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G = C7×C11⋊C5order 385 = 5·7·11

Direct product of C7 and C11⋊C5

direct product, metacyclic, supersoluble, monomial, Z-group, 5-hyperelementary

Aliases: C7×C11⋊C5, C77⋊C5, C11⋊C35, SmallGroup(385,1)

Series: Derived Chief Lower central Upper central

C1C11 — C7×C11⋊C5
C1C11C77 — C7×C11⋊C5
C11 — C7×C11⋊C5
C1C7

Generators and relations for C7×C11⋊C5
 G = < a,b,c | a7=b11=c5=1, ab=ba, ac=ca, cbc-1=b3 >

11C5
11C35

Smallest permutation representation of C7×C11⋊C5
On 77 points
Generators in S77
(1 67 56 45 34 23 12)(2 68 57 46 35 24 13)(3 69 58 47 36 25 14)(4 70 59 48 37 26 15)(5 71 60 49 38 27 16)(6 72 61 50 39 28 17)(7 73 62 51 40 29 18)(8 74 63 52 41 30 19)(9 75 64 53 42 31 20)(10 76 65 54 43 32 21)(11 77 66 55 44 33 22)
(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)
(2 5 6 10 4)(3 9 11 8 7)(13 16 17 21 15)(14 20 22 19 18)(24 27 28 32 26)(25 31 33 30 29)(35 38 39 43 37)(36 42 44 41 40)(46 49 50 54 48)(47 53 55 52 51)(57 60 61 65 59)(58 64 66 63 62)(68 71 72 76 70)(69 75 77 74 73)

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

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

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

49 conjugacy classes

class 1 5A5B5C5D7A···7F11A11B35A···35X77A···77L
order155557···7111135···3577···77
size1111111111···15511···115···5

49 irreducible representations

dim111155
type+
imageC1C5C7C35C11⋊C5C7×C11⋊C5
kernelC7×C11⋊C5C77C11⋊C5C11C7C1
# reps14624212

Matrix representation of C7×C11⋊C5 in GL5(𝔽2311)

1590000
0159000
0015900
0001590
0000159
,
1711259717121
1712259717121
1711359717121
1711259817121
1711259717131
,
00100
171360023091714599
171459917112598
10000
00010

G:=sub<GL(5,GF(2311))| [159,0,0,0,0,0,159,0,0,0,0,0,159,0,0,0,0,0,159,0,0,0,0,0,159],[1711,1712,1711,1711,1711,2,2,3,2,2,597,597,597,598,597,1712,1712,1712,1712,1713,1,1,1,1,1],[0,1713,1714,1,0,0,600,599,0,0,1,2309,1711,0,0,0,1714,2,0,1,0,599,598,0,0] >;

C7×C11⋊C5 in GAP, Magma, Sage, TeX

C_7\times C_{11}\rtimes C_5
% in TeX

G:=Group("C7xC11:C5");
// GroupNames label

G:=SmallGroup(385,1);
// by ID

G=gap.SmallGroup(385,1);
# by ID

G:=PCGroup([3,-5,-7,-11,1262]);
// Polycyclic

G:=Group<a,b,c|a^7=b^11=c^5=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^3>;
// generators/relations

Export

Subgroup lattice of C7×C11⋊C5 in TeX

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