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G = D7×3- 1+2  order 378 = 2·33·7

Direct product of D7 and 3- 1+2

direct product, metacyclic, supersoluble, monomial

Aliases: D7×3- 1+2, C63⋊12C6, C9⋊2(C3×D7), (C9×D7)⋊4C3, C32.(C3×D7), (C3×C21).2C6, C21.15(C3×C6), C3.3(C32×D7), (C32×D7).1C3, (C3×D7).9C32, C7⋊7(C2×3- 1+2), (C7×3- 1+2)⋊3C2, SmallGroup(378,31)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C21 — D7×3- 1+2
Chief series
C1 — C7 — C21 — C63 — C7×3- 1+2 — D7×3- 1+2
Lower central
C7 — C21 — D7×3- 1+2
Upper central
C1 — C3 — 3- 1+2

Generators and relations for D7×3- 1+2
 G = < a,b,c,d | a7=b2=c9=d3=1, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c4 >

C1
7C2
C3
3C3
C7
7C6
21C6
C32
C9
C9
C9
D7
C21
3C21
7C18
7C18
7C3×C6
7C18
3- 1+2
C3×D7
3C3×D7
C63
C63
C63
C3×C21
7C2×3- 1+2
C9×D7
C9×D7
C32×D7
C9×D7
C7×3- 1+2
D7×3- 1+2

Smallest permutation representation of D7×3- 1+2
►On 63 points
Generators in S63
(1 39 17 24 63 49 33)(2 40 18 25 55 50 34)(3 41 10 26 56 51 35)(4 42 11 27 57 52 36)(5 43 12 19 58 53 28)(6 44 13 20 59 54 29)(7 45 14 21 60 46 30)(8 37 15 22 61 47 31)(9 38 16 23 62 48 32)

(1 33)(2 34)(3 35)(4 36)(5 28)(6 29)(7 30)(8 31)(9 32)(10 56)(11 57)(12 58)(13 59)(14 60)(15 61)(16 62)(17 63)(18 55)(37 47)(38 48)(39 49)(40 50)(41 51)(42 52)(43 53)(44 54)(45 46)

(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)

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

55 conjugacy classes

class 1  2 3A3B3C3D6A6B6C6D7A7B7C9A···9F18A···18F21A···21F21G···21L63A···63R
order12333366667779···918···1821···2121···2163···63
size1711337721212223···321···212···26···66···6

55 irreducible representations

dim111111222336
type+++
imageC1C2C3C3C6C6D7C3×D7C3×D73- 1+2C2×3- 1+2D7×3- 1+2
kernelD7×3- 1+2C7×3- 1+2C9×D7C32×D7C63C3×C213- 1+2C9C32D7C7C1
# reps1162623186226

Matrix representation of D7×3- 1+2 ►in GL5(𝔽127)

361000
1260000
00100
00010
00001
,
01000
10000
0012600
0001260
0000126
,
190000
019000
0011050
0076126107
00761260
,
190000
019000
00100
0080190
00760107

G:=sub<GL(5,GF(127))| [36,126,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[0,1,0,0,0,1,0,0,0,0,0,0,126,0,0,0,0,0,126,0,0,0,0,0,126],[19,0,0,0,0,0,19,0,0,0,0,0,1,76,76,0,0,105,126,126,0,0,0,107,0],[19,0,0,0,0,0,19,0,0,0,0,0,1,80,76,0,0,0,19,0,0,0,0,0,107] >;
 

D7×3- 1+2 in GAP, Magma, Sage, TeX 

D_7\times 3_-^{1+2}
 
% in TeX
 
G:=Group("D7xES-(3,1)");
 
// GroupNames label
 
G:=SmallGroup(378,31);
 
// by ID
 
G=gap.SmallGroup(378,31);
 
# by ID
 
G:=PCGroup([5,-2,-3,-3,-3,-7,187,57,8104]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^7=b^2=c^9=d^3=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^4>;
 
// generators/relations
 

Export

Subgroup lattice of D7×3- 1+2 in TeX

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