D7xHe3 - GroupNames

G = D₇×He₃ order 378 = 2·3³·7

Direct product of D₇ and He₃

direct product, metabelian, supersoluble, monomial

Aliases: D₇×He₃, C₇⋊₃(C₂×He₃), (C₃×C₂₁)⋊₅C₆, (C₇×He₃)⋊₃C₂, C₃²⋊₂(C₃×D₇), C₂₁.₁₄(C₃×C₆), (C₃²×D₇)⋊₁C₃, C₃.₂(C₃²×D₇), (C₃×D₇).₈C₃², SmallGroup(378,30)

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 — He₃

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, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ece^-1=cd^-1, de=ed >

C₁

₇C₂

C₃

₃C₃

C₇

₇C₆

₂₁C₆

C₃²

D₇

C₂₁

₃C₂₁

₇C₃×C₆

He₃

C₃×D₇

₃C₃×D₇

C₃×C₂₁

₇C₂×He₃

C₃²×D₇

C₇×He₃

D₇×He₃

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)

(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)

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), (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)>;

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), (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) );

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)], [(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)]])

55 conjugacy classes

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

55 irreducible representations

dim	1	1	1	1	2	2	3	3	6
type	+	+			+
image	C₁	C₂	C₃	C₆	D₇	C₃×D₇	He₃	C₂×He₃	D₇×He₃
kernel	D₇×He₃	C₇×He₃	C₃²×D₇	C₃×C₂₁	He₃	C₃²	D₇	C₇	C₁
# reps	1	1	8	8	3	24	2	2	6

Matrix representation of D₇×He₃ ►in GL₅(𝔽₄₃)

0	1	0	0	0
42	19	0	0	0
0	0	1	0	0
0	0	0	1	0
0	0	0	0	1

16	24	0	0	0
27	27	0	0	0
0	0	42	0	0
0	0	0	42	0
0	0	0	0	42

6	0	0	0	0
0	6	0	0	0
0	0	0	6	0
0	0	0	0	6
0	0	6	0	0

1	0	0	0	0
0	1	0	0	0
0	0	6	0	0
0	0	0	6	0
0	0	0	0	6

6	0	0	0	0
0	6	0	0	0
0	0	1	0	0
0	0	0	36	0
0	0	0	0	6

G:=sub<GL(5,GF(43))| [0,42,0,0,0,1,19,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[16,27,0,0,0,24,27,0,0,0,0,0,42,0,0,0,0,0,42,0,0,0,0,0,42],[6,0,0,0,0,0,6,0,0,0,0,0,0,0,6,0,0,6,0,0,0,0,0,6,0],[1,0,0,0,0,0,1,0,0,0,0,0,6,0,0,0,0,0,6,0,0,0,0,0,6],[6,0,0,0,0,0,6,0,0,0,0,0,1,0,0,0,0,0,36,0,0,0,0,0,6] >;

D₇×He₃ in GAP, Magma, Sage, TeX

D_7\times {\rm He}_3

% in TeX

G:=Group("D7xHe3");

// GroupNames label

G:=SmallGroup(378,30);

// by ID

G=gap.SmallGroup(378,30);

# by ID

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

// 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,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,e*c*e^-1=c*d^-1,d*e=e*d>;

// generators/relations

Export

Subgroup lattice of D₇×He₃ in TeX

G = D7×He3 order 378 = 2·33·7

Direct product of D7 and He3

G = D₇×He₃ order 378 = 2·3³·7

Direct product of D₇ and He₃