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## G = D7⋊He3order 378 = 2·33·7

### The semidirect product of D7 and He3 acting via He3/C32=C3

Aliases: D7⋊He3, C322F7, C7⋊(C2×He3), (C3×F7)⋊C3, C7⋊He32C2, (C3×C21)⋊6C6, C3.6(C3×F7), C21.6(C3×C6), (C32×D7)⋊2C3, (C3×D7).6C32, (C3×C7⋊C3)⋊C6, SmallGroup(378,12)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C21 — D7⋊He3
 Chief series C1 — C7 — C21 — C3×C21 — C7⋊He3 — D7⋊He3
 Lower central C7 — C21 — D7⋊He3
 Upper central C1 — C3 — C32

Generators and relations for D7⋊He3
G = < a,b,c,d,e | a7=b2=c3=d3=e3=1, bab=a-1, ac=ca, ad=da, eae-1=a2, bc=cb, bd=db, ebe-1=ab, cd=dc, ece-1=cd-1, de=ed >

Smallest permutation representation of D7⋊He3
On 63 points
Generators in S63
(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)
(1 7)(2 6)(3 5)(8 10)(11 14)(12 13)(15 17)(18 21)(19 20)(22 24)(25 28)(26 27)(29 31)(32 35)(33 34)(36 38)(39 42)(40 41)(43 45)(46 49)(47 48)(50 52)(53 56)(54 55)(57 59)(60 63)(61 62)
(1 48 27)(2 49 28)(3 43 22)(4 44 23)(5 45 24)(6 46 25)(7 47 26)(8 50 29)(9 51 30)(10 52 31)(11 53 32)(12 54 33)(13 55 34)(14 56 35)(15 57 36)(16 58 37)(17 59 38)(18 60 39)(19 61 40)(20 62 41)(21 63 42)
(1 20 13)(2 21 14)(3 15 8)(4 16 9)(5 17 10)(6 18 11)(7 19 12)(22 36 29)(23 37 30)(24 38 31)(25 39 32)(26 40 33)(27 41 34)(28 42 35)(43 57 50)(44 58 51)(45 59 52)(46 60 53)(47 61 54)(48 62 55)(49 63 56)
(2 5 3)(4 6 7)(8 14 10)(9 11 12)(15 21 17)(16 18 19)(22 35 38)(23 32 40)(24 29 42)(25 33 37)(26 30 39)(27 34 41)(28 31 36)(43 63 52)(44 60 54)(45 57 56)(46 61 51)(47 58 53)(48 62 55)(49 59 50)

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

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

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

31 conjugacy classes

 class 1 2 3A 3B 3C 3D 3E ··· 3J 6A 6B 6C ··· 6J 7 21A ··· 21H order 1 2 3 3 3 3 3 ··· 3 6 6 6 ··· 6 7 21 ··· 21 size 1 7 1 1 3 3 21 ··· 21 7 7 21 ··· 21 6 6 ··· 6

31 irreducible representations

 dim 1 1 1 1 1 1 3 3 6 6 6 type + + + image C1 C2 C3 C3 C6 C6 He3 C2×He3 F7 C3×F7 D7⋊He3 kernel D7⋊He3 C7⋊He3 C3×F7 C32×D7 C3×C7⋊C3 C3×C21 D7 C7 C32 C3 C1 # reps 1 1 6 2 6 2 2 2 1 2 6

Matrix representation of D7⋊He3 in GL6(𝔽43)

 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 42 42 42 42 42 42
,
 0 1 0 0 0 0 1 0 0 0 0 0 42 42 42 42 42 42 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 0 0
,
 10 0 37 36 36 37 6 16 6 0 42 42 1 7 17 7 1 0 0 1 7 17 7 1 42 42 0 6 16 6 37 36 36 37 0 10
,
 36 0 0 0 0 0 0 36 0 0 0 0 0 0 36 0 0 0 0 0 0 36 0 0 0 0 0 0 36 0 0 0 0 0 0 36
,
 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 42 42 42 42 42 42 0 1 0 0 0 0 0 0 0 1 0 0

G:=sub<GL(6,GF(43))| [0,0,0,0,0,42,1,0,0,0,0,42,0,1,0,0,0,42,0,0,1,0,0,42,0,0,0,1,0,42,0,0,0,0,1,42],[0,1,42,0,0,0,1,0,42,0,0,0,0,0,42,0,0,0,0,0,42,0,0,1,0,0,42,0,1,0,0,0,42,1,0,0],[10,6,1,0,42,37,0,16,7,1,42,36,37,6,17,7,0,36,36,0,7,17,6,37,36,42,1,7,16,0,37,42,0,1,6,10],[36,0,0,0,0,0,0,36,0,0,0,0,0,0,36,0,0,0,0,0,0,36,0,0,0,0,0,0,36,0,0,0,0,0,0,36],[1,0,0,42,0,0,0,0,0,42,1,0,0,1,0,42,0,0,0,0,0,42,0,1,0,0,1,42,0,0,0,0,0,42,0,0] >;

D7⋊He3 in GAP, Magma, Sage, TeX

D_7\rtimes {\rm He}_3
% in TeX

G:=Group("D7:He3");
// GroupNames label

G:=SmallGroup(378,12);
// by ID

G=gap.SmallGroup(378,12);
# by ID

G:=PCGroup([5,-2,-3,-3,-3,-7,187,8104,2709]);
// Polycyclic

G:=Group<a,b,c,d,e|a^7=b^2=c^3=d^3=e^3=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,e*a*e^-1=a^2,b*c=c*b,b*d=d*b,e*b*e^-1=a*b,c*d=d*c,e*c*e^-1=c*d^-1,d*e=e*d>;
// generators/relations

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