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G = C24.C22order 64 = 26

2nd non-split extension by C24 of C22 acting faithfully

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

Aliases: C24.2C22, C23.64C23, C2.7(C4×D4), C22⋊C45C4, (C2×C42)⋊2C2, (C2×C4).100D4, C23.7(C2×C4), C2.3(C4⋊D4), C22.37(C2×D4), C2.C423C2, C2.2(C4.4D4), C2.3(C422C2), C22.22(C4○D4), C22.37(C22×C4), (C22×C4).23C22, C2.10(C42⋊C2), C2.5(C22.D4), (C2×C4⋊C4)⋊3C2, (C2×C4).17(C2×C4), (C2×C22⋊C4).6C2, SmallGroup(64,69)

Series: Derived Chief Lower central Upper central Jennings

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

Generators and relations for C24.C22
 G = < a,b,c,d,e,f | a2=b2=c2=d2=1, e2=d, f2=b, 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: 161 in 95 conjugacy classes, 45 normal (31 characteristic)
C1, C2 [×7], C2 [×2], C4 [×10], C22 [×7], C22 [×10], C2×C4 [×8], C2×C4 [×14], C23, C23 [×2], C23 [×6], C42 [×2], C22⋊C4 [×4], C22⋊C4 [×4], C4⋊C4 [×2], C22×C4 [×6], C24, C2.C42 [×2], C2×C42, C2×C22⋊C4 [×3], C2×C4⋊C4, C24.C22
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×4], C23, C22×C4, C2×D4 [×2], C4○D4 [×4], C42⋊C2, C4×D4 [×2], C4⋊D4, C22.D4, C4.4D4, C422C2, C24.C22

Character table of C24.C22

 class 12A2B2C2D2E2F2G2H2I4A4B4C4D4E4F4G4H4I4J4K4L4M4N4O4P4Q4R
 size 1111111144222222222222444444
ρ11111111111111111111111111111    trivial
ρ211111111111-1111-1-1-1-1-1-1-1-11-1-1-11    linear of order 2
ρ311111111-1-1-1-1-1-1-1-1-1-11111111-1-11    linear of order 2
ρ411111111-1-1-11-1-1-1111-1-1-1-1-11-1111    linear of order 2
ρ51111111111-11-1-1-1111-1-1-1-11-11-1-1-1    linear of order 2
ρ61111111111-1-1-1-1-1-1-1-11111-1-1-111-1    linear of order 2
ρ711111111-1-1111111111111-1-1-1-1-1-1    linear of order 2
ρ811111111-1-11-1111-1-1-1-1-1-1-11-1111-1    linear of order 2
ρ91-11-11-11-1-111i-11-1-ii-i-ii-ii-i1ii-i-1    linear of order 4
ρ101-11-11-11-1-11-1i1-11-ii-ii-ii-i-i-1i-ii1    linear of order 4
ρ111-11-11-11-11-11-i-11-1i-iii-ii-i-i-1ii-i1    linear of order 4
ρ121-11-11-11-11-1-1-i1-11i-ii-ii-ii-i1i-ii-1    linear of order 4
ρ131-11-11-11-1-11-1-i1-11i-ii-ii-iii-1-ii-i1    linear of order 4
ρ141-11-11-11-1-111-i-11-1i-iii-ii-ii1-i-ii-1    linear of order 4
ρ151-11-11-11-11-1-1i1-11-ii-ii-ii-ii1-ii-i-1    linear of order 4
ρ161-11-11-11-11-11i-11-1-ii-i-ii-iii-1-i-ii1    linear of order 4
ρ172-2-2-222-22000-2000-2220000000000    orthogonal lifted from D4
ρ182-2-2-222-2200020002-2-20000000000    orthogonal lifted from D4
ρ1922-2-2-2-22200000000002-2-22000000    orthogonal lifted from D4
ρ2022-2-2-2-2220000000000-222-2000000    orthogonal lifted from D4
ρ21222-2-22-2-200-2i02i2i-2i0000000000000    complex lifted from C4○D4
ρ222-222-2-2-2200-2i0-2i2i2i0000000000000    complex lifted from C4○D4
ρ232-2-22-222-200000000002i2i-2i-2i000000    complex lifted from C4○D4
ρ242-222-2-2-22002i02i-2i-2i0000000000000    complex lifted from C4○D4
ρ252-2-22-222-20000000000-2i-2i2i2i000000    complex lifted from C4○D4
ρ2622-222-2-2-20002i000-2i-2i2i0000000000    complex lifted from C4○D4
ρ27222-2-22-2-2002i0-2i-2i2i0000000000000    complex lifted from C4○D4
ρ2822-222-2-2-2000-2i0002i2i-2i0000000000    complex lifted from C4○D4

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

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

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

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

C24.C22 is a maximal subgroup of
C432C2  C4×C4⋊D4  C4×C22.D4  C4×C422C2  C23.194C24  C24.192C23  C24.547C23  C23.201C24  C23.203C24  C24.195C23  C24.198C23  C23.214C24  C23.215C24  C24.203C23  C24.204C23  C24.205C23  C23.224C24  C24.208C23  C23.229C24  C23.235C24  C24.212C23  C23.240C24  C23.241C24  C24.215C23  C24.218C23  C24.221C23  C23.255C24  C24.223C23  C23.257C24  C24.225C23  C23.259C24  C24.227C23  C23.261C24  C24.230C23  C23.311C24  C23.313C24  C24.249C23  C23.315C24  C23.318C24  C23.321C24  C23.323C24  C23.324C24  C24.258C23  C24.259C23  C23.329C24  C24.269C23  C23.344C24  C23.348C24  C24.276C23  C23.356C24  C24.278C23  C24.279C23  C23.359C24  C23.360C24  C24.282C23  C24.283C23  C23.364C24  C24.285C23  C24.286C23  C23.367C24  C23.368C24  C23.369C24  C24.289C23  C24.290C23  C23.372C24  C23.374C24  C23.375C24  C24.293C23  C23.377C24  C24.295C23  C23.379C24  C23.380C24  C24.573C23  C23.382C24  C23.385C24  C23.388C24  C24.301C23  C23.390C24  C23.391C24  C23.392C24  C24.304C23  C23.395C24  C23.397C24  C23.398C24  C23.400C24  C23.410C24  C23.412C24  C23.413C24  C24.309C23  C23.416C24  C23.418C24  C23.419C24  C24.311C23  C23.422C24  C23.425C24  C23.426C24  C24.315C23  C23.429C24  C23.430C24  C23.431C24  C23.432C24  C23.434C24  C42.165D4  C23.443C24  C42.168D4  C24.326C23  C24.327C23  C23.455C24  C23.456C24  C23.457C24  C23.458C24  C24.331C23  C24.332C23  C23.461C24  C42.172D4  C23.472C24  C23.473C24  C24.338C23  C24.339C23  C24.340C23  C24.341C23  C23.478C24  C24.345C23  C24.346C23  C23.491C24  C42.182D4  C23.493C24  C23.494C24  C24.347C23  C23.496C24  C24.348C23  C4222D4  C42.183D4  C23.500C24  C4223D4  C23.502C24  C4224D4  C42.184D4  C4225D4  C4226D4  C42.185D4  C23.530C24  C42.189D4  C23.535C24  C42.192D4  C23.548C24  C24.375C23  C23.550C24  C23.551C24  C23.553C24  C23.554C24  C24.377C23  C4232D4  C24.378C23  C42.198D4  C23.569C24  C23.570C24  C23.572C24  C23.573C24  C23.574C24  C24.385C23  C23.578C24  C23.580C24  C23.581C24  C23.584C24  C23.585C24  C24.393C23  C24.394C23  C23.589C24  C23.590C24  C24.401C23  C23.595C24  C24.403C23  C23.597C24  C24.405C23  C24.406C23  C23.600C24  C24.407C23  C23.602C24  C23.603C24  C24.408C23  C23.605C24  C23.606C24  C23.607C24  C24.412C23  C23.611C24  C23.612C24  C24.413C23  C23.615C24  C23.616C24  C23.618C24  C23.620C24  C23.621C24  C24.418C23  C23.625C24  C23.627C24  C24.420C23  C24.421C23  C23.630C24  C23.631C24  C23.633C24  C23.635C24  C23.636C24  C23.637C24  C24.426C23  C24.427C23  C23.640C24  C23.641C24  C24.428C23  C23.643C24  C24.430C23  C23.645C24  C24.432C23  C23.647C24  C23.649C24  C24.435C23  C23.651C24  C23.652C24  C24.437C23  C23.654C24  C23.656C24  C24.438C23  C23.659C24  C24.440C23  C23.663C24  C23.664C24  C24.443C23  C23.668C24  C24.445C23  C23.671C24  C23.672C24  C23.673C24  C23.675C24  C23.677C24  C23.678C24  C23.679C24  C23.681C24  C23.682C24  C23.683C24  C23.685C24  C23.686C24  C23.687C24  C23.688C24  C24.454C23  C23.693C24  C23.695C24  C23.696C24  C23.697C24  C23.698C24  C23.700C24  C23.701C24  C23.703C24  C23.707C24  C23.708C24  C23.728C24  C23.729C24  C23.731C24  C23.732C24  C23.736C24  C23.737C24  C23.738C24  C4246D4  C4243D4  C23.753C24  C24.598C23  C4313C2  C4314C2  C434C2  C435C2
 C2p.(C4×D4): C4242D4  C439C2  C4×C4.4D4  C42.159D4  C4213D4  C42.160D4  C23.234C24  C23.236C24 ...
C24.C22 is a maximal quotient of
C24.624C23  C24.626C23  C232C42  C24.632C23  C24.633C23  C22⋊C44C8  C23.9M4(2)  C4.68(C4×D4)  C2.(C4×Q16)  C4.10D43C4  C4.D43C4  C42.427D4  C2.(C88D4)  C2.(C8⋊D4)
 C24.D2p: C24.5Q8  C24.52D4  C24.14D6  C24.15D6  C24.19D6  C24.3D10  C24.4D10  C24.8D10 ...
 D2p⋊C4⋊C4: D4⋊C4⋊C4  C4.67(C4×D4)  M4(2).24D4  C2.(C87D4)  C2.(C82D4)  C42.428D4  C42.107D4  D6⋊C45C4 ...

Matrix representation of C24.C22 in GL5(𝔽5)

40000
01000
01400
00010
00024
,
10000
04000
00400
00010
00001
,
10000
04000
00400
00040
00004
,
40000
04000
00400
00010
00001
,
20000
03400
00200
00010
00024
,
10000
01300
01400
00014
00004

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

C24.C22 in GAP, Magma, Sage, TeX

C_2^4.C_2^2
% in TeX

G:=Group("C2^4.C2^2");
// GroupNames label

G:=SmallGroup(64,69);
// by ID

G=gap.SmallGroup(64,69);
# by ID

G:=PCGroup([6,-2,2,2,-2,2,2,192,121,247,362,50]);
// Polycyclic

G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=1,e^2=d,f^2=b,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.C22 in TeX

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