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G = He3⋊D7order 378 = 2·33·7

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

metabelian, supersoluble, monomial

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

C1C3×C21 — He3⋊D7
C1C7C21C3×C21C7×He3 — He3⋊D7
C3×C21 — He3⋊D7
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 >

63C2
3C3
3C3
6C3
21S3
63C6
63S3
2C32
9D7
3C21
3C21
6C21
7C3⋊S3
21C3×S3
3D21
9C3×D7
9D21
2C3×C21
7C32⋊C6
3C3×D21

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 3A3B3C3D3E3F6A6B7A7B7C21A···21F21G···21AD
order123333336677721···2121···21
size16323366663632222···26···6

43 irreducible representations

dim111122222266
type+++++++
imageC1C2C3C6S3D7C3×S3C3×D7D21C3×D21C32⋊C6He3⋊D7
kernelHe3⋊D7C7×He3C3⋊D21C3×C21C3×C21He3C21C32C32C3C7C1
# reps1122132661216

Matrix representation of He3⋊D7 in GL6(𝔽43)

008100
11414200
0042010
008001
0042000
108000
,
26120000
31160000
310381200
22129400
310003812
22120094
,
100000
010000
383843100
2828343800
2626003812
19190094
,
810000
4200000
4209100
91334200
4200091
91003342
,
37140000
1960000
37000256
1914002518
37025600
1914251800

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

Export

Subgroup lattice of He3⋊D7 in TeX

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