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G = C7⋊D7  order 98 = 2·72

The semidirect product of C7 and D7 acting via D7/C7=C2

metabelian, supersoluble, monomial, A-group

Aliases: C7⋊D7, C72⋊2C2, SmallGroup(98,4)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C72 — C7⋊D7
Chief series
C1 — C7 — C72 — C7⋊D7
Lower central
C72 — C7⋊D7
Upper central
C1

Generators and relations for C7⋊D7
 G = < a,b,c | a7=b7=c2=1, ab=ba, cac=a-1, cbc=b-1 >

C1
49C2
C7
C7
C7
C7
C7
C7
C7
C7
7D7
7D7
7D7
7D7
7D7
7D7
7D7
7D7
C72
C7⋊D7

Character table of C7⋊D7

 class 127A7B7C7D7E7F7G7H7I7J7K7L7M7N7O7P7Q7R7S7T7U7V7W7X
 size 149222222222222222222222222
ρ111111111111111111111111111    trivial
ρ21-1111111111111111111111111    linear of order 2
ρ320ζ76+ζ7ζ75+ζ72ζ76+ζ7ζ74+ζ732ζ74+ζ73ζ76+ζ7ζ75+ζ72ζ75+ζ72ζ75+ζ72ζ75+ζ72ζ76+ζ7ζ74+ζ732ζ74+ζ73ζ76+ζ7ζ74+ζ73ζ76+ζ7ζ75+ζ72ζ75+ζ72ζ76+ζ7ζ74+ζ732ζ74+ζ73    orthogonal lifted from D7
ρ420ζ76+ζ7ζ75+ζ72ζ74+ζ73ζ76+ζ7ζ75+ζ72ζ75+ζ72ζ76+ζ7ζ74+ζ732ζ76+ζ7ζ75+ζ72ζ75+ζ72ζ76+ζ7ζ74+ζ732ζ74+ζ73ζ75+ζ72ζ75+ζ72ζ76+ζ7ζ74+ζ732ζ74+ζ73ζ76+ζ7ζ74+ζ73    orthogonal lifted from D7
ρ520ζ75+ζ72ζ74+ζ73ζ74+ζ73ζ74+ζ73ζ75+ζ72ζ76+ζ72ζ76+ζ7ζ75+ζ72ζ76+ζ72ζ76+ζ7ζ75+ζ72ζ74+ζ73ζ74+ζ73ζ75+ζ72ζ75+ζ72ζ74+ζ73ζ74+ζ73ζ75+ζ72ζ76+ζ72ζ76+ζ7ζ76+ζ7    orthogonal lifted from D7
ρ62022ζ75+ζ72ζ75+ζ72ζ75+ζ72ζ75+ζ72ζ75+ζ72ζ75+ζ72ζ75+ζ72ζ74+ζ73ζ74+ζ73ζ74+ζ73ζ74+ζ73ζ74+ζ73ζ74+ζ73ζ74+ζ73ζ76+ζ7ζ76+ζ7ζ76+ζ7ζ76+ζ7ζ76+ζ7ζ76+ζ7ζ76+ζ72    orthogonal lifted from D7
ρ720ζ74+ζ73ζ76+ζ7ζ74+ζ73ζ75+ζ722ζ75+ζ72ζ74+ζ73ζ76+ζ7ζ76+ζ7ζ76+ζ7ζ76+ζ7ζ74+ζ73ζ75+ζ722ζ75+ζ72ζ74+ζ73ζ75+ζ72ζ74+ζ73ζ76+ζ7ζ76+ζ7ζ74+ζ73ζ75+ζ722ζ75+ζ72    orthogonal lifted from D7
ρ82022ζ74+ζ73ζ74+ζ73ζ74+ζ73ζ74+ζ73ζ74+ζ73ζ74+ζ73ζ74+ζ73ζ76+ζ7ζ76+ζ7ζ76+ζ7ζ76+ζ7ζ76+ζ7ζ76+ζ7ζ76+ζ7ζ75+ζ72ζ75+ζ72ζ75+ζ72ζ75+ζ72ζ75+ζ72ζ75+ζ72ζ75+ζ722    orthogonal lifted from D7
ρ920ζ74+ζ73ζ76+ζ7ζ75+ζ72ζ74+ζ73ζ76+ζ7ζ76+ζ7ζ74+ζ73ζ75+ζ722ζ74+ζ73ζ76+ζ7ζ76+ζ7ζ74+ζ73ζ75+ζ722ζ75+ζ72ζ76+ζ7ζ76+ζ7ζ74+ζ73ζ75+ζ722ζ75+ζ72ζ74+ζ73ζ75+ζ72    orthogonal lifted from D7
ρ1020ζ74+ζ73ζ76+ζ7ζ76+ζ7ζ74+ζ73ζ75+ζ722ζ75+ζ72ζ74+ζ73ζ76+ζ7ζ75+ζ72ζ74+ζ73ζ76+ζ7ζ76+ζ7ζ74+ζ73ζ75+ζ722ζ74+ζ73ζ75+ζ722ζ75+ζ72ζ74+ζ73ζ76+ζ7ζ76+ζ7ζ75+ζ72    orthogonal lifted from D7
ρ1120ζ75+ζ72ζ74+ζ73ζ76+ζ7ζ75+ζ72ζ74+ζ73ζ74+ζ73ζ75+ζ72ζ76+ζ72ζ75+ζ72ζ74+ζ73ζ74+ζ73ζ75+ζ72ζ76+ζ72ζ76+ζ7ζ74+ζ73ζ74+ζ73ζ75+ζ72ζ76+ζ72ζ76+ζ7ζ75+ζ72ζ76+ζ7    orthogonal lifted from D7
ρ1220ζ76+ζ7ζ75+ζ722ζ74+ζ73ζ76+ζ7ζ75+ζ72ζ75+ζ72ζ76+ζ7ζ74+ζ732ζ74+ζ73ζ76+ζ7ζ75+ζ72ζ75+ζ72ζ76+ζ7ζ74+ζ732ζ74+ζ73ζ76+ζ7ζ75+ζ72ζ75+ζ72ζ76+ζ7ζ74+ζ73ζ74+ζ73    orthogonal lifted from D7
ρ1320ζ76+ζ7ζ75+ζ72ζ75+ζ72ζ76+ζ7ζ74+ζ732ζ74+ζ73ζ76+ζ7ζ75+ζ72ζ74+ζ73ζ76+ζ7ζ75+ζ72ζ75+ζ72ζ76+ζ7ζ74+ζ732ζ76+ζ7ζ74+ζ732ζ74+ζ73ζ76+ζ7ζ75+ζ72ζ75+ζ72ζ74+ζ73    orthogonal lifted from D7
ρ1420ζ74+ζ73ζ76+ζ7ζ75+ζ722ζ75+ζ72ζ74+ζ73ζ76+ζ7ζ76+ζ7ζ74+ζ73ζ74+ζ73ζ75+ζ722ζ75+ζ72ζ74+ζ73ζ76+ζ7ζ76+ζ7ζ76+ζ7ζ74+ζ73ζ75+ζ722ζ75+ζ72ζ74+ζ73ζ76+ζ7ζ75+ζ72    orthogonal lifted from D7
ρ1520ζ75+ζ72ζ74+ζ73ζ75+ζ72ζ74+ζ73ζ74+ζ73ζ75+ζ72ζ76+ζ72ζ76+ζ7ζ74+ζ73ζ75+ζ72ζ76+ζ72ζ76+ζ7ζ75+ζ72ζ74+ζ73ζ76+ζ72ζ76+ζ7ζ75+ζ72ζ74+ζ73ζ74+ζ73ζ75+ζ72ζ76+ζ7    orthogonal lifted from D7
ρ1620ζ76+ζ7ζ75+ζ72ζ76+ζ7ζ75+ζ72ζ75+ζ72ζ76+ζ7ζ74+ζ732ζ74+ζ73ζ75+ζ72ζ76+ζ7ζ74+ζ732ζ74+ζ73ζ76+ζ7ζ75+ζ72ζ74+ζ732ζ74+ζ73ζ76+ζ7ζ75+ζ72ζ75+ζ72ζ76+ζ7ζ74+ζ73    orthogonal lifted from D7
ρ1720ζ75+ζ72ζ74+ζ73ζ76+ζ72ζ76+ζ7ζ75+ζ72ζ74+ζ73ζ74+ζ73ζ75+ζ72ζ75+ζ72ζ76+ζ72ζ76+ζ7ζ75+ζ72ζ74+ζ73ζ74+ζ73ζ74+ζ73ζ75+ζ72ζ76+ζ72ζ76+ζ7ζ75+ζ72ζ74+ζ73ζ76+ζ7    orthogonal lifted from D7
ρ1820ζ76+ζ7ζ75+ζ72ζ75+ζ72ζ75+ζ72ζ76+ζ7ζ74+ζ732ζ74+ζ73ζ76+ζ7ζ74+ζ732ζ74+ζ73ζ76+ζ7ζ75+ζ72ζ75+ζ72ζ76+ζ7ζ76+ζ7ζ75+ζ72ζ75+ζ72ζ76+ζ7ζ74+ζ732ζ74+ζ73ζ74+ζ73    orthogonal lifted from D7
ρ1920ζ74+ζ73ζ76+ζ7ζ76+ζ7ζ76+ζ7ζ74+ζ73ζ75+ζ722ζ75+ζ72ζ74+ζ73ζ75+ζ722ζ75+ζ72ζ74+ζ73ζ76+ζ7ζ76+ζ7ζ74+ζ73ζ74+ζ73ζ76+ζ7ζ76+ζ7ζ74+ζ73ζ75+ζ722ζ75+ζ72ζ75+ζ72    orthogonal lifted from D7
ρ2020ζ74+ζ73ζ76+ζ72ζ75+ζ72ζ74+ζ73ζ76+ζ7ζ76+ζ7ζ74+ζ73ζ75+ζ722ζ75+ζ72ζ74+ζ73ζ76+ζ7ζ76+ζ7ζ74+ζ73ζ75+ζ722ζ75+ζ72ζ74+ζ73ζ76+ζ7ζ76+ζ7ζ74+ζ73ζ75+ζ72ζ75+ζ72    orthogonal lifted from D7
ρ2120ζ75+ζ72ζ74+ζ73ζ74+ζ73ζ75+ζ72ζ76+ζ72ζ76+ζ7ζ75+ζ72ζ74+ζ73ζ76+ζ7ζ75+ζ72ζ74+ζ73ζ74+ζ73ζ75+ζ72ζ76+ζ72ζ75+ζ72ζ76+ζ72ζ76+ζ7ζ75+ζ72ζ74+ζ73ζ74+ζ73ζ76+ζ7    orthogonal lifted from D7
ρ2220ζ76+ζ7ζ75+ζ72ζ74+ζ732ζ74+ζ73ζ76+ζ7ζ75+ζ72ζ75+ζ72ζ76+ζ7ζ76+ζ7ζ74+ζ732ζ74+ζ73ζ76+ζ7ζ75+ζ72ζ75+ζ72ζ75+ζ72ζ76+ζ7ζ74+ζ732ζ74+ζ73ζ76+ζ7ζ75+ζ72ζ74+ζ73    orthogonal lifted from D7
ρ2320ζ75+ζ72ζ74+ζ732ζ76+ζ7ζ75+ζ72ζ74+ζ73ζ74+ζ73ζ75+ζ72ζ76+ζ72ζ76+ζ7ζ75+ζ72ζ74+ζ73ζ74+ζ73ζ75+ζ72ζ76+ζ72ζ76+ζ7ζ75+ζ72ζ74+ζ73ζ74+ζ73ζ75+ζ72ζ76+ζ7ζ76+ζ7    orthogonal lifted from D7
ρ2420ζ75+ζ72ζ74+ζ73ζ75+ζ72ζ76+ζ72ζ76+ζ7ζ75+ζ72ζ74+ζ73ζ74+ζ73ζ74+ζ73ζ74+ζ73ζ75+ζ72ζ76+ζ72ζ76+ζ7ζ75+ζ72ζ76+ζ7ζ75+ζ72ζ74+ζ73ζ74+ζ73ζ75+ζ72ζ76+ζ72ζ76+ζ7    orthogonal lifted from D7
ρ252022ζ76+ζ7ζ76+ζ7ζ76+ζ7ζ76+ζ7ζ76+ζ7ζ76+ζ7ζ76+ζ7ζ75+ζ72ζ75+ζ72ζ75+ζ72ζ75+ζ72ζ75+ζ72ζ75+ζ72ζ75+ζ72ζ74+ζ73ζ74+ζ73ζ74+ζ73ζ74+ζ73ζ74+ζ73ζ74+ζ73ζ74+ζ732    orthogonal lifted from D7
ρ2620ζ74+ζ73ζ76+ζ7ζ74+ζ73ζ76+ζ7ζ76+ζ7ζ74+ζ73ζ75+ζ722ζ75+ζ72ζ76+ζ7ζ74+ζ73ζ75+ζ722ζ75+ζ72ζ74+ζ73ζ76+ζ7ζ75+ζ722ζ75+ζ72ζ74+ζ73ζ76+ζ7ζ76+ζ7ζ74+ζ73ζ75+ζ72    orthogonal lifted from D7

Smallest permutation representation of C7⋊D7
►On 49 points
Generators in S49
(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)

(1 31 9 17 25 46 41)(2 32 10 18 26 47 42)(3 33 11 19 27 48 36)(4 34 12 20 28 49 37)(5 35 13 21 22 43 38)(6 29 14 15 23 44 39)(7 30 8 16 24 45 40)

(1 41)(2 40)(3 39)(4 38)(5 37)(6 36)(7 42)(8 26)(9 25)(10 24)(11 23)(12 22)(13 28)(14 27)(15 19)(16 18)(20 21)(29 48)(30 47)(31 46)(32 45)(33 44)(34 43)(35 49)
 
G:=sub<Sym(49)| (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), (1,31,9,17,25,46,41)(2,32,10,18,26,47,42)(3,33,11,19,27,48,36)(4,34,12,20,28,49,37)(5,35,13,21,22,43,38)(6,29,14,15,23,44,39)(7,30,8,16,24,45,40), (1,41)(2,40)(3,39)(4,38)(5,37)(6,36)(7,42)(8,26)(9,25)(10,24)(11,23)(12,22)(13,28)(14,27)(15,19)(16,18)(20,21)(29,48)(30,47)(31,46)(32,45)(33,44)(34,43)(35,49)>;
 
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), (1,31,9,17,25,46,41)(2,32,10,18,26,47,42)(3,33,11,19,27,48,36)(4,34,12,20,28,49,37)(5,35,13,21,22,43,38)(6,29,14,15,23,44,39)(7,30,8,16,24,45,40), (1,41)(2,40)(3,39)(4,38)(5,37)(6,36)(7,42)(8,26)(9,25)(10,24)(11,23)(12,22)(13,28)(14,27)(15,19)(16,18)(20,21)(29,48)(30,47)(31,46)(32,45)(33,44)(34,43)(35,49) );
 
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)], [(1,31,9,17,25,46,41),(2,32,10,18,26,47,42),(3,33,11,19,27,48,36),(4,34,12,20,28,49,37),(5,35,13,21,22,43,38),(6,29,14,15,23,44,39),(7,30,8,16,24,45,40)], [(1,41),(2,40),(3,39),(4,38),(5,37),(6,36),(7,42),(8,26),(9,25),(10,24),(11,23),(12,22),(13,28),(14,27),(15,19),(16,18),(20,21),(29,48),(30,47),(31,46),(32,45),(33,44),(34,43),(35,49)]])
 

C7⋊D7 is a maximal subgroup of   C72⋊C4  D72  C7⋊5F7  C7⋊F7  C72⋊C6  C7⋊D21  C7⋊D35
C7⋊D7 is a maximal quotient of   C7⋊Dic7  C7⋊D21  C7⋊D35

Matrix representation of C7⋊D7 ►in GL4(𝔽29) generated by

22100
161000
002219
001010
,
22100
161000
0001
00287
,
102800
121900
00107
001919
G:=sub<GL(4,GF(29))| [22,16,0,0,1,10,0,0,0,0,22,10,0,0,19,10],[22,16,0,0,1,10,0,0,0,0,0,28,0,0,1,7],[10,12,0,0,28,19,0,0,0,0,10,19,0,0,7,19] >;
 

C7⋊D7 in GAP, Magma, Sage, TeX 

C_7\rtimes D_7
 
% in TeX
 
G:=Group("C7:D7");
 
// GroupNames label
 
G:=SmallGroup(98,4);
 
// by ID
 
G=gap.SmallGroup(98,4);
 
# by ID
 
G:=PCGroup([3,-2,-7,-7,73,758]);
 
// Polycyclic
 
G:=Group<a,b,c|a^7=b^7=c^2=1,a*b=b*a,c*a*c=a^-1,c*b*c=b^-1>;
 
// generators/relations
 

Export

Subgroup lattice of C7⋊D7 in TeX
Character table of C7⋊D7 in TeX

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