6 | 6 | 6 | 6 | 24 | 1 | 1 | 2 | 2 | 6 | 6 | 12 | 12 | 2 | 12 |
Matrix representation of C7×M5(2)⋊C2 ►in GL4(𝔽113) generated by
| 28 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 |
| 0 | 0 | 28 | 0 |
| 0 | 0 | 0 | 28 |
| 13 | 15 | 111 | 0 |
| 100 | 98 | 1 | 112 |
| 100 | 16 | 51 | 49 |
| 95 | 97 | 62 | 64 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 13 | 15 | 112 | 0 |
| 100 | 98 | 0 | 112 |
| 1 | 0 | 0 | 0 |
| 112 | 112 | 0 | 0 |
| 48 | 36 | 31 | 82 |
| 1 | 64 | 82 | 82 |
G:=sub<GL(4,GF(113))| [28,0,0,0,0,28,0,0,0,0,28,0,0,0,0,28],[13,100,100,95,15,98,16,97,111,1,51,62,0,112,49,64],[1,0,13,100,0,1,15,98,0,0,112,0,0,0,0,112],[1,112,48,1,0,112,36,64,0,0,31,82,0,0,82,82] >;
C7×M5(2)⋊C2 in GAP, Magma, Sage, TeX
C_7\times M_5(2)\rtimes C_2
% in TeX
G:=Group("C7xM5(2):C2"); // GroupNames label
G:=SmallGroup(448,165);
// by ID
G=gap.SmallGroup(448,165);
# by ID
G:=PCGroup([7,-2,-2,-7,-2,-2,-2,-2,392,421,3923,5106,136,4911,172,14117,7068,124]);
// Polycyclic
G:=Group<a,b,c,d|a^7=b^16=c^2=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c=b^9,d*b*d=b^3*c,c*d=d*c>;
// generators/relations