direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: C2×C7⋊C8, C14⋊C8, C28.3C4, C4.14D14, C4.3Dic7, C28.14C22, C22.2Dic7, C4○(C7⋊C8), C7⋊2(C2×C8), (C2×C4).5D7, C14.6(C2×C4), (C2×C14).2C4, (C2×C28).6C2, C2.1(C2×Dic7), SmallGroup(112,8)
Series: Derived ►Chief ►Lower central ►Upper central
| C7 — C2×C7⋊C8 |
Generators and relations for C2×C7⋊C8
G = < a,b,c | a2=b7=c8=1, ab=ba, ac=ca, cbc-1=b-1 >
Smallest permutation representation of C2×C7⋊C8
►Regular action on 112 points
Generators in S112
(1 25)(2 26)(3 27)(4 28)(5 29)(6 30)(7 31)(8 32)(9 53)(10 54)(11 55)(12 56)(13 49)(14 50)(15 51)(16 52)(17 61)(18 62)(19 63)(20 64)(21 57)(22 58)(23 59)(24 60)(33 96)(34 89)(35 90)(36 91)(37 92)(38 93)(39 94)(40 95)(41 104)(42 97)(43 98)(44 99)(45 100)(46 101)(47 102)(48 103)(65 85)(66 86)(67 87)(68 88)(69 81)(70 82)(71 83)(72 84)(73 108)(74 109)(75 110)(76 111)(77 112)(78 105)(79 106)(80 107)
(1 47 36 86 19 10 105)(2 106 11 20 87 37 48)(3 41 38 88 21 12 107)(4 108 13 22 81 39 42)(5 43 40 82 23 14 109)(6 110 15 24 83 33 44)(7 45 34 84 17 16 111)(8 112 9 18 85 35 46)(25 102 91 66 63 54 78)(26 79 55 64 67 92 103)(27 104 93 68 57 56 80)(28 73 49 58 69 94 97)(29 98 95 70 59 50 74)(30 75 51 60 71 96 99)(31 100 89 72 61 52 76)(32 77 53 62 65 90 101)
(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)
G:=sub<Sym(112)| (1,25)(2,26)(3,27)(4,28)(5,29)(6,30)(7,31)(8,32)(9,53)(10,54)(11,55)(12,56)(13,49)(14,50)(15,51)(16,52)(17,61)(18,62)(19,63)(20,64)(21,57)(22,58)(23,59)(24,60)(33,96)(34,89)(35,90)(36,91)(37,92)(38,93)(39,94)(40,95)(41,104)(42,97)(43,98)(44,99)(45,100)(46,101)(47,102)(48,103)(65,85)(66,86)(67,87)(68,88)(69,81)(70,82)(71,83)(72,84)(73,108)(74,109)(75,110)(76,111)(77,112)(78,105)(79,106)(80,107), (1,47,36,86,19,10,105)(2,106,11,20,87,37,48)(3,41,38,88,21,12,107)(4,108,13,22,81,39,42)(5,43,40,82,23,14,109)(6,110,15,24,83,33,44)(7,45,34,84,17,16,111)(8,112,9,18,85,35,46)(25,102,91,66,63,54,78)(26,79,55,64,67,92,103)(27,104,93,68,57,56,80)(28,73,49,58,69,94,97)(29,98,95,70,59,50,74)(30,75,51,60,71,96,99)(31,100,89,72,61,52,76)(32,77,53,62,65,90,101), (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)>;
G:=Group( (1,25)(2,26)(3,27)(4,28)(5,29)(6,30)(7,31)(8,32)(9,53)(10,54)(11,55)(12,56)(13,49)(14,50)(15,51)(16,52)(17,61)(18,62)(19,63)(20,64)(21,57)(22,58)(23,59)(24,60)(33,96)(34,89)(35,90)(36,91)(37,92)(38,93)(39,94)(40,95)(41,104)(42,97)(43,98)(44,99)(45,100)(46,101)(47,102)(48,103)(65,85)(66,86)(67,87)(68,88)(69,81)(70,82)(71,83)(72,84)(73,108)(74,109)(75,110)(76,111)(77,112)(78,105)(79,106)(80,107), (1,47,36,86,19,10,105)(2,106,11,20,87,37,48)(3,41,38,88,21,12,107)(4,108,13,22,81,39,42)(5,43,40,82,23,14,109)(6,110,15,24,83,33,44)(7,45,34,84,17,16,111)(8,112,9,18,85,35,46)(25,102,91,66,63,54,78)(26,79,55,64,67,92,103)(27,104,93,68,57,56,80)(28,73,49,58,69,94,97)(29,98,95,70,59,50,74)(30,75,51,60,71,96,99)(31,100,89,72,61,52,76)(32,77,53,62,65,90,101), (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) );
G=PermutationGroup([[(1,25),(2,26),(3,27),(4,28),(5,29),(6,30),(7,31),(8,32),(9,53),(10,54),(11,55),(12,56),(13,49),(14,50),(15,51),(16,52),(17,61),(18,62),(19,63),(20,64),(21,57),(22,58),(23,59),(24,60),(33,96),(34,89),(35,90),(36,91),(37,92),(38,93),(39,94),(40,95),(41,104),(42,97),(43,98),(44,99),(45,100),(46,101),(47,102),(48,103),(65,85),(66,86),(67,87),(68,88),(69,81),(70,82),(71,83),(72,84),(73,108),(74,109),(75,110),(76,111),(77,112),(78,105),(79,106),(80,107)], [(1,47,36,86,19,10,105),(2,106,11,20,87,37,48),(3,41,38,88,21,12,107),(4,108,13,22,81,39,42),(5,43,40,82,23,14,109),(6,110,15,24,83,33,44),(7,45,34,84,17,16,111),(8,112,9,18,85,35,46),(25,102,91,66,63,54,78),(26,79,55,64,67,92,103),(27,104,93,68,57,56,80),(28,73,49,58,69,94,97),(29,98,95,70,59,50,74),(30,75,51,60,71,96,99),(31,100,89,72,61,52,76),(32,77,53,62,65,90,101)], [(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)]])
C2×C7⋊C8 is a maximal subgroup of
C42.D7 C28⋊C8 C28.Q8 C4.Dic14 C14.D8 C14.Q16 C8×Dic7 Dic7⋊C8 C56⋊C4 D14⋊C8 C28.53D4 C28.55D4 D4⋊Dic7 Q8⋊Dic7 D7×C2×C8 D28.C4 Q8.Dic7 D4.8D14
C2×C7⋊C8 is a maximal quotient of
C28⋊C8 C28.C8 C28.55D4
40 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 7A | 7B | 7C | 8A | ··· | 8H | 14A | ··· | 14I | 28A | ··· | 28L |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 8 | ··· | 8 | 14 | ··· | 14 | 28 | ··· | 28 |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 7 | ··· | 7 | 2 | ··· | 2 | 2 | ··· | 2 |
40 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | - | + | - | ||||
| image | C1 | C2 | C2 | C4 | C4 | C8 | D7 | Dic7 | D14 | Dic7 | C7⋊C8 |
| kernel | C2×C7⋊C8 | C7⋊C8 | C2×C28 | C28 | C2×C14 | C14 | C2×C4 | C4 | C4 | C22 | C2 |
| # reps | 1 | 2 | 1 | 2 | 2 | 8 | 3 | 3 | 3 | 3 | 12 |
Matrix representation of C2×C7⋊C8 ►in GL3(𝔽113) generated by
| 112 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 0 | 1 |
| 1 | 0 | 0 |
| 0 | 10 | 112 |
| 0 | 11 | 112 |
| 112 | 0 | 0 |
| 0 | 48 | 7 |
| 0 | 40 | 65 |
G:=sub<GL(3,GF(113))| [112,0,0,0,1,0,0,0,1],[1,0,0,0,10,11,0,112,112],[112,0,0,0,48,40,0,7,65] >;
C2×C7⋊C8 in GAP, Magma, Sage, TeX
C_2\times C_7\rtimes C_8
% in TeX
G:=Group("C2xC7:C8"); // GroupNames label
G:=SmallGroup(112,8);
// by ID
G=gap.SmallGroup(112,8);
# by ID
G:=PCGroup([5,-2,-2,-2,-2,-7,20,42,2404]);
// Polycyclic
G:=Group<a,b,c|a^2=b^7=c^8=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations
Export