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G = C4.Q16order 64 = 26

3rd non-split extension by C4 of Q16 acting via Q16/Q8=C2

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

Aliases: Q82Q8, C4.7Q16, C42.25C22, C4⋊C8.7C2, C4⋊Q8.6C2, (C4×Q8).7C2, C2.7(C2×Q16), C4.14(C2×Q8), C2.D8.4C2, (C2×C4).133D4, (C2×C8).6C22, C4.26(C4○D4), C4⋊C4.14C22, Q8⋊C4.3C2, C22.98(C2×D4), C2.15(C8⋊C22), (C2×C4).102C23, C2.15(C22⋊Q8), (C2×Q8).52C22, SmallGroup(64,158)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C4.Q16
C1C2C4C2×C4C2×Q8C4×Q8 — C4.Q16
C1C2C2×C4 — C4.Q16
C1C22C42 — C4.Q16
C1C2C2C2×C4 — C4.Q16

Generators and relations for C4.Q16
 G = < a,b,c | a4=b8=1, c2=b4, bab-1=a-1, ac=ca, cbc-1=a2b-1 >

2C4
2C4
2C4
4C4
4C4
4C4
2C8
2C8
2C2×C4
2C2×C4
2Q8
2C2×C4
2C2×C4
4Q8
4Q8
2C4⋊C4
2C2×Q8
2C42
2C4⋊C4

Character table of C4.Q16

 class 12A2B2C4A4B4C4D4E4F4G4H4I4J4K8A8B8C8D
 size 1111222244444884444
ρ11111111111111111111    trivial
ρ211111111-11-1-1-1-1-11111    linear of order 2
ρ31111-111-1-1-1-1111-11-1-11    linear of order 2
ρ41111-111-11-11-1-1-111-1-11    linear of order 2
ρ51111-111-1-1-1-111-11-111-1    linear of order 2
ρ61111-111-11-11-1-11-1-111-1    linear of order 2
ρ71111111111111-1-1-1-1-1-1    linear of order 2
ρ811111111-11-1-1-111-1-1-1-1    linear of order 2
ρ922222-2-220-2000000000    orthogonal lifted from D4
ρ102222-2-2-2-202000000000    orthogonal lifted from D4
ρ112-22-202-200002-2000000    symplectic lifted from Q8, Schur index 2
ρ122-22-202-20000-22000000    symplectic lifted from Q8, Schur index 2
ρ132-2-22200-20000000-2-222    symplectic lifted from Q16, Schur index 2
ρ142-2-22200-2000000022-2-2    symplectic lifted from Q16, Schur index 2
ρ152-2-22-200200000002-22-2    symplectic lifted from Q16, Schur index 2
ρ162-2-22-20020000000-22-22    symplectic lifted from Q16, Schur index 2
ρ172-22-20-2202i0-2i00000000    complex lifted from C4○D4
ρ182-22-20-220-2i02i00000000    complex lifted from C4○D4
ρ1944-4-4000000000000000    orthogonal lifted from C8⋊C22

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

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

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

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

C4.Q16 is a maximal subgroup of
C42.191C23  Q82SD16  D4⋊Q16  Q8⋊Q16  Q8.Q16  D4.3Q16  C42.199C23  C88Q16  Q8.2SD16  Q8.2Q16  C8⋊Q16  C42.249C23  C42.255C23  C42.447D4  C42.448D4  C42.22C23  C42.224D4  C42.450D4  C42.230D4  C42.234D4  C42.355C23  C42.358C23  C42.282D4  C42.285D4  C42.289D4  C42.290D4  C42.424C23  C42.425C23  C42.295D4  C42.297D4  C42.300D4  C42.302D4  C42.25C23  C42.29C23  D45Q16  C42.470C23  C42.43C23  C42.48C23  C42.478C23  C42.482C23  D46Q16  C42.489C23  C42.57C23  C42.63C23  C42.497C23  C42.498C23  C42.502C23  Q85Q16  C42.508C23  C42.513C23  C42.516C23  C42.518C23  SD164Q8  Q8×Q16  SD16⋊Q8  SD162Q8  SL2(𝔽3)⋊Q8
 C4p.Q16: Q8.1Q16  C8.3Q16  Dic63Q8  Q85Dic6  Dic65Q8  C20.7Q16  C20.23Q16  Dic105Q8 ...
 (Cp×Q8)⋊Q8: C42.220D4  C42.21C23  Q83Dic6  Dic5.9Q16  Dic7.Q16 ...
 C4p⋊Q8.C2: Q164Q8  Q165Q8  Dic3.Q16  Dic102Q8  Dic142Q8 ...
C4.Q16 is a maximal quotient of
Q8⋊(C4⋊C4)  C2.D85C4  C42.29Q8  C4⋊C4⋊Q8  (C2×C4).23Q16
 (Cp×Q8)⋊Q8: (C2×Q8)⋊Q8  Q83Dic6  Q85Dic6  Dic5.9Q16  C20.23Q16  Dic7.Q16  C28.23Q16 ...
 C42.D2p: C42.99D4  C42.121D4  Dic63Q8  Dic65Q8  C20.7Q16  Dic105Q8  C28.7Q16  Dic145Q8 ...
 (C2×C8).D2p: (C2×C8).52D4  (C2×C4).21Q16  (C2×C8).60D4  Dic3.Q16  Dic102Q8  Dic142Q8 ...

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

4200
01300
00160
00016
,
5300
141200
00150
0008
,
1000
0100
0001
00160
G:=sub<GL(4,GF(17))| [4,0,0,0,2,13,0,0,0,0,16,0,0,0,0,16],[5,14,0,0,3,12,0,0,0,0,15,0,0,0,0,8],[1,0,0,0,0,1,0,0,0,0,0,16,0,0,1,0] >;

C4.Q16 in GAP, Magma, Sage, TeX

C_4.Q_{16}
% in TeX

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

G:=SmallGroup(64,158);
// by ID

G=gap.SmallGroup(64,158);
# by ID

G:=PCGroup([6,-2,2,2,-2,2,-2,240,121,55,362,158,1444,376,88]);
// Polycyclic

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

Export

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

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