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G = C7⋊C3×D11order 462 = 2·3·7·11

Direct product of C7⋊C3 and D11

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

Aliases: C7⋊C3×D11, C772C6, (C7×D11)⋊C3, C72(C3×D11), C11⋊(C2×C7⋊C3), (C11×C7⋊C3)⋊2C2, SmallGroup(462,1)

Series: Derived Chief Lower central Upper central

C1C77 — C7⋊C3×D11
C1C11C77C11×C7⋊C3 — C7⋊C3×D11
C77 — C7⋊C3×D11
C1

Generators and relations for C7⋊C3×D11
 G = < a,b,c,d | a7=b3=c11=d2=1, bab-1=a4, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >

11C2
7C3
77C6
11C14
7C33
11C2×C7⋊C3
7C3×D11

Smallest permutation representation of C7⋊C3×D11
On 77 points
Generators in S77
(1 21 32 43 54 65 76)(2 22 33 44 55 66 77)(3 12 23 34 45 56 67)(4 13 24 35 46 57 68)(5 14 25 36 47 58 69)(6 15 26 37 48 59 70)(7 16 27 38 49 60 71)(8 17 28 39 50 61 72)(9 18 29 40 51 62 73)(10 19 30 41 52 63 74)(11 20 31 42 53 64 75)
(12 23 45)(13 24 46)(14 25 47)(15 26 48)(16 27 49)(17 28 50)(18 29 51)(19 30 52)(20 31 53)(21 32 54)(22 33 55)(34 67 56)(35 68 57)(36 69 58)(37 70 59)(38 71 60)(39 72 61)(40 73 62)(41 74 63)(42 75 64)(43 76 65)(44 77 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)| (1,21,32,43,54,65,76)(2,22,33,44,55,66,77)(3,12,23,34,45,56,67)(4,13,24,35,46,57,68)(5,14,25,36,47,58,69)(6,15,26,37,48,59,70)(7,16,27,38,49,60,71)(8,17,28,39,50,61,72)(9,18,29,40,51,62,73)(10,19,30,41,52,63,74)(11,20,31,42,53,64,75), (12,23,45)(13,24,46)(14,25,47)(15,26,48)(16,27,49)(17,28,50)(18,29,51)(19,30,52)(20,31,53)(21,32,54)(22,33,55)(34,67,56)(35,68,57)(36,69,58)(37,70,59)(38,71,60)(39,72,61)(40,73,62)(41,74,63)(42,75,64)(43,76,65)(44,77,66), (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( (1,21,32,43,54,65,76)(2,22,33,44,55,66,77)(3,12,23,34,45,56,67)(4,13,24,35,46,57,68)(5,14,25,36,47,58,69)(6,15,26,37,48,59,70)(7,16,27,38,49,60,71)(8,17,28,39,50,61,72)(9,18,29,40,51,62,73)(10,19,30,41,52,63,74)(11,20,31,42,53,64,75), (12,23,45)(13,24,46)(14,25,47)(15,26,48)(16,27,49)(17,28,50)(18,29,51)(19,30,52)(20,31,53)(21,32,54)(22,33,55)(34,67,56)(35,68,57)(36,69,58)(37,70,59)(38,71,60)(39,72,61)(40,73,62)(41,74,63)(42,75,64)(43,76,65)(44,77,66), (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([(1,21,32,43,54,65,76),(2,22,33,44,55,66,77),(3,12,23,34,45,56,67),(4,13,24,35,46,57,68),(5,14,25,36,47,58,69),(6,15,26,37,48,59,70),(7,16,27,38,49,60,71),(8,17,28,39,50,61,72),(9,18,29,40,51,62,73),(10,19,30,41,52,63,74),(11,20,31,42,53,64,75)], [(12,23,45),(13,24,46),(14,25,47),(15,26,48),(16,27,49),(17,28,50),(18,29,51),(19,30,52),(20,31,53),(21,32,54),(22,33,55),(34,67,56),(35,68,57),(36,69,58),(37,70,59),(38,71,60),(39,72,61),(40,73,62),(41,74,63),(42,75,64),(43,76,65),(44,77,66)], [(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  2 3A3B6A6B7A7B11A···11E14A14B33A···33J77A···77J
order1233667711···11141433···3377···77
size111777777332···2333314···146···6

35 irreducible representations

dim111122336
type+++
imageC1C2C3C6D11C3×D11C7⋊C3C2×C7⋊C3C7⋊C3×D11
kernelC7⋊C3×D11C11×C7⋊C3C7×D11C77C7⋊C3C7D11C11C1
# reps11225102210

Matrix representation of C7⋊C3×D11 in GL5(𝔽463)

10000
01000
0010181
00347462382
0001382
,
210000
021000
0010336
0000462
0001462
,
2151000
442366000
00100
00010
00001
,
379334000
34984000
00100
00010
00001

G:=sub<GL(5,GF(463))| [1,0,0,0,0,0,1,0,0,0,0,0,1,347,0,0,0,0,462,1,0,0,181,382,382],[21,0,0,0,0,0,21,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,336,462,462],[215,442,0,0,0,1,366,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[379,349,0,0,0,334,84,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1] >;

C7⋊C3×D11 in GAP, Magma, Sage, TeX

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

G:=Group("C7:C3xD11");
// GroupNames label

G:=SmallGroup(462,1);
// by ID

G=gap.SmallGroup(462,1);
# by ID

G:=PCGroup([4,-2,-3,-7,-11,78,6723]);
// Polycyclic

G:=Group<a,b,c,d|a^7=b^3=c^11=d^2=1,b*a*b^-1=a^4,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 C7⋊C3×D11 in TeX

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