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## G = He3⋊D7order 378 = 2·33·7

### 1st semidirect product of He3 and D7 acting via D7/C7=C2

Aliases: He31D7, C321D21, C32⋊(C3×D7), (C3×C21)⋊1C6, (C3×C21)⋊1S3, C3⋊D211C3, (C7×He3)⋊1C2, C73(C32⋊C6), C3.2(C3×D21), C21.12(C3×S3), SmallGroup(378,38)

Series: Derived Chief Lower central Upper central

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

Generators and relations for He3⋊D7
G = < a,b,c,d,e | a3=b3=c3=d7=e2=1, ab=ba, cac-1=ab-1, ad=da, eae=a-1, bc=cb, bd=db, ebe=b-1, cd=dc, ce=ec, ede=d-1 >

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

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

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

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

43 conjugacy classes

 class 1 2 3A 3B 3C 3D 3E 3F 6A 6B 7A 7B 7C 21A ··· 21F 21G ··· 21AD order 1 2 3 3 3 3 3 3 6 6 7 7 7 21 ··· 21 21 ··· 21 size 1 63 2 3 3 6 6 6 63 63 2 2 2 2 ··· 2 6 ··· 6

43 irreducible representations

 dim 1 1 1 1 2 2 2 2 2 2 6 6 type + + + + + + + image C1 C2 C3 C6 S3 D7 C3×S3 C3×D7 D21 C3×D21 C32⋊C6 He3⋊D7 kernel He3⋊D7 C7×He3 C3⋊D21 C3×C21 C3×C21 He3 C21 C32 C32 C3 C7 C1 # reps 1 1 2 2 1 3 2 6 6 12 1 6

Matrix representation of He3⋊D7 in GL6(𝔽43)

 0 0 8 1 0 0 1 1 41 42 0 0 0 0 42 0 1 0 0 0 8 0 0 1 0 0 42 0 0 0 1 0 8 0 0 0
,
 26 12 0 0 0 0 31 16 0 0 0 0 31 0 38 12 0 0 22 12 9 4 0 0 31 0 0 0 38 12 22 12 0 0 9 4
,
 1 0 0 0 0 0 0 1 0 0 0 0 38 38 4 31 0 0 28 28 34 38 0 0 26 26 0 0 38 12 19 19 0 0 9 4
,
 8 1 0 0 0 0 42 0 0 0 0 0 42 0 9 1 0 0 9 1 33 42 0 0 42 0 0 0 9 1 9 1 0 0 33 42
,
 37 14 0 0 0 0 19 6 0 0 0 0 37 0 0 0 25 6 19 14 0 0 25 18 37 0 25 6 0 0 19 14 25 18 0 0

G:=sub<GL(6,GF(43))| [0,1,0,0,0,1,0,1,0,0,0,0,8,41,42,8,42,8,1,42,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0],[26,31,31,22,31,22,12,16,0,12,0,12,0,0,38,9,0,0,0,0,12,4,0,0,0,0,0,0,38,9,0,0,0,0,12,4],[1,0,38,28,26,19,0,1,38,28,26,19,0,0,4,34,0,0,0,0,31,38,0,0,0,0,0,0,38,9,0,0,0,0,12,4],[8,42,42,9,42,9,1,0,0,1,0,1,0,0,9,33,0,0,0,0,1,42,0,0,0,0,0,0,9,33,0,0,0,0,1,42],[37,19,37,19,37,19,14,6,0,14,0,14,0,0,0,0,25,25,0,0,0,0,6,18,0,0,25,25,0,0,0,0,6,18,0,0] >;

He3⋊D7 in GAP, Magma, Sage, TeX

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

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

G:=SmallGroup(378,38);
// by ID

G=gap.SmallGroup(378,38);
# by ID

G:=PCGroup([5,-2,-3,-3,-3,-7,182,187,723,8104]);
// Polycyclic

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

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