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G = C24.3C22order 64 = 26

3rd non-split extension by C24 of C22 acting faithfully

p-group, metabelian, nilpotent (class 2), monomial

Aliases: C24.3C22, C23.66C23, (C2×C4)⋊6D4, (C2×D4)⋊4C4, C2.9(C4×D4), (C2×C42)⋊5C2, C41(C22⋊C4), C23.8(C2×C4), C2.2(C41D4), C2.5(C4⋊D4), (C22×D4).2C2, C22.39(C2×D4), C2.3(C4.4D4), (C22×C4).5C22, C22.24(C4○D4), C22.39(C22×C4), (C2×C4⋊C4)⋊4C2, (C2×C22⋊C4)⋊3C2, (C2×C4).41(C2×C4), C2.8(C2×C22⋊C4), SmallGroup(64,71)

Series: Derived Chief Lower central Upper central Jennings

C1C22 — C24.3C22
C1C2C22C23C22×C4C2×C42 — C24.3C22
C1C22 — C24.3C22
C1C23 — C24.3C22
C1C23 — C24.3C22

Generators and relations for C24.3C22
 G = < a,b,c,d,e,f | a2=b2=c2=d2=1, e2=d, f2=c, eae-1=ab=ba, faf-1=ac=ca, ad=da, bc=cb, bd=db, be=eb, bf=fb, cd=dc, fef-1=ce=ec, cf=fc, de=ed, df=fd >

Subgroups: 233 in 129 conjugacy classes, 53 normal (13 characteristic)
C1, C2 [×3], C2 [×4], C2 [×4], C4 [×4], C4 [×6], C22 [×3], C22 [×4], C22 [×20], C2×C4 [×10], C2×C4 [×10], D4 [×8], C23, C23 [×4], C23 [×12], C42 [×2], C22⋊C4 [×8], C4⋊C4 [×2], C22×C4, C22×C4 [×4], C2×D4 [×4], C2×D4 [×4], C24 [×2], C2×C42, C2×C22⋊C4 [×4], C2×C4⋊C4, C22×D4, C24.3C22
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×8], C23, C22⋊C4 [×4], C22×C4, C2×D4 [×4], C4○D4 [×2], C2×C22⋊C4, C4×D4 [×2], C4⋊D4 [×2], C4.4D4, C41D4, C24.3C22

Character table of C24.3C22

 class 12A2B2C2D2E2F2G2H2I2J2K4A4B4C4D4E4F4G4H4I4J4K4L4M4N4O4P
 size 1111111144442222222222224444
ρ11111111111111111111111111111    trivial
ρ2111111111-11-11-1-1-1-111-1-1-11-1-111-1    linear of order 2
ρ311111111-1-1-1-1-11111-1-1-1-1-1-1-11111    linear of order 2
ρ411111111-11-11-1-1-1-1-1-1-1111-11-111-1    linear of order 2
ρ511111111-11-111-1-1-1-111-1-1-11-11-1-11    linear of order 2
ρ6111111111111-11111-1-1-1-1-1-1-1-1-1-1-1    linear of order 2
ρ711111111-1-1-1-1111111111111-1-1-1-1    linear of order 2
ρ8111111111-11-1-1-1-1-1-1-1-1111-111-1-11    linear of order 2
ρ91-11-11-11-11-1-11i-11-11-iii-ii-i-iii-i-i    linear of order 4
ρ101-11-11-11-1-1-111-i1-11-1i-ii-iii-i-ii-ii    linear of order 4
ρ111-11-11-11-1-111-1i-11-11-iii-ii-i-i-i-iii    linear of order 4
ρ121-11-11-11-111-1-1-i1-11-1i-ii-iii-ii-ii-i    linear of order 4
ρ131-11-11-11-1-111-1-i-11-11i-i-ii-iiiii-i-i    linear of order 4
ρ141-11-11-11-111-1-1i1-11-1-ii-ii-i-ii-ii-ii    linear of order 4
ρ151-11-11-11-11-1-11-i-11-11i-i-ii-iii-i-iii    linear of order 4
ρ161-11-11-11-1-1-111i1-11-1-ii-ii-i-iii-ii-i    linear of order 4
ρ17222-2-22-2-20000022-2-200000000000    orthogonal lifted from D4
ρ18222-2-22-2-200000-2-22200000000000    orthogonal lifted from D4
ρ192-2-2-222-22000020000-2-2000200000    orthogonal lifted from D4
ρ2022-2-2-2-222000000000002-2-2020000    orthogonal lifted from D4
ρ2122-2-2-2-22200000000000-2220-20000    orthogonal lifted from D4
ρ222-222-2-2-22000002-2-2200000000000    orthogonal lifted from D4
ρ232-222-2-2-2200000-222-200000000000    orthogonal lifted from D4
ρ242-2-2-222-220000-2000022000-200000    orthogonal lifted from D4
ρ252-2-22-222-200000000000-2i-2i2i02i0000    complex lifted from C4○D4
ρ2622-222-2-2-200002i00002i-2i000-2i00000    complex lifted from C4○D4
ρ272-2-22-222-2000000000002i2i-2i0-2i0000    complex lifted from C4○D4
ρ2822-222-2-2-20000-2i0000-2i2i0002i00000    complex lifted from C4○D4

Smallest permutation representation of C24.3C22
On 32 points
Generators in S32
(1 14)(2 17)(3 16)(4 19)(5 30)(6 28)(7 32)(8 26)(9 13)(10 20)(11 15)(12 18)(21 29)(22 27)(23 31)(24 25)
(1 10)(2 11)(3 12)(4 9)(5 22)(6 23)(7 24)(8 21)(13 19)(14 20)(15 17)(16 18)(25 32)(26 29)(27 30)(28 31)
(1 29)(2 30)(3 31)(4 32)(5 17)(6 18)(7 19)(8 20)(9 25)(10 26)(11 27)(12 28)(13 24)(14 21)(15 22)(16 23)
(1 3)(2 4)(5 7)(6 8)(9 11)(10 12)(13 15)(14 16)(17 19)(18 20)(21 23)(22 24)(25 27)(26 28)(29 31)(30 32)
(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)
(1 21 29 14)(2 15 30 22)(3 23 31 16)(4 13 32 24)(5 11 17 27)(6 28 18 12)(7 9 19 25)(8 26 20 10)

G:=sub<Sym(32)| (1,14)(2,17)(3,16)(4,19)(5,30)(6,28)(7,32)(8,26)(9,13)(10,20)(11,15)(12,18)(21,29)(22,27)(23,31)(24,25), (1,10)(2,11)(3,12)(4,9)(5,22)(6,23)(7,24)(8,21)(13,19)(14,20)(15,17)(16,18)(25,32)(26,29)(27,30)(28,31), (1,29)(2,30)(3,31)(4,32)(5,17)(6,18)(7,19)(8,20)(9,25)(10,26)(11,27)(12,28)(13,24)(14,21)(15,22)(16,23), (1,3)(2,4)(5,7)(6,8)(9,11)(10,12)(13,15)(14,16)(17,19)(18,20)(21,23)(22,24)(25,27)(26,28)(29,31)(30,32), (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), (1,21,29,14)(2,15,30,22)(3,23,31,16)(4,13,32,24)(5,11,17,27)(6,28,18,12)(7,9,19,25)(8,26,20,10)>;

G:=Group( (1,14)(2,17)(3,16)(4,19)(5,30)(6,28)(7,32)(8,26)(9,13)(10,20)(11,15)(12,18)(21,29)(22,27)(23,31)(24,25), (1,10)(2,11)(3,12)(4,9)(5,22)(6,23)(7,24)(8,21)(13,19)(14,20)(15,17)(16,18)(25,32)(26,29)(27,30)(28,31), (1,29)(2,30)(3,31)(4,32)(5,17)(6,18)(7,19)(8,20)(9,25)(10,26)(11,27)(12,28)(13,24)(14,21)(15,22)(16,23), (1,3)(2,4)(5,7)(6,8)(9,11)(10,12)(13,15)(14,16)(17,19)(18,20)(21,23)(22,24)(25,27)(26,28)(29,31)(30,32), (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), (1,21,29,14)(2,15,30,22)(3,23,31,16)(4,13,32,24)(5,11,17,27)(6,28,18,12)(7,9,19,25)(8,26,20,10) );

G=PermutationGroup([(1,14),(2,17),(3,16),(4,19),(5,30),(6,28),(7,32),(8,26),(9,13),(10,20),(11,15),(12,18),(21,29),(22,27),(23,31),(24,25)], [(1,10),(2,11),(3,12),(4,9),(5,22),(6,23),(7,24),(8,21),(13,19),(14,20),(15,17),(16,18),(25,32),(26,29),(27,30),(28,31)], [(1,29),(2,30),(3,31),(4,32),(5,17),(6,18),(7,19),(8,20),(9,25),(10,26),(11,27),(12,28),(13,24),(14,21),(15,22),(16,23)], [(1,3),(2,4),(5,7),(6,8),(9,11),(10,12),(13,15),(14,16),(17,19),(18,20),(21,23),(22,24),(25,27),(26,28),(29,31),(30,32)], [(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)], [(1,21,29,14),(2,15,30,22),(3,23,31,16),(4,13,32,24),(5,11,17,27),(6,28,18,12),(7,9,19,25),(8,26,20,10)])

C24.3C22 is a maximal subgroup of
(C2×C4).98D8  (C2×D4)⋊C8  (C2×SD16)⋊15C4  C8⋊C22⋊C4  D4⋊C4⋊C4  C4.67(C4×D4)  M4(2)⋊D4  (C2×D4)⋊Q8  C4⋊C4.84D4  (C2×D4)⋊2Q8  (C2×C4)⋊5SD16  (C2×C4).24D8  C428C4⋊C2  (C2×C4).27D8  C439C2  C23.179C24  C4×C4⋊D4  C4×C4.4D4  C4×C41D4  C23.191C24  C24.542C23  C23.201C24  C23.203C24  C4213D4  C42.160D4  C4214D4  C23.215C24  C24.204C23  D4×C22⋊C4  C23.223C24  C23.236C24  C24.212C23  C23.240C24  C24.215C23  C24.217C23  C24.219C23  C24.220C23  C24.223C23  C23.257C24  C24.225C23  C23.259C24  C24.244C23  C23.308C24  C23.311C24  C24.249C23  C23.316C24  C24.254C23  C23.322C24  C23.324C24  C24.259C23  C23.328C24  C24.263C23  C24.264C23  C23.333C24  C23.335C24  C23.345C24  C24.271C23  C23.348C24  C23.352C24  C23.354C24  C23.356C24  C24.279C23  C24.282C23  C23.364C24  C23.367C24  C24.290C23  C23.372C24  C24.293C23  C23.379C24  C23.390C24  C23.391C24  C23.400C24  C23.401C24  C23.404C24  C23.412C24  C23.413C24  C23.416C24  C23.426C24  C23.431C24  C42.166D4  C23.439C24  C4219D4  C4220D4  C42.167D4  C4221D4  C42.168D4  C42.171D4  C24.327C23  C23.455C24  C23.457C24  C23.458C24  C24.331C23  C24.332C23  C42.173D4  C42.175D4  C42.178D4  C42.182D4  C23.493C24  C24.347C23  C24.348C23  C4223D4  C23.502C24  C4224D4  C4226D4  C4227D4  C4228D4  C23.524C24  C42.193D4  C42.194D4  C23.544C24  C23.548C24  C23.551C24  C24.377C23  C4232D4  C23.570C24  C23.576C24  C23.578C24  C25⋊C22  C23.581C24  C24.389C23  C23.583C24  C23.585C24  C23.591C24  C23.592C24  C24.401C23  C24.403C23  C23.597C24  C24.406C23  C24.407C23  C23.603C24  C23.605C24  C23.606C24  C23.607C24  C24.412C23  C23.611C24  C24.413C23  C24.418C23  C23.624C24  C23.627C24  C23.630C24  C23.632C24  C23.633C24  C23.637C24  C23.640C24  C24.434C23  C23.649C24  C23.651C24  C23.652C24  C24.437C23  C23.656C24  C24.438C23  C24.440C23  C24.448C23  C24.450C23  C23.686C24  C23.696C24  C23.697C24  C23.700C24  C23.703C24  C24.456C23  C23.708C24  C23.728C24  C23.729C24  C23.730C24  C4246D4  C24.598C23  C4247D4  C4312C2  C4313C2  C4314C2
 C24.D2p: C24.6D4  2+ 1+43C4  C24.21D4  M4(2)⋊6D4  C24.24D6  C24.30D6  C24.13D10  C24.19D10 ...
 (C2×D4p)⋊C4: (C2×C4)⋊9D8  (C2×C4)⋊6D8  (C2×D8)⋊10C4  (C2×C4)⋊6D12  (C2×D12)⋊10C4  (C2×C4)⋊6D20  (C2×D20)⋊22C4  C2.(D4×F5) ...
 C2.(C8pD4): C2.(C87D4)  C2.(C82D4)  (C2×C4)⋊9SD16  C8⋊(C22⋊C4)  (C2×C4)⋊2D8  (C22×D8).C2  (C2×C4)⋊3SD16  (C2×C4)⋊3D8 ...
C24.3C22 is a maximal quotient of
C24.625C23  C232C42  C24.635C23  C42.325D4  C42.109D4  C42.431D4  C42.432D4  C42.433D4  C42.110D4  C42.111D4  C42.112D4  C43⋊C2  C428D4  C24.175C23  M4(2)⋊12D4  C42.114D4  C42.115D4  (C2×C4)⋊9SD16  (C2×C4)⋊6Q16  (C2×Q16)⋊10C4  C8⋊(C22⋊C4)  M4(2).31D4  M4(2).33D4  M4(2)⋊13D4  C42.117D4  C42.118D4  C42.119D4
 (C2×D4p)⋊C4: (C2×C4)⋊6D8  (C2×D8)⋊10C4  C42.326D4  C42.116D4  M4(2).30D4  M4(2).32D4  (C2×C4)⋊6D12  (C2×D12)⋊10C4 ...
 C24.D2p: C24.50D4  C24.5Q8  C24.24D6  C24.30D6  C24.13D10  C24.19D10  C24.13D14  C24.19D14 ...

Matrix representation of C24.3C22 in GL5(𝔽5)

40000
00400
04000
00001
00010
,
10000
04000
00400
00010
00001
,
10000
04000
00400
00040
00004
,
40000
01000
00100
00040
00004
,
20000
01000
00400
00003
00030
,
10000
00100
04000
00001
00040

G:=sub<GL(5,GF(5))| [4,0,0,0,0,0,0,4,0,0,0,4,0,0,0,0,0,0,0,1,0,0,0,1,0],[1,0,0,0,0,0,4,0,0,0,0,0,4,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,4,0,0,0,0,0,4,0,0,0,0,0,4,0,0,0,0,0,4],[4,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,4,0,0,0,0,0,4],[2,0,0,0,0,0,1,0,0,0,0,0,4,0,0,0,0,0,0,3,0,0,0,3,0],[1,0,0,0,0,0,0,4,0,0,0,1,0,0,0,0,0,0,0,4,0,0,0,1,0] >;

C24.3C22 in GAP, Magma, Sage, TeX

C_2^4._3C_2^2
% in TeX

G:=Group("C2^4.3C2^2");
// GroupNames label

G:=SmallGroup(64,71);
// by ID

G=gap.SmallGroup(64,71);
# by ID

G:=PCGroup([6,-2,2,2,-2,2,2,192,121,247,362,86]);
// Polycyclic

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

Export

Character table of C24.3C22 in TeX

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