metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: C16⋊2D7, C112⋊2C2, C7⋊1SD32, C14.2D8, C2.4D56, C4.2D28, D56.1C2, C28.25D4, C8.14D14, Dic28⋊1C2, C56.15C22, SmallGroup(224,6)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C112⋊C2
G = < a,b | a112=b2=1, bab=a55 >
Smallest permutation representation of C112⋊C2
►On 112 points
Generators in S112
(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 56)(3 111)(4 54)(5 109)(6 52)(7 107)(8 50)(9 105)(10 48)(11 103)(12 46)(13 101)(14 44)(15 99)(16 42)(17 97)(18 40)(19 95)(20 38)(21 93)(22 36)(23 91)(24 34)(25 89)(26 32)(27 87)(28 30)(29 85)(31 83)(33 81)(35 79)(37 77)(39 75)(41 73)(43 71)(45 69)(47 67)(49 65)(51 63)(53 61)(55 59)(58 112)(60 110)(62 108)(64 106)(66 104)(68 102)(70 100)(72 98)(74 96)(76 94)(78 92)(80 90)(82 88)(84 86)
G:=sub<Sym(112)| (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), (2,56)(3,111)(4,54)(5,109)(6,52)(7,107)(8,50)(9,105)(10,48)(11,103)(12,46)(13,101)(14,44)(15,99)(16,42)(17,97)(18,40)(19,95)(20,38)(21,93)(22,36)(23,91)(24,34)(25,89)(26,32)(27,87)(28,30)(29,85)(31,83)(33,81)(35,79)(37,77)(39,75)(41,73)(43,71)(45,69)(47,67)(49,65)(51,63)(53,61)(55,59)(58,112)(60,110)(62,108)(64,106)(66,104)(68,102)(70,100)(72,98)(74,96)(76,94)(78,92)(80,90)(82,88)(84,86)>;
G:=Group( (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), (2,56)(3,111)(4,54)(5,109)(6,52)(7,107)(8,50)(9,105)(10,48)(11,103)(12,46)(13,101)(14,44)(15,99)(16,42)(17,97)(18,40)(19,95)(20,38)(21,93)(22,36)(23,91)(24,34)(25,89)(26,32)(27,87)(28,30)(29,85)(31,83)(33,81)(35,79)(37,77)(39,75)(41,73)(43,71)(45,69)(47,67)(49,65)(51,63)(53,61)(55,59)(58,112)(60,110)(62,108)(64,106)(66,104)(68,102)(70,100)(72,98)(74,96)(76,94)(78,92)(80,90)(82,88)(84,86) );
G=PermutationGroup([[(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)], [(2,56),(3,111),(4,54),(5,109),(6,52),(7,107),(8,50),(9,105),(10,48),(11,103),(12,46),(13,101),(14,44),(15,99),(16,42),(17,97),(18,40),(19,95),(20,38),(21,93),(22,36),(23,91),(24,34),(25,89),(26,32),(27,87),(28,30),(29,85),(31,83),(33,81),(35,79),(37,77),(39,75),(41,73),(43,71),(45,69),(47,67),(49,65),(51,63),(53,61),(55,59),(58,112),(60,110),(62,108),(64,106),(66,104),(68,102),(70,100),(72,98),(74,96),(76,94),(78,92),(80,90),(82,88),(84,86)]])
C112⋊C2 is a maximal subgroup of
D112⋊7C2 C16⋊D14 C16.D14 D8⋊D14 D7×SD32 SD32⋊3D7 Q32⋊D7
C112⋊C2 is a maximal quotient of C56.78D4 C112⋊6C4 C2.D112
59 conjugacy classes
| class | 1 | 2A | 2B | 4A | 4B | 7A | 7B | 7C | 8A | 8B | 14A | 14B | 14C | 16A | 16B | 16C | 16D | 28A | ··· | 28F | 56A | ··· | 56L | 112A | ··· | 112X |
| order | 1 | 2 | 2 | 4 | 4 | 7 | 7 | 7 | 8 | 8 | 14 | 14 | 14 | 16 | 16 | 16 | 16 | 28 | ··· | 28 | 56 | ··· | 56 | 112 | ··· | 112 |
| size | 1 | 1 | 56 | 2 | 56 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
59 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | D4 | D7 | D8 | D14 | SD32 | D28 | D56 | C112⋊C2 |
| kernel | C112⋊C2 | C112 | D56 | Dic28 | C28 | C16 | C14 | C8 | C7 | C4 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 3 | 2 | 3 | 4 | 6 | 12 | 24 |
Matrix representation of C112⋊C2 ►in GL2(𝔽113) generated by
| 93 | 73 |
| 40 | 97 |
| 1 | 0 |
| 79 | 112 |
G:=sub<GL(2,GF(113))| [93,40,73,97],[1,79,0,112] >;
C112⋊C2 in GAP, Magma, Sage, TeX
C_{112}\rtimes C_2 % in TeX
G:=Group("C112:C2"); // GroupNames label
G:=SmallGroup(224,6);
// by ID
G=gap.SmallGroup(224,6);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-7,73,79,506,50,579,69,6917]);
// Polycyclic
G:=Group<a,b|a^112=b^2=1,b*a*b=a^55>;
// generators/relations
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