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G = C112⋊C2order 224 = 25·7

2nd semidirect product of C112 and C2 acting faithfully

metacyclic, supersoluble, monomial, 2-hyperelementary

Aliases: C162D7, C1122C2, C71SD32, C14.2D8, C2.4D56, C4.2D28, D56.1C2, C28.25D4, C8.14D14, Dic281C2, C56.15C22, SmallGroup(224,6)

Series: Derived Chief Lower central Upper central

C1C56 — C112⋊C2
C1C7C14C28C56D56 — C112⋊C2
C7C14C28C56 — C112⋊C2
C1C2C4C8C16

Generators and relations for C112⋊C2
 G = < a,b | a112=b2=1, bab=a55 >

56C2
28C4
28C22
8D7
14Q8
14D4
4D14
4Dic7
7Q16
7D8
2Dic14
2D28
7SD32

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   D1127C2  C16⋊D14  C16.D14  D8⋊D14  D7×SD32  SD323D7  Q32⋊D7
C112⋊C2 is a maximal quotient of   C56.78D4  C1126C4  C2.D112

59 conjugacy classes

class 1 2A2B4A4B7A7B7C8A8B14A14B14C16A16B16C16D28A···28F56A···56L112A···112X
order12244777881414141616161628···2856···56112···112
size11562562222222222222···22···22···2

59 irreducible representations

dim111122222222
type++++++++++
imageC1C2C2C2D4D7D8D14SD32D28D56C112⋊C2
kernelC112⋊C2C112D56Dic28C28C16C14C8C7C4C2C1
# reps11111323461224

Matrix representation of C112⋊C2 in GL2(𝔽113) generated by

9373
4097
,
10
79112
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

Export

Subgroup lattice of C112⋊C2 in TeX

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