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G = Dic7⋊C6  order 168 = 23·3·7

The semidirect product of Dic7 and C6 acting faithfully

metabelian, supersoluble, monomial

Aliases: Dic7⋊C6, D14⋊2C6, C22⋊2F7, C7⋊C12⋊C2, C7⋊C3⋊2D4, C7⋊D4⋊C3, C7⋊2(C3×D4), (C2×C14)⋊3C6, (C2×F7)⋊2C2, C2.5(C2×F7), C14.5(C2×C6), (C22×C7⋊C3)⋊1C2, (C2×C7⋊C3).5C22, SmallGroup(168,11)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C14 — Dic7⋊C6
Chief series
C1 — C7 — C14 — C2×C7⋊C3 — C2×F7 — Dic7⋊C6
Lower central
C7 — C14 — Dic7⋊C6
Upper central
C1 — C2 — C22

Generators and relations for Dic7⋊C6
 G = < a,b,c | a14=c6=1, b2=a7, bab-1=a-1, cac-1=a11, cbc-1=a7b >

C1
C2
2C2
14C2
7C3
C7
C22
7C22
7C4
7C6
14C6
14C6
C14
2D7
2C14
C7⋊C3
7D4
7C2×C6
7C2×C6
7C12
D14
C2×C14
Dic7
C2×C7⋊C3
2C2×C7⋊C3
2F7
7C3×D4
C7⋊D4
C2×F7
C7⋊C12
C22×C7⋊C3
Dic7⋊C6

Character table of Dic7⋊C6

 class 12A2B2C3A3B46A6B6C6D6E6F712A12B14A14B14C
 size 112147714771414141461414666
ρ11111111111111111111    trivial
ρ2111-111-11111-1-11-1-1111    linear of order 2
ρ311-1-111111-1-1-1-1111-11-1    linear of order 2
ρ411-1111-111-1-1111-1-1-11-1    linear of order 2
ρ51111ζ3ζ321ζ3ζ32ζ3ζ32ζ32ζ31ζ3ζ32111    linear of order 3
ρ611-1-1ζ3ζ321ζ3ζ32ζ65ζ6ζ6ζ651ζ3ζ32-11-1    linear of order 6
ρ711-1-1ζ32ζ31ζ32ζ3ζ6ζ65ζ65ζ61ζ32ζ3-11-1    linear of order 6
ρ8111-1ζ3ζ32-1ζ3ζ32ζ3ζ32ζ6ζ651ζ65ζ6111    linear of order 6
ρ911-11ζ3ζ32-1ζ3ζ32ζ65ζ6ζ32ζ31ζ65ζ6-11-1    linear of order 6
ρ1011-11ζ32ζ3-1ζ32ζ3ζ6ζ65ζ3ζ321ζ6ζ65-11-1    linear of order 6
ρ11111-1ζ32ζ3-1ζ32ζ3ζ32ζ3ζ65ζ61ζ6ζ65111    linear of order 6
ρ121111ζ32ζ31ζ32ζ3ζ32ζ3ζ3ζ321ζ32ζ3111    linear of order 3
ρ132-200220-2-200002000-20    orthogonal lifted from D4
ρ142-200-1+√-3-1-√-301-√-31+√-300002000-20    complex lifted from C3×D4
ρ152-200-1-√-3-1+√-301+√-31-√-300002000-20    complex lifted from C3×D4
ρ166660000000000-100-1-1-1    orthogonal lifted from F7
ρ1766-60000000000-1001-11    orthogonal lifted from C2×F7
ρ186-600000000000-100-√-71√-7    complex faithful
ρ196-600000000000-100√-71-√-7    complex faithful

Permutation representations of Dic7⋊C6
►On 28 points - transitive group 28T21
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)

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

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

►On 28 points - transitive group 28T25
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)

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

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

Dic7⋊C6 is a maximal subgroup of   D28⋊6C6  D4×F7  D4⋊2F7
Dic7⋊C6 is a maximal quotient of   Dic7⋊C12  D14⋊C12  D4⋊F7  D4.F7  Q8⋊2F7  Q8.2F7  C23.2F7

Matrix representation of Dic7⋊C6 ►in GL6(𝔽337)

03360000
00336000
336212213000
000125124336
00033600
00003360
,
000100
000010
000001
33600000
03360000
00336000
,
100000
124336336000
010000
00033600
00021311
00003360

G:=sub<GL(6,GF(337))| [0,0,336,0,0,0,336,0,212,0,0,0,0,336,213,0,0,0,0,0,0,125,336,0,0,0,0,124,0,336,0,0,0,336,0,0],[0,0,0,336,0,0,0,0,0,0,336,0,0,0,0,0,0,336,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0],[1,124,0,0,0,0,0,336,1,0,0,0,0,336,0,0,0,0,0,0,0,336,213,0,0,0,0,0,1,336,0,0,0,0,1,0] >;
 

Dic7⋊C6 in GAP, Magma, Sage, TeX 

{\rm Dic}_7\rtimes C_6
 
% in TeX
 
G:=Group("Dic7:C6");
 
// GroupNames label
 
G:=SmallGroup(168,11);
 
// by ID
 
G=gap.SmallGroup(168,11);
 
# by ID
 
G:=PCGroup([5,-2,-2,-3,-2,-7,141,3604,614]);
 
// Polycyclic
 
G:=Group<a,b,c|a^14=c^6=1,b^2=a^7,b*a*b^-1=a^-1,c*a*c^-1=a^11,c*b*c^-1=a^7*b>;
 
// generators/relations
 

Export

Subgroup lattice of Dic7⋊C6 in TeX
Character table of Dic7⋊C6 in TeX

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