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G = C7×D11order 154 = 2·7·11

Direct product of C7 and D11

direct product, metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary

Aliases: C7×D11, C11⋊C14, C772C2, SmallGroup(154,2)

Series: Derived Chief Lower central Upper central

C1C11 — C7×D11
C1C11C77 — C7×D11
C11 — C7×D11
C1C7

Generators and relations for C7×D11
 G = < a,b,c | a7=b11=c2=1, ab=ba, ac=ca, cbc=b-1 >

11C2
11C14

Smallest permutation representation of C7×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 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), (1,2,3,4,5,6,7,8,9,10,11)(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), (1,2,3,4,5,6,7,8,9,10,11)(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)], [(1,2,3,4,5,6,7,8,9,10,11),(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)])

49 conjugacy classes

class 1  2 7A···7F11A···11E14A···14F77A···77AD
order127···711···1114···1477···77
size1111···12···211···112···2

49 irreducible representations

dim111122
type+++
imageC1C2C7C14D11C7×D11
kernelC7×D11C77D11C11C7C1
# reps1166530

Matrix representation of C7×D11 in GL2(𝔽43) generated by

40
04
,
422
3515
,
1515
828
G:=sub<GL(2,GF(43))| [4,0,0,4],[42,35,2,15],[15,8,15,28] >;

C7×D11 in GAP, Magma, Sage, TeX

C_7\times D_{11}
% in TeX

G:=Group("C7xD11");
// GroupNames label

G:=SmallGroup(154,2);
// by ID

G=gap.SmallGroup(154,2);
# by ID

G:=PCGroup([3,-2,-7,-11,1262]);
// Polycyclic

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

Export

Subgroup lattice of C7×D11 in TeX

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