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## G = D7×3- 1+2order 378 = 2·33·7

### Direct product of D7 and 3- 1+2

Aliases: D7×3- 1+2, C6312C6, C92(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, C77(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 >

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 3A 3B 3C 3D 6A 6B 6C 6D 7A 7B 7C 9A ··· 9F 18A ··· 18F 21A ··· 21F 21G ··· 21L 63A ··· 63R order 1 2 3 3 3 3 6 6 6 6 7 7 7 9 ··· 9 18 ··· 18 21 ··· 21 21 ··· 21 63 ··· 63 size 1 7 1 1 3 3 7 7 21 21 2 2 2 3 ··· 3 21 ··· 21 2 ··· 2 6 ··· 6 6 ··· 6

55 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 3 3 6 type + + + image C1 C2 C3 C3 C6 C6 D7 C3×D7 C3×D7 3- 1+2 C2×3- 1+2 D7×3- 1+2 kernel D7×3- 1+2 C7×3- 1+2 C9×D7 C32×D7 C63 C3×C21 3- 1+2 C9 C32 D7 C7 C1 # reps 1 1 6 2 6 2 3 18 6 2 2 6

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

 36 1 0 0 0 126 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 1 105 0 0 0 76 126 107 0 0 76 126 0
,
 19 0 0 0 0 0 19 0 0 0 0 0 1 0 0 0 0 80 19 0 0 0 76 0 107

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

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