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## G = C7⋊C3×D11order 462 = 2·3·7·11

### Direct product of C7⋊C3 and D11

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

 Derived series C1 — C77 — C7⋊C3×D11
 Chief series C1 — C11 — C77 — C11×C7⋊C3 — C7⋊C3×D11
 Lower central C77 — C7⋊C3×D11
 Upper central 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 >

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 3A 3B 6A 6B 7A 7B 11A ··· 11E 14A 14B 33A ··· 33J 77A ··· 77J order 1 2 3 3 6 6 7 7 11 ··· 11 14 14 33 ··· 33 77 ··· 77 size 1 11 7 7 77 77 3 3 2 ··· 2 33 33 14 ··· 14 6 ··· 6

35 irreducible representations

 dim 1 1 1 1 2 2 3 3 6 type + + + image C1 C2 C3 C6 D11 C3×D11 C7⋊C3 C2×C7⋊C3 C7⋊C3×D11 kernel C7⋊C3×D11 C11×C7⋊C3 C7×D11 C77 C7⋊C3 C7 D11 C11 C1 # reps 1 1 2 2 5 10 2 2 10

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

 1 0 0 0 0 0 1 0 0 0 0 0 1 0 181 0 0 347 462 382 0 0 0 1 382
,
 21 0 0 0 0 0 21 0 0 0 0 0 1 0 336 0 0 0 0 462 0 0 0 1 462
,
 215 1 0 0 0 442 366 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1
,
 379 334 0 0 0 349 84 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1

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

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