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G = C11×C7⋊C3order 231 = 3·7·11

Direct product of C11 and C7⋊C3

direct product, metacyclic, supersoluble, monomial, Z-group, 3-hyperelementary

Aliases: C11×C7⋊C3, C7⋊C33, C77⋊C3, SmallGroup(231,1)

Series: Derived Chief Lower central Upper central

C1C7 — C11×C7⋊C3
C1C7C77 — C11×C7⋊C3
C7 — C11×C7⋊C3
C1C11

Generators and relations for C11×C7⋊C3
 G = < a,b,c | a11=b7=c3=1, ab=ba, ac=ca, cbc-1=b4 >

7C3
7C33

Smallest permutation representation of C11×C7⋊C3
On 77 points
Generators in S77
(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 24 13 49 40 60 70)(2 25 14 50 41 61 71)(3 26 15 51 42 62 72)(4 27 16 52 43 63 73)(5 28 17 53 44 64 74)(6 29 18 54 34 65 75)(7 30 19 55 35 66 76)(8 31 20 45 36 56 77)(9 32 21 46 37 57 67)(10 33 22 47 38 58 68)(11 23 12 48 39 59 69)
(12 39 23)(13 40 24)(14 41 25)(15 42 26)(16 43 27)(17 44 28)(18 34 29)(19 35 30)(20 36 31)(21 37 32)(22 38 33)(45 77 56)(46 67 57)(47 68 58)(48 69 59)(49 70 60)(50 71 61)(51 72 62)(52 73 63)(53 74 64)(54 75 65)(55 76 66)

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

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

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

C11×C7⋊C3 is a maximal subgroup of   C11⋊F7

55 conjugacy classes

class 1 3A3B7A7B11A···11J33A···33T77A···77T
order1337711···1133···3377···77
size177331···17···73···3

55 irreducible representations

dim111133
type+
imageC1C3C11C33C7⋊C3C11×C7⋊C3
kernelC11×C7⋊C3C77C7⋊C3C7C11C1
# reps121020220

Matrix representation of C11×C7⋊C3 in GL3(𝔽463) generated by

15800
01580
00158
,
001
10383
01382
,
10382
00462
01462
G:=sub<GL(3,GF(463))| [158,0,0,0,158,0,0,0,158],[0,1,0,0,0,1,1,383,382],[1,0,0,0,0,1,382,462,462] >;

C11×C7⋊C3 in GAP, Magma, Sage, TeX

C_{11}\times C_7\rtimes C_3
% in TeX

G:=Group("C11xC7:C3");
// GroupNames label

G:=SmallGroup(231,1);
// by ID

G=gap.SmallGroup(231,1);
# by ID

G:=PCGroup([3,-3,-11,-7,596]);
// Polycyclic

G:=Group<a,b,c|a^11=b^7=c^3=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^4>;
// generators/relations

Export

Subgroup lattice of C11×C7⋊C3 in TeX

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