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G = D56⋊C2order 224 = 25·7

6th semidirect product of D56 and C2 acting faithfully

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C83D14, D566C2, Q82D14, C563C22, SD161D7, D4.3D14, D14.7D4, D282C22, C28.5C23, Dic7.9D4, D4⋊D73C2, (D4×D7)⋊3C2, C7⋊C82C22, Q8⋊D72C2, C8⋊D71C2, C73(C8⋊C22), C2.19(D4×D7), Q82D71C2, (C7×SD16)⋊1C2, C14.31(C2×D4), (C7×Q8)⋊2C22, C4.5(C22×D7), (C7×D4).3C22, (C4×D7).2C22, SmallGroup(224,109)

Series: Derived Chief Lower central Upper central

C1C28 — D56⋊C2
C1C7C14C28C4×D7D4×D7 — D56⋊C2
C7C14C28 — D56⋊C2
C1C2C4SD16

Generators and relations for D56⋊C2
 G = < a,b,c | a56=b2=c2=1, bab=a-1, cac=a43, bc=cb >

Subgroups: 374 in 68 conjugacy classes, 27 normal (all characteristic)
C1, C2, C2 [×4], C4, C4 [×2], C22 [×6], C7, C8, C8, C2×C4 [×2], D4, D4 [×4], Q8, C23, D7 [×3], C14, C14, M4(2), D8 [×2], SD16, SD16, C2×D4, C4○D4, Dic7, C28, C28, D14, D14 [×4], C2×C14, C8⋊C22, C7⋊C8, C56, C4×D7, C4×D7, D28 [×2], D28, C7⋊D4, C7×D4, C7×Q8, C22×D7, C8⋊D7, D56, D4⋊D7, Q8⋊D7, C7×SD16, D4×D7, Q82D7, D56⋊C2
Quotients: C1, C2 [×7], C22 [×7], D4 [×2], C23, D7, C2×D4, D14 [×3], C8⋊C22, C22×D7, D4×D7, D56⋊C2

Smallest permutation representation of D56⋊C2
On 56 points
Generators in S56
(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)
(1 21)(2 20)(3 19)(4 18)(5 17)(6 16)(7 15)(8 14)(9 13)(10 12)(22 56)(23 55)(24 54)(25 53)(26 52)(27 51)(28 50)(29 49)(30 48)(31 47)(32 46)(33 45)(34 44)(35 43)(36 42)(37 41)(38 40)
(2 44)(3 31)(4 18)(6 48)(7 35)(8 22)(10 52)(11 39)(12 26)(14 56)(15 43)(16 30)(19 47)(20 34)(23 51)(24 38)(27 55)(28 42)(32 46)(36 50)(40 54)

G:=sub<Sym(56)| (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), (1,21)(2,20)(3,19)(4,18)(5,17)(6,16)(7,15)(8,14)(9,13)(10,12)(22,56)(23,55)(24,54)(25,53)(26,52)(27,51)(28,50)(29,49)(30,48)(31,47)(32,46)(33,45)(34,44)(35,43)(36,42)(37,41)(38,40), (2,44)(3,31)(4,18)(6,48)(7,35)(8,22)(10,52)(11,39)(12,26)(14,56)(15,43)(16,30)(19,47)(20,34)(23,51)(24,38)(27,55)(28,42)(32,46)(36,50)(40,54)>;

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), (1,21)(2,20)(3,19)(4,18)(5,17)(6,16)(7,15)(8,14)(9,13)(10,12)(22,56)(23,55)(24,54)(25,53)(26,52)(27,51)(28,50)(29,49)(30,48)(31,47)(32,46)(33,45)(34,44)(35,43)(36,42)(37,41)(38,40), (2,44)(3,31)(4,18)(6,48)(7,35)(8,22)(10,52)(11,39)(12,26)(14,56)(15,43)(16,30)(19,47)(20,34)(23,51)(24,38)(27,55)(28,42)(32,46)(36,50)(40,54) );

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)], [(1,21),(2,20),(3,19),(4,18),(5,17),(6,16),(7,15),(8,14),(9,13),(10,12),(22,56),(23,55),(24,54),(25,53),(26,52),(27,51),(28,50),(29,49),(30,48),(31,47),(32,46),(33,45),(34,44),(35,43),(36,42),(37,41),(38,40)], [(2,44),(3,31),(4,18),(6,48),(7,35),(8,22),(10,52),(11,39),(12,26),(14,56),(15,43),(16,30),(19,47),(20,34),(23,51),(24,38),(27,55),(28,42),(32,46),(36,50),(40,54)])

D56⋊C2 is a maximal subgroup of
D28.29D4  D810D14  D815D14  D7×C8⋊C22  D85D14  D56⋊C22  C56.C23
D56⋊C2 is a maximal quotient of
C4⋊C4.D14  D4.2Dic14  (D4×D7)⋊C4  D4⋊D28  C7⋊C8⋊D4  C561C4⋊C2  D4⋊D7⋊C4  D283D4  Dic7.Q16  Q8⋊C4⋊D7  Q8⋊(C4×D7)  D14.Q16  D284D4  C7⋊(C8⋊D4)  Q8⋊D7⋊C4  D28.12D4  C563Q8  C8⋊(C4×D7)  D14.4SD16  C567D4  C4.Q8⋊D7  D569C4  D28⋊Q8  D28.Q8  Dic75SD16  SD16⋊Dic7  (C7×D4).D4  D146SD16  D287D4  C568D4  C569D4

32 conjugacy classes

class 1 2A2B2C2D2E4A4B4C7A7B7C8A8B14A14B14C14D14E14F28A28B28C28D28E28F56A···56F
order1222224447778814141414141428282828282856···56
size11414282824142224282228884448884···4

32 irreducible representations

dim11111111222222444
type+++++++++++++++++
imageC1C2C2C2C2C2C2C2D4D4D7D14D14D14C8⋊C22D4×D7D56⋊C2
kernelD56⋊C2C8⋊D7D56D4⋊D7Q8⋊D7C7×SD16D4×D7Q82D7Dic7D14SD16C8D4Q8C7C2C1
# reps11111111113333136

Matrix representation of D56⋊C2 in GL4(𝔽113) generated by

987987
1068610686
645300
607000
,
34100
887900
7911279112
25342534
,
1000
0100
11201120
01120112
G:=sub<GL(4,GF(113))| [98,106,64,60,7,86,53,70,98,106,0,0,7,86,0,0],[34,88,79,25,1,79,112,34,0,0,79,25,0,0,112,34],[1,0,112,0,0,1,0,112,0,0,112,0,0,0,0,112] >;

D56⋊C2 in GAP, Magma, Sage, TeX

D_{56}\rtimes C_2
% in TeX

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

G:=SmallGroup(224,109);
// by ID

G=gap.SmallGroup(224,109);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-7,362,116,86,297,159,69,6917]);
// Polycyclic

G:=Group<a,b,c|a^56=b^2=c^2=1,b*a*b=a^-1,c*a*c=a^43,b*c=c*b>;
// generators/relations

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