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G = C72⋊Q8order 392 = 23·72

The semidirect product of C72 and Q8 acting faithfully

non-abelian, soluble, monomial

Aliases: C72⋊Q8, C72⋊C4.2C2, C7⋊D7.2C22, SmallGroup(392,38)

Series: Derived Chief Lower central Upper central

C1C72C7⋊D7 — C72⋊Q8
C1C72C7⋊D7C72⋊C4 — C72⋊Q8
C72C7⋊D7 — C72⋊Q8
C1

Generators and relations for C72⋊Q8
 G = < a,b,c,d | a7=b7=c4=1, d2=c2, ab=ba, cac-1=a4b2, dad-1=a3b-1, cbc-1=a2b3, dbd-1=a3b4, dcd-1=c-1 >

49C2
4C7
4C7
49C4
49C4
49C4
28D7
28D7
49Q8

Character table of C72⋊Q8

 class 124A4B4C7A7B7C7D7E7F
 size 149989898888888
ρ111111111111    trivial
ρ211-1-11111111    linear of order 2
ρ311-11-1111111    linear of order 2
ρ4111-1-1111111    linear of order 2
ρ52-2000222222    symplectic lifted from Q8, Schur index 2
ρ680000757473+3ζ72ζ76+3ζ74+3ζ737111767572+3ζ7    orthogonal faithful
ρ780000767572+3ζ7757473+3ζ72111ζ76+3ζ74+3ζ737    orthogonal faithful
ρ88000011767572+3ζ7757473+3ζ72ζ76+3ζ74+3ζ7371    orthogonal faithful
ρ98000011757473+3ζ72ζ76+3ζ74+3ζ737767572+3ζ71    orthogonal faithful
ρ108000011ζ76+3ζ74+3ζ737767572+3ζ7757473+3ζ721    orthogonal faithful
ρ1180000ζ76+3ζ74+3ζ737767572+3ζ7111757473+3ζ72    orthogonal faithful

Permutation representations of C72⋊Q8
On 28 points - transitive group 28T58
Generators in S28
(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)
(8 12 9 13 10 14 11)(15 16 17 18 19 20 21)(22 24 26 28 23 25 27)
(1 9)(2 11 7 14)(3 13 6 12)(4 8 5 10)(15 27 16 26)(17 25 21 28)(18 24 20 22)(19 23)
(1 19)(2 17 7 21)(3 15 6 16)(4 20 5 18)(8 24 10 22)(9 23)(11 28 14 25)(12 27 13 26)

G:=sub<Sym(28)| (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), (8,12,9,13,10,14,11)(15,16,17,18,19,20,21)(22,24,26,28,23,25,27), (1,9)(2,11,7,14)(3,13,6,12)(4,8,5,10)(15,27,16,26)(17,25,21,28)(18,24,20,22)(19,23), (1,19)(2,17,7,21)(3,15,6,16)(4,20,5,18)(8,24,10,22)(9,23)(11,28,14,25)(12,27,13,26)>;

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), (8,12,9,13,10,14,11)(15,16,17,18,19,20,21)(22,24,26,28,23,25,27), (1,9)(2,11,7,14)(3,13,6,12)(4,8,5,10)(15,27,16,26)(17,25,21,28)(18,24,20,22)(19,23), (1,19)(2,17,7,21)(3,15,6,16)(4,20,5,18)(8,24,10,22)(9,23)(11,28,14,25)(12,27,13,26) );

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)], [(8,12,9,13,10,14,11),(15,16,17,18,19,20,21),(22,24,26,28,23,25,27)], [(1,9),(2,11,7,14),(3,13,6,12),(4,8,5,10),(15,27,16,26),(17,25,21,28),(18,24,20,22),(19,23)], [(1,19),(2,17,7,21),(3,15,6,16),(4,20,5,18),(8,24,10,22),(9,23),(11,28,14,25),(12,27,13,26)])

G:=TransitiveGroup(28,58);

Matrix representation of C72⋊Q8 in GL8(𝔽29)

281000000
919000000
24101140000
9625250000
202300252500
5190041100
24100000114
9600002525
,
10000000
01000000
23104110000
182818280000
212300252500
6190041100
182800001828
00000010
,
002810000
182827180000
00100000
101100000
002800010
1828280001828
00101000
00100100
,
000028100
11100271800
00001010
00001001
000028000
1010028000
00101000
182818281000

G:=sub<GL(8,GF(29))| [28,9,24,9,20,5,24,9,1,19,10,6,23,19,10,6,0,0,11,25,0,0,0,0,0,0,4,25,0,0,0,0,0,0,0,0,25,4,0,0,0,0,0,0,25,11,0,0,0,0,0,0,0,0,11,25,0,0,0,0,0,0,4,25],[1,0,23,18,21,6,18,0,0,1,10,28,23,19,28,0,0,0,4,18,0,0,0,0,0,0,11,28,0,0,0,0,0,0,0,0,25,4,0,0,0,0,0,0,25,11,0,0,0,0,0,0,0,0,18,1,0,0,0,0,0,0,28,0],[0,18,0,10,0,18,0,0,0,28,0,1,0,28,0,0,28,27,1,1,28,28,1,1,1,18,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,18,0,0,0,0,0,0,0,28,0,0],[0,11,0,0,0,10,0,18,0,1,0,0,0,1,0,28,0,0,0,0,0,0,1,18,0,0,0,0,0,0,0,28,28,27,1,1,28,28,1,1,1,18,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0] >;

C72⋊Q8 in GAP, Magma, Sage, TeX

C_7^2\rtimes Q_8
% in TeX

G:=Group("C7^2:Q8");
// GroupNames label

G:=SmallGroup(392,38);
// by ID

G=gap.SmallGroup(392,38);
# by ID

G:=PCGroup([5,-2,-2,-2,-7,7,20,61,26,1763,3048,253,5004,3309,2114]);
// Polycyclic

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

Export

Subgroup lattice of C72⋊Q8 in TeX
Character table of C72⋊Q8 in TeX

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