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G = C4.6Q16order 64 = 26

2nd non-split extension by C4 of Q16 acting via Q16/Q8=C2

p-group, metabelian, nilpotent (class 3), monomial

Aliases: C4.6Q16, C4.7SD16, C42.5C22, C4⋊C8.3C2, C4⋊Q8.2C2, (C2×Q8).2C4, (C2×C4).110D4, C2.5(Q8⋊C4), C2.5(C4.D4), C22.41(C22⋊C4), (C2×C4).14(C2×C4), SmallGroup(64,14)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C4.6Q16
C1C2C22C2×C4C42C4⋊Q8 — C4.6Q16
C1C22C2×C4 — C4.6Q16
C1C22C42 — C4.6Q16
C1C22C22C42 — C4.6Q16

Generators and relations for C4.6Q16
 G = < a,b,c | a4=b8=1, c2=a2b4, bab-1=cac-1=a-1, cbc-1=a-1b-1 >

2C4
4C4
4C4
2C2×C4
2C2×C4
4Q8
4Q8
4C8
4C8
2C2×C8
2C2×C8
2C4⋊C4
2C4⋊C4

Character table of C4.6Q16

 class 12A2B2C4A4B4C4D4E4F4G8A8B8C8D8E8F8G8H
 size 1111222248844444444
ρ11111111111111111111    trivial
ρ211111111111-1-1-1-1-1-1-1-1    linear of order 2
ρ3111111111-1-11-1-111-1-11    linear of order 2
ρ4111111111-1-1-111-1-111-1    linear of order 2
ρ51111-1-1-1-11-11-i-iii-ii-ii    linear of order 4
ρ61111-1-1-1-11-11ii-i-ii-ii-i    linear of order 4
ρ71111-1-1-1-111-1-ii-ii-i-iii    linear of order 4
ρ81111-1-1-1-111-1i-ii-iii-i-i    linear of order 4
ρ92222-222-2-20000000000    orthogonal lifted from D4
ρ1022222-2-22-20000000000    orthogonal lifted from D4
ρ112-22-2200-20000-2-200220    symplectic lifted from Q16, Schur index 2
ρ122-2-220-220000-2002200-2    symplectic lifted from Q16, Schur index 2
ρ132-2-220-220000200-2-2002    symplectic lifted from Q16, Schur index 2
ρ142-22-2200-200002200-2-20    symplectic lifted from Q16, Schur index 2
ρ152-22-2-20020000--2-200--2-20    complex lifted from SD16
ρ162-22-2-20020000-2--200-2--20    complex lifted from SD16
ρ172-2-2202-20000-200-2--200--2    complex lifted from SD16
ρ182-2-2202-20000--200--2-200-2    complex lifted from SD16
ρ1944-4-4000000000000000    orthogonal lifted from C4.D4

Smallest permutation representation of C4.6Q16
Regular action on 64 points
Generators in S64
(1 25 11 45)(2 46 12 26)(3 27 13 47)(4 48 14 28)(5 29 15 41)(6 42 16 30)(7 31 9 43)(8 44 10 32)(17 35 61 55)(18 56 62 36)(19 37 63 49)(20 50 64 38)(21 39 57 51)(22 52 58 40)(23 33 59 53)(24 54 60 34)
(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 64)
(1 19 15 59)(2 56 16 40)(3 17 9 57)(4 54 10 38)(5 23 11 63)(6 52 12 36)(7 21 13 61)(8 50 14 34)(18 30 58 46)(20 28 60 44)(22 26 62 42)(24 32 64 48)(25 49 41 33)(27 55 43 39)(29 53 45 37)(31 51 47 35)

G:=sub<Sym(64)| (1,25,11,45)(2,46,12,26)(3,27,13,47)(4,48,14,28)(5,29,15,41)(6,42,16,30)(7,31,9,43)(8,44,10,32)(17,35,61,55)(18,56,62,36)(19,37,63,49)(20,50,64,38)(21,39,57,51)(22,52,58,40)(23,33,59,53)(24,54,60,34), (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)(57,58,59,60,61,62,63,64), (1,19,15,59)(2,56,16,40)(3,17,9,57)(4,54,10,38)(5,23,11,63)(6,52,12,36)(7,21,13,61)(8,50,14,34)(18,30,58,46)(20,28,60,44)(22,26,62,42)(24,32,64,48)(25,49,41,33)(27,55,43,39)(29,53,45,37)(31,51,47,35)>;

G:=Group( (1,25,11,45)(2,46,12,26)(3,27,13,47)(4,48,14,28)(5,29,15,41)(6,42,16,30)(7,31,9,43)(8,44,10,32)(17,35,61,55)(18,56,62,36)(19,37,63,49)(20,50,64,38)(21,39,57,51)(22,52,58,40)(23,33,59,53)(24,54,60,34), (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)(57,58,59,60,61,62,63,64), (1,19,15,59)(2,56,16,40)(3,17,9,57)(4,54,10,38)(5,23,11,63)(6,52,12,36)(7,21,13,61)(8,50,14,34)(18,30,58,46)(20,28,60,44)(22,26,62,42)(24,32,64,48)(25,49,41,33)(27,55,43,39)(29,53,45,37)(31,51,47,35) );

G=PermutationGroup([(1,25,11,45),(2,46,12,26),(3,27,13,47),(4,48,14,28),(5,29,15,41),(6,42,16,30),(7,31,9,43),(8,44,10,32),(17,35,61,55),(18,56,62,36),(19,37,63,49),(20,50,64,38),(21,39,57,51),(22,52,58,40),(23,33,59,53),(24,54,60,34)], [(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),(57,58,59,60,61,62,63,64)], [(1,19,15,59),(2,56,16,40),(3,17,9,57),(4,54,10,38),(5,23,11,63),(6,52,12,36),(7,21,13,61),(8,50,14,34),(18,30,58,46),(20,28,60,44),(22,26,62,42),(24,32,64,48),(25,49,41,33),(27,55,43,39),(29,53,45,37),(31,51,47,35)])

C4.6Q16 is a maximal subgroup of
C42.4D4  C42.410D4  C42.411D4  C42.415D4  C42.79D4  C42.80D4  C42.418D4  C42.85D4  C42.86D4  C42.87D4  D4⋊SD16  Q8⋊SD16  C42.185C23  D4⋊Q16  Q8⋊Q16  C42.195C23  D4.5SD16  D43Q16  Q83Q16  C42.207C23  C811SD16  C88Q16  D4.1Q16  D4.2SD16  Q8.2SD16  D4.Q16  Q8.2Q16  C83SD16  C8⋊Q16  C8.8SD16  C42.248C23  C42.249C23  C42.254C23  C42.255C23  Dic5.Q16
 C4p.Q16: Q8.1Q16  C8.3Q16  C4.Dic12  C12.5Q16  C4.Dic20  C20.5Q16  C4.Dic28  C28.5Q16 ...
C4.6Q16 is a maximal quotient of
(C2×Q8)⋊C8  C42.8Q8  Dic5.Q16
 C4p.Q16: C8.1Q16  C4.Dic12  C12.5Q16  C4.Dic20  C20.5Q16  C4.Dic28  C28.5Q16 ...

Matrix representation of C4.6Q16 in GL4(𝔽17) generated by

0100
16000
00160
00016
,
101600
16700
00512
0055
,
1700
71600
00101
0017
G:=sub<GL(4,GF(17))| [0,16,0,0,1,0,0,0,0,0,16,0,0,0,0,16],[10,16,0,0,16,7,0,0,0,0,5,5,0,0,12,5],[1,7,0,0,7,16,0,0,0,0,10,1,0,0,1,7] >;

C4.6Q16 in GAP, Magma, Sage, TeX

C_4._6Q_{16}
% in TeX

G:=Group("C4.6Q16");
// GroupNames label

G:=SmallGroup(64,14);
// by ID

G=gap.SmallGroup(64,14);
# by ID

G:=PCGroup([6,-2,2,-2,2,-2,2,48,73,199,362,332,158,681,165]);
// Polycyclic

G:=Group<a,b,c|a^4=b^8=1,c^2=a^2*b^4,b*a*b^-1=c*a*c^-1=a^-1,c*b*c^-1=a^-1*b^-1>;
// generators/relations

Export

Subgroup lattice of C4.6Q16 in TeX
Character table of C4.6Q16 in TeX

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