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G = D7×D11order 308 = 22·7·11

Direct product of D7 and D11

direct product, metabelian, supersoluble, monomial, A-group, 2-hyperelementary

Aliases: D7×D11, D77⋊C2, C71D22, C77⋊C22, C111D14, (C7×D11)⋊C2, (C11×D7)⋊C2, SmallGroup(308,5)

Series: Derived Chief Lower central Upper central

C1C77 — D7×D11
C1C11C77C7×D11 — D7×D11
C77 — D7×D11
C1

Generators and relations for D7×D11
 G = < a,b,c,d | a7=b2=c11=d2=1, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >

7C2
11C2
77C2
77C22
11C14
11D7
7C22
7D11
11D14
7D22

Smallest permutation representation of D7×D11
On 77 points
Generators in S77
(1 76 65 54 43 32 21)(2 77 66 55 44 33 22)(3 67 56 45 34 23 12)(4 68 57 46 35 24 13)(5 69 58 47 36 25 14)(6 70 59 48 37 26 15)(7 71 60 49 38 27 16)(8 72 61 50 39 28 17)(9 73 62 51 40 29 18)(10 74 63 52 41 30 19)(11 75 64 53 42 31 20)
(1 21)(2 22)(3 12)(4 13)(5 14)(6 15)(7 16)(8 17)(9 18)(10 19)(11 20)(23 67)(24 68)(25 69)(26 70)(27 71)(28 72)(29 73)(30 74)(31 75)(32 76)(33 77)(34 56)(35 57)(36 58)(37 59)(38 60)(39 61)(40 62)(41 63)(42 64)(43 65)(44 66)
(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)
(1 11)(2 10)(3 9)(4 8)(5 7)(12 18)(13 17)(14 16)(19 22)(20 21)(23 29)(24 28)(25 27)(30 33)(31 32)(34 40)(35 39)(36 38)(41 44)(42 43)(45 51)(46 50)(47 49)(52 55)(53 54)(56 62)(57 61)(58 60)(63 66)(64 65)(67 73)(68 72)(69 71)(74 77)(75 76)

G:=sub<Sym(77)| (1,76,65,54,43,32,21)(2,77,66,55,44,33,22)(3,67,56,45,34,23,12)(4,68,57,46,35,24,13)(5,69,58,47,36,25,14)(6,70,59,48,37,26,15)(7,71,60,49,38,27,16)(8,72,61,50,39,28,17)(9,73,62,51,40,29,18)(10,74,63,52,41,30,19)(11,75,64,53,42,31,20), (1,21)(2,22)(3,12)(4,13)(5,14)(6,15)(7,16)(8,17)(9,18)(10,19)(11,20)(23,67)(24,68)(25,69)(26,70)(27,71)(28,72)(29,73)(30,74)(31,75)(32,76)(33,77)(34,56)(35,57)(36,58)(37,59)(38,60)(39,61)(40,62)(41,63)(42,64)(43,65)(44,66), (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), (1,11)(2,10)(3,9)(4,8)(5,7)(12,18)(13,17)(14,16)(19,22)(20,21)(23,29)(24,28)(25,27)(30,33)(31,32)(34,40)(35,39)(36,38)(41,44)(42,43)(45,51)(46,50)(47,49)(52,55)(53,54)(56,62)(57,61)(58,60)(63,66)(64,65)(67,73)(68,72)(69,71)(74,77)(75,76)>;

G:=Group( (1,76,65,54,43,32,21)(2,77,66,55,44,33,22)(3,67,56,45,34,23,12)(4,68,57,46,35,24,13)(5,69,58,47,36,25,14)(6,70,59,48,37,26,15)(7,71,60,49,38,27,16)(8,72,61,50,39,28,17)(9,73,62,51,40,29,18)(10,74,63,52,41,30,19)(11,75,64,53,42,31,20), (1,21)(2,22)(3,12)(4,13)(5,14)(6,15)(7,16)(8,17)(9,18)(10,19)(11,20)(23,67)(24,68)(25,69)(26,70)(27,71)(28,72)(29,73)(30,74)(31,75)(32,76)(33,77)(34,56)(35,57)(36,58)(37,59)(38,60)(39,61)(40,62)(41,63)(42,64)(43,65)(44,66), (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), (1,11)(2,10)(3,9)(4,8)(5,7)(12,18)(13,17)(14,16)(19,22)(20,21)(23,29)(24,28)(25,27)(30,33)(31,32)(34,40)(35,39)(36,38)(41,44)(42,43)(45,51)(46,50)(47,49)(52,55)(53,54)(56,62)(57,61)(58,60)(63,66)(64,65)(67,73)(68,72)(69,71)(74,77)(75,76) );

G=PermutationGroup([(1,76,65,54,43,32,21),(2,77,66,55,44,33,22),(3,67,56,45,34,23,12),(4,68,57,46,35,24,13),(5,69,58,47,36,25,14),(6,70,59,48,37,26,15),(7,71,60,49,38,27,16),(8,72,61,50,39,28,17),(9,73,62,51,40,29,18),(10,74,63,52,41,30,19),(11,75,64,53,42,31,20)], [(1,21),(2,22),(3,12),(4,13),(5,14),(6,15),(7,16),(8,17),(9,18),(10,19),(11,20),(23,67),(24,68),(25,69),(26,70),(27,71),(28,72),(29,73),(30,74),(31,75),(32,76),(33,77),(34,56),(35,57),(36,58),(37,59),(38,60),(39,61),(40,62),(41,63),(42,64),(43,65),(44,66)], [(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)], [(1,11),(2,10),(3,9),(4,8),(5,7),(12,18),(13,17),(14,16),(19,22),(20,21),(23,29),(24,28),(25,27),(30,33),(31,32),(34,40),(35,39),(36,38),(41,44),(42,43),(45,51),(46,50),(47,49),(52,55),(53,54),(56,62),(57,61),(58,60),(63,66),(64,65),(67,73),(68,72),(69,71),(74,77),(75,76)])

35 conjugacy classes

class 1 2A2B2C7A7B7C11A···11E14A14B14C22A···22E77A···77O
order122277711···1114141422···2277···77
size1711772222···222222214···144···4

35 irreducible representations

dim111122224
type+++++++++
imageC1C2C2C2D7D11D14D22D7×D11
kernelD7×D11C11×D7C7×D11D77D11D7C11C7C1
# reps1111353515

Matrix representation of D7×D11 in GL4(𝔽463) generated by

1000
0100
0001
00462320
,
1000
0100
0001
0010
,
391100
40416800
0010
0001
,
2154500
8424800
0010
0001
G:=sub<GL(4,GF(463))| [1,0,0,0,0,1,0,0,0,0,0,462,0,0,1,320],[1,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0],[391,404,0,0,1,168,0,0,0,0,1,0,0,0,0,1],[215,84,0,0,45,248,0,0,0,0,1,0,0,0,0,1] >;

D7×D11 in GAP, Magma, Sage, TeX

D_7\times D_{11}
% in TeX

G:=Group("D7xD11");
// GroupNames label

G:=SmallGroup(308,5);
// by ID

G=gap.SmallGroup(308,5);
# by ID

G:=PCGroup([4,-2,-2,-7,-11,150,4483]);
// Polycyclic

G:=Group<a,b,c,d|a^7=b^2=c^11=d^2=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations

Export

Subgroup lattice of D7×D11 in TeX

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