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## G = C7⋊He3⋊C2order 378 = 2·33·7

### 3rd semidirect product of C7⋊He3 and C2 acting faithfully

Aliases: C7⋊He33C2, (C3×C21)⋊9C6, C21.9(C3×S3), C72(C32⋊C6), C3⋊S3⋊(C7⋊C3), C32⋊(C2×C7⋊C3), (C3×C7⋊C3)⋊2S3, C3.4(S3×C7⋊C3), (C7×C3⋊S3)⋊2C3, SmallGroup(378,17)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C21 — C7⋊He3⋊C2
 Chief series C1 — C7 — C21 — C3×C21 — C7⋊He3 — C7⋊He3⋊C2
 Lower central C3×C21 — C7⋊He3⋊C2
 Upper central C1

Generators and relations for C7⋊He3⋊C2
G = < a,b,c,d,e | a7=b3=c3=d3=e2=1, ab=ba, ac=ca, dad-1=a4, ae=ea, bc=cb, dbd-1=bc-1, ebe=b-1, cd=dc, ece=c-1, de=ed >

Character table of C7⋊He3⋊C2

 class 1 2 3A 3B 3C 3D 3E 3F 6A 6B 7A 7B 14A 14B 21A 21B 21C 21D 21E 21F 21G 21H size 1 9 2 6 21 21 42 42 63 63 3 3 27 27 6 6 6 6 6 6 6 6 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 -1 1 1 1 1 1 1 -1 -1 1 1 -1 -1 1 1 1 1 1 1 1 1 linear of order 2 ρ3 1 -1 1 1 ζ3 ζ32 ζ3 ζ32 ζ65 ζ6 1 1 -1 -1 1 1 1 1 1 1 1 1 linear of order 6 ρ4 1 -1 1 1 ζ32 ζ3 ζ32 ζ3 ζ6 ζ65 1 1 -1 -1 1 1 1 1 1 1 1 1 linear of order 6 ρ5 1 1 1 1 ζ32 ζ3 ζ32 ζ3 ζ32 ζ3 1 1 1 1 1 1 1 1 1 1 1 1 linear of order 3 ρ6 1 1 1 1 ζ3 ζ32 ζ3 ζ32 ζ3 ζ32 1 1 1 1 1 1 1 1 1 1 1 1 linear of order 3 ρ7 2 0 2 -1 2 2 -1 -1 0 0 2 2 0 0 -1 2 -1 -1 -1 2 -1 -1 orthogonal lifted from S3 ρ8 2 0 2 -1 -1+√-3 -1-√-3 ζ65 ζ6 0 0 2 2 0 0 -1 2 -1 -1 -1 2 -1 -1 complex lifted from C3×S3 ρ9 2 0 2 -1 -1-√-3 -1+√-3 ζ6 ζ65 0 0 2 2 0 0 -1 2 -1 -1 -1 2 -1 -1 complex lifted from C3×S3 ρ10 3 3 3 3 0 0 0 0 0 0 -1-√-7/2 -1+√-7/2 -1+√-7/2 -1-√-7/2 -1-√-7/2 -1+√-7/2 -1+√-7/2 -1-√-7/2 -1+√-7/2 -1-√-7/2 -1-√-7/2 -1+√-7/2 complex lifted from C7⋊C3 ρ11 3 3 3 3 0 0 0 0 0 0 -1+√-7/2 -1-√-7/2 -1-√-7/2 -1+√-7/2 -1+√-7/2 -1-√-7/2 -1-√-7/2 -1+√-7/2 -1-√-7/2 -1+√-7/2 -1+√-7/2 -1-√-7/2 complex lifted from C7⋊C3 ρ12 3 -3 3 3 0 0 0 0 0 0 -1+√-7/2 -1-√-7/2 1+√-7/2 1-√-7/2 -1+√-7/2 -1-√-7/2 -1-√-7/2 -1+√-7/2 -1-√-7/2 -1+√-7/2 -1+√-7/2 -1-√-7/2 complex lifted from C2×C7⋊C3 ρ13 3 -3 3 3 0 0 0 0 0 0 -1-√-7/2 -1+√-7/2 1-√-7/2 1+√-7/2 -1-√-7/2 -1+√-7/2 -1+√-7/2 -1-√-7/2 -1+√-7/2 -1-√-7/2 -1-√-7/2 -1+√-7/2 complex lifted from C2×C7⋊C3 ρ14 6 0 -3 0 0 0 0 0 0 0 6 6 0 0 0 -3 0 0 0 -3 0 0 orthogonal lifted from C32⋊C6 ρ15 6 0 -3 0 0 0 0 0 0 0 -1+√-7 -1-√-7 0 0 -ζ74+2ζ72-ζ7 1+√-7/2 -ζ76-ζ75+2ζ73 2ζ74-ζ72-ζ7 2ζ76-ζ75-ζ73 1-√-7/2 -ζ74-ζ72+2ζ7 -ζ76+2ζ75-ζ73 complex faithful ρ16 6 0 -3 0 0 0 0 0 0 0 -1-√-7 -1+√-7 0 0 -ζ76-ζ75+2ζ73 1-√-7/2 -ζ74-ζ72+2ζ7 2ζ76-ζ75-ζ73 -ζ74+2ζ72-ζ7 1+√-7/2 -ζ76+2ζ75-ζ73 2ζ74-ζ72-ζ7 complex faithful ρ17 6 0 -3 0 0 0 0 0 0 0 -1-√-7 -1+√-7 0 0 2ζ76-ζ75-ζ73 1-√-7/2 -ζ74+2ζ72-ζ7 -ζ76+2ζ75-ζ73 2ζ74-ζ72-ζ7 1+√-7/2 -ζ76-ζ75+2ζ73 -ζ74-ζ72+2ζ7 complex faithful ρ18 6 0 -3 0 0 0 0 0 0 0 -1+√-7 -1-√-7 0 0 2ζ74-ζ72-ζ7 1+√-7/2 2ζ76-ζ75-ζ73 -ζ74-ζ72+2ζ7 -ζ76+2ζ75-ζ73 1-√-7/2 -ζ74+2ζ72-ζ7 -ζ76-ζ75+2ζ73 complex faithful ρ19 6 0 6 -3 0 0 0 0 0 0 -1+√-7 -1-√-7 0 0 1-√-7/2 -1-√-7 1+√-7/2 1-√-7/2 1+√-7/2 -1+√-7 1-√-7/2 1+√-7/2 complex lifted from S3×C7⋊C3 ρ20 6 0 -3 0 0 0 0 0 0 0 -1+√-7 -1-√-7 0 0 -ζ74-ζ72+2ζ7 1+√-7/2 -ζ76+2ζ75-ζ73 -ζ74+2ζ72-ζ7 -ζ76-ζ75+2ζ73 1-√-7/2 2ζ74-ζ72-ζ7 2ζ76-ζ75-ζ73 complex faithful ρ21 6 0 6 -3 0 0 0 0 0 0 -1-√-7 -1+√-7 0 0 1+√-7/2 -1+√-7 1-√-7/2 1+√-7/2 1-√-7/2 -1-√-7 1+√-7/2 1-√-7/2 complex lifted from S3×C7⋊C3 ρ22 6 0 -3 0 0 0 0 0 0 0 -1-√-7 -1+√-7 0 0 -ζ76+2ζ75-ζ73 1-√-7/2 2ζ74-ζ72-ζ7 -ζ76-ζ75+2ζ73 -ζ74-ζ72+2ζ7 1+√-7/2 2ζ76-ζ75-ζ73 -ζ74+2ζ72-ζ7 complex faithful

Smallest permutation representation of C7⋊He3⋊C2
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 43 22)(2 44 23)(3 45 24)(4 46 25)(5 47 26)(6 48 27)(7 49 28)(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 15 8)(2 16 9)(3 17 10)(4 18 11)(5 19 12)(6 20 13)(7 21 14)(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 3 5)(4 7 6)(9 10 12)(11 14 13)(16 17 19)(18 21 20)(22 29 36)(23 31 40)(24 33 37)(25 35 41)(26 30 38)(27 32 42)(28 34 39)(43 57 50)(44 59 54)(45 61 51)(46 63 55)(47 58 52)(48 60 56)(49 62 53)
(8 15)(9 16)(10 17)(11 18)(12 19)(13 20)(14 21)(22 43)(23 44)(24 45)(25 46)(26 47)(27 48)(28 49)(29 57)(30 58)(31 59)(32 60)(33 61)(34 62)(35 63)(36 50)(37 51)(38 52)(39 53)(40 54)(41 55)(42 56)

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

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

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

Matrix representation of C7⋊He3⋊C2 in GL6(𝔽43)

 18 19 1 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 18 19 1 0 0 0 1 0 0 0 0 0 0 1 0
,
 30 15 35 33 35 20 35 2 38 20 17 42 38 39 11 42 38 36 10 8 23 40 23 15 23 26 1 15 28 39 1 5 7 39 1 18
,
 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 42 0 0 42 0 0 0 42 0 0 42 0 0 0 42 0 0 42
,
 1 0 0 0 0 0 24 42 42 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 24 42 42 0 0 0 0 1 0
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 42 0 0 42 0 0 0 42 0 0 42 0 0 0 42 0 0 42

G:=sub<GL(6,GF(43))| [18,1,0,0,0,0,19,0,1,0,0,0,1,0,0,0,0,0,0,0,0,18,1,0,0,0,0,19,0,1,0,0,0,1,0,0],[30,35,38,10,23,1,15,2,39,8,26,5,35,38,11,23,1,7,33,20,42,40,15,39,35,17,38,23,28,1,20,42,36,15,39,18],[0,0,0,42,0,0,0,0,0,0,42,0,0,0,0,0,0,42,1,0,0,42,0,0,0,1,0,0,42,0,0,0,1,0,0,42],[1,24,0,0,0,0,0,42,1,0,0,0,0,42,0,0,0,0,0,0,0,1,24,0,0,0,0,0,42,1,0,0,0,0,42,0],[1,0,0,42,0,0,0,1,0,0,42,0,0,0,1,0,0,42,0,0,0,42,0,0,0,0,0,0,42,0,0,0,0,0,0,42] >;

C7⋊He3⋊C2 in GAP, Magma, Sage, TeX

C_7\rtimes {\rm He}_3\rtimes C_2
% in TeX

G:=Group("C7:He3:C2");
// GroupNames label

G:=SmallGroup(378,17);
// by ID

G=gap.SmallGroup(378,17);
# by ID

G:=PCGroup([5,-2,-3,-3,-3,-7,182,187,723,1359]);
// Polycyclic

G:=Group<a,b,c,d,e|a^7=b^3=c^3=d^3=e^2=1,a*b=b*a,a*c=c*a,d*a*d^-1=a^4,a*e=e*a,b*c=c*b,d*b*d^-1=b*c^-1,e*b*e=b^-1,c*d=d*c,e*c*e=c^-1,d*e=e*d>;
// generators/relations

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