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## G = D7×D11order 308 = 22·7·11

### Direct product of D7 and D11

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

 Derived series C1 — C77 — D7×D11
 Chief series C1 — C11 — C77 — C7×D11 — D7×D11
 Lower central C77 — D7×D11
 Upper central 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 2A 2B 2C 7A 7B 7C 11A ··· 11E 14A 14B 14C 22A ··· 22E 77A ··· 77O order 1 2 2 2 7 7 7 11 ··· 11 14 14 14 22 ··· 22 77 ··· 77 size 1 7 11 77 2 2 2 2 ··· 2 22 22 22 14 ··· 14 4 ··· 4

35 irreducible representations

 dim 1 1 1 1 2 2 2 2 4 type + + + + + + + + + image C1 C2 C2 C2 D7 D11 D14 D22 D7×D11 kernel D7×D11 C11×D7 C7×D11 D77 D11 D7 C11 C7 C1 # reps 1 1 1 1 3 5 3 5 15

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

 1 0 0 0 0 1 0 0 0 0 0 1 0 0 462 320
,
 1 0 0 0 0 1 0 0 0 0 0 1 0 0 1 0
,
 391 1 0 0 404 168 0 0 0 0 1 0 0 0 0 1
,
 215 45 0 0 84 248 0 0 0 0 1 0 0 0 0 1
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

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