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G = C2.D56order 224 = 25·7

2nd central extension by C2 of D56

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D282C4, C14.5D8, C2.2D56, C28.45D4, C14.3SD16, C22.10D28, (C2×C8)⋊2D7, (C2×C56)⋊2C2, C4.8(C4×D7), C4⋊Dic71C2, C72(D4⋊C4), C28.18(C2×C4), (C2×D28).1C2, (C2×C14).15D4, (C2×C4).71D14, C2.8(D14⋊C4), C2.3(C56⋊C2), C4.20(C7⋊D4), C14.7(C22⋊C4), (C2×C28).83C22, SmallGroup(224,27)

Series: Derived Chief Lower central Upper central

C1C28 — C2.D56
C1C7C14C28C2×C28C2×D28 — C2.D56
C7C14C28 — C2.D56
C1C22C2×C4C2×C8

Generators and relations for C2.D56
 G = < a,b,c | a2=b56=1, c2=a, ab=ba, ac=ca, cbc-1=ab-1 >

28C2
28C2
14C22
14C22
28C22
28C22
28C4
4D7
4D7
2C8
7D4
7D4
14C2×C4
14C23
14D4
2D14
2D14
4Dic7
4D14
4D14
7C4⋊C4
7C2×D4
2C22×D7
2C2×Dic7
2C56
2D28
7D4⋊C4

Smallest permutation representation of C2.D56
On 112 points
Generators in S112
(1 98)(2 99)(3 100)(4 101)(5 102)(6 103)(7 104)(8 105)(9 106)(10 107)(11 108)(12 109)(13 110)(14 111)(15 112)(16 57)(17 58)(18 59)(19 60)(20 61)(21 62)(22 63)(23 64)(24 65)(25 66)(26 67)(27 68)(28 69)(29 70)(30 71)(31 72)(32 73)(33 74)(34 75)(35 76)(36 77)(37 78)(38 79)(39 80)(40 81)(41 82)(42 83)(43 84)(44 85)(45 86)(46 87)(47 88)(48 89)(49 90)(50 91)(51 92)(52 93)(53 94)(54 95)(55 96)(56 97)
(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)
(1 97 98 56)(2 55 99 96)(3 95 100 54)(4 53 101 94)(5 93 102 52)(6 51 103 92)(7 91 104 50)(8 49 105 90)(9 89 106 48)(10 47 107 88)(11 87 108 46)(12 45 109 86)(13 85 110 44)(14 43 111 84)(15 83 112 42)(16 41 57 82)(17 81 58 40)(18 39 59 80)(19 79 60 38)(20 37 61 78)(21 77 62 36)(22 35 63 76)(23 75 64 34)(24 33 65 74)(25 73 66 32)(26 31 67 72)(27 71 68 30)(28 29 69 70)

G:=sub<Sym(112)| (1,98)(2,99)(3,100)(4,101)(5,102)(6,103)(7,104)(8,105)(9,106)(10,107)(11,108)(12,109)(13,110)(14,111)(15,112)(16,57)(17,58)(18,59)(19,60)(20,61)(21,62)(22,63)(23,64)(24,65)(25,66)(26,67)(27,68)(28,69)(29,70)(30,71)(31,72)(32,73)(33,74)(34,75)(35,76)(36,77)(37,78)(38,79)(39,80)(40,81)(41,82)(42,83)(43,84)(44,85)(45,86)(46,87)(47,88)(48,89)(49,90)(50,91)(51,92)(52,93)(53,94)(54,95)(55,96)(56,97), (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), (1,97,98,56)(2,55,99,96)(3,95,100,54)(4,53,101,94)(5,93,102,52)(6,51,103,92)(7,91,104,50)(8,49,105,90)(9,89,106,48)(10,47,107,88)(11,87,108,46)(12,45,109,86)(13,85,110,44)(14,43,111,84)(15,83,112,42)(16,41,57,82)(17,81,58,40)(18,39,59,80)(19,79,60,38)(20,37,61,78)(21,77,62,36)(22,35,63,76)(23,75,64,34)(24,33,65,74)(25,73,66,32)(26,31,67,72)(27,71,68,30)(28,29,69,70)>;

G:=Group( (1,98)(2,99)(3,100)(4,101)(5,102)(6,103)(7,104)(8,105)(9,106)(10,107)(11,108)(12,109)(13,110)(14,111)(15,112)(16,57)(17,58)(18,59)(19,60)(20,61)(21,62)(22,63)(23,64)(24,65)(25,66)(26,67)(27,68)(28,69)(29,70)(30,71)(31,72)(32,73)(33,74)(34,75)(35,76)(36,77)(37,78)(38,79)(39,80)(40,81)(41,82)(42,83)(43,84)(44,85)(45,86)(46,87)(47,88)(48,89)(49,90)(50,91)(51,92)(52,93)(53,94)(54,95)(55,96)(56,97), (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), (1,97,98,56)(2,55,99,96)(3,95,100,54)(4,53,101,94)(5,93,102,52)(6,51,103,92)(7,91,104,50)(8,49,105,90)(9,89,106,48)(10,47,107,88)(11,87,108,46)(12,45,109,86)(13,85,110,44)(14,43,111,84)(15,83,112,42)(16,41,57,82)(17,81,58,40)(18,39,59,80)(19,79,60,38)(20,37,61,78)(21,77,62,36)(22,35,63,76)(23,75,64,34)(24,33,65,74)(25,73,66,32)(26,31,67,72)(27,71,68,30)(28,29,69,70) );

G=PermutationGroup([(1,98),(2,99),(3,100),(4,101),(5,102),(6,103),(7,104),(8,105),(9,106),(10,107),(11,108),(12,109),(13,110),(14,111),(15,112),(16,57),(17,58),(18,59),(19,60),(20,61),(21,62),(22,63),(23,64),(24,65),(25,66),(26,67),(27,68),(28,69),(29,70),(30,71),(31,72),(32,73),(33,74),(34,75),(35,76),(36,77),(37,78),(38,79),(39,80),(40,81),(41,82),(42,83),(43,84),(44,85),(45,86),(46,87),(47,88),(48,89),(49,90),(50,91),(51,92),(52,93),(53,94),(54,95),(55,96),(56,97)], [(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)], [(1,97,98,56),(2,55,99,96),(3,95,100,54),(4,53,101,94),(5,93,102,52),(6,51,103,92),(7,91,104,50),(8,49,105,90),(9,89,106,48),(10,47,107,88),(11,87,108,46),(12,45,109,86),(13,85,110,44),(14,43,111,84),(15,83,112,42),(16,41,57,82),(17,81,58,40),(18,39,59,80),(19,79,60,38),(20,37,61,78),(21,77,62,36),(22,35,63,76),(23,75,64,34),(24,33,65,74),(25,73,66,32),(26,31,67,72),(27,71,68,30),(28,29,69,70)])

C2.D56 is a maximal subgroup of
C4×C56⋊C2  C4×D56  C4.5D56  C42.264D14  C42.16D14  D56⋊C4  C42.19D14  C42.20D14  D28.31D4  D2813D4  D28.32D4  D2814D4  C23.38D28  C22.D56  C23.13D28  Dic74D8  Dic7.SD16  C4⋊C4.D14  D7×D4⋊C4  (D4×D7)⋊C4  D14⋊D8  C7⋊C8⋊D4  D4⋊D7⋊C4  Dic77SD16  Q8⋊C4⋊D7  Q8⋊Dic7⋊C2  Q8⋊(C4×D7)  Q82D7⋊C4  D142SD16  C7⋊(C8⋊D4)  Q8⋊D7⋊C4  D283Q8  C4⋊D56  D28.19D4  C42.36D14  D284Q8  D28.3Q8  Dic148D4  D14.4SD16  C4.Q8⋊D7  D28⋊Q8  D28.Q8  D14.5D8  C2.D8⋊D7  D282Q8  D28.2Q8  C23.23D28  C5630D4  C5629D4  C23.48D28  C23.49D28  C562D4  C563D4  Dic7⋊D8  D28⋊D4  Dic75SD16  (C7×D4).D4  D146SD16  D287D4  (C2×Q16)⋊D7  D28.17D4
C2.D56 is a maximal quotient of
C4.17D56  C22.2D56  C4.D56  C28.2D8  C56.78D4  C2.D112  D56.1C4  C28.3D8  C28.4D8  D562C4  C28.9C42

62 conjugacy classes

class 1 2A2B2C2D2E4A4B4C4D7A7B7C8A8B8C8D14A···14I28A···28L56A···56X
order1222224444777888814···1428···2856···56
size1111282822282822222222···22···22···2

62 irreducible representations

dim1111122222222222
type+++++++++++
imageC1C2C2C2C4D4D4D7D8SD16D14C4×D7C7⋊D4D28C56⋊C2D56
kernelC2.D56C4⋊Dic7C2×C56C2×D28D28C28C2×C14C2×C8C14C14C2×C4C4C4C22C2C2
# reps111141132236661212

Matrix representation of C2.D56 in GL3(𝔽113) generated by

11200
01120
00112
,
1500
08124
08971
,
9800
08124
07532
G:=sub<GL(3,GF(113))| [112,0,0,0,112,0,0,0,112],[15,0,0,0,81,89,0,24,71],[98,0,0,0,81,75,0,24,32] >;

C2.D56 in GAP, Magma, Sage, TeX

C_2.D_{56}
% in TeX

G:=Group("C2.D56");
// GroupNames label

G:=SmallGroup(224,27);
// by ID

G=gap.SmallGroup(224,27);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-7,73,79,362,86,6917]);
// Polycyclic

G:=Group<a,b,c|a^2=b^56=1,c^2=a,a*b=b*a,a*c=c*a,c*b*c^-1=a*b^-1>;
// generators/relations

Export

Subgroup lattice of C2.D56 in TeX

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