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G = C23.10D4order 64 = 26

3rd non-split extension by C23 of D4 acting via D4/C2=C22

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

Aliases: C23.10D4, C24.6C22, C23.77C23, (C2×C4)⋊3D4, C2.6C22≀C2, C2.7(C4⋊D4), (C22×D4).3C2, C22.70(C2×D4), C2.6(C4.4D4), C2.C4211C2, C22.37(C4○D4), (C22×C4).26C22, C2.6(C22.D4), (C2×C4⋊C4)⋊5C2, (C2×C22⋊C4)⋊7C2, SmallGroup(64,75)

Series: Derived Chief Lower central Upper central Jennings

C1C23 — C23.10D4
C1C2C22C23C22×C4C2×C22⋊C4 — C23.10D4
C1C23 — C23.10D4
C1C23 — C23.10D4
C1C23 — C23.10D4

Generators and relations for C23.10D4
 G = < a,b,c,d,e | a2=b2=c2=d4=e2=1, eae=ab=ba, ac=ca, dad-1=abc, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede=bd-1 >

Subgroups: 229 in 119 conjugacy classes, 39 normal (15 characteristic)
C1, C2 [×3], C2 [×4], C2 [×4], C4 [×7], C22 [×3], C22 [×4], C22 [×20], C2×C4 [×4], C2×C4 [×13], D4 [×8], C23, C23 [×4], C23 [×12], C22⋊C4 [×8], C4⋊C4 [×2], C22×C4 [×3], C22×C4 [×2], C2×D4 [×6], C24 [×2], C2.C42, C2×C22⋊C4 [×4], C2×C4⋊C4, C22×D4, C23.10D4
Quotients: C1, C2 [×7], C22 [×7], D4 [×8], C23, C2×D4 [×4], C4○D4 [×3], C22≀C2, C4⋊D4 [×3], C22.D4 [×2], C4.4D4, C23.10D4

Character table of C23.10D4

 class 12A2B2C2D2E2F2G2H2I2J2K4A4B4C4D4E4F4G4H4I4J
 size 1111111144444444444444
ρ11111111111111111111111    trivial
ρ211111111-1-1-1-11111-1-1-1-111    linear of order 2
ρ311111111-1-1-1-1-11-1-111111-1    linear of order 2
ρ4111111111111-11-1-1-1-1-1-11-1    linear of order 2
ρ5111111111-11-11-1-11-1-111-1-1    linear of order 2
ρ611111111-11-111-1-1111-1-1-1-1    linear of order 2
ρ711111111-11-11-1-11-1-1-111-11    linear of order 2
ρ8111111111-11-1-1-11-111-1-1-11    linear of order 2
ρ922-2-222-2-2020-20000000000    orthogonal lifted from D4
ρ102-2-22-222-20000002000000-2    orthogonal lifted from D4
ρ1122-22-2-2-22000002000000-20    orthogonal lifted from D4
ρ12222-2-2-22-220-200000000000    orthogonal lifted from D4
ρ1322-22-2-2-2200000-200000020    orthogonal lifted from D4
ρ142-2-22-222-2000000-20000002    orthogonal lifted from D4
ρ15222-2-2-22-2-20200000000000    orthogonal lifted from D4
ρ1622-2-222-2-20-2020000000000    orthogonal lifted from D4
ρ172-2-2-22-222000000002i-2i0000    complex lifted from C4○D4
ρ182-2-2-22-22200000000-2i2i0000    complex lifted from C4○D4
ρ192-2222-2-2-200002i00-2i000000    complex lifted from C4○D4
ρ202-2222-2-2-20000-2i002i000000    complex lifted from C4○D4
ρ212-22-2-22-2200000000002i-2i00    complex lifted from C4○D4
ρ222-22-2-22-220000000000-2i2i00    complex lifted from C4○D4

Smallest permutation representation of C23.10D4
On 32 points
Generators in S32
(1 5)(2 11)(3 7)(4 9)(6 19)(8 17)(10 18)(12 20)(13 25)(14 24)(15 27)(16 22)(21 31)(23 29)(26 30)(28 32)
(1 22)(2 23)(3 24)(4 21)(5 16)(6 13)(7 14)(8 15)(9 31)(10 32)(11 29)(12 30)(17 27)(18 28)(19 25)(20 26)
(1 28)(2 25)(3 26)(4 27)(5 32)(6 29)(7 30)(8 31)(9 15)(10 16)(11 13)(12 14)(17 21)(18 22)(19 23)(20 24)
(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)
(2 21)(4 23)(5 16)(6 8)(7 14)(9 11)(10 32)(12 30)(13 15)(17 25)(19 27)(29 31)

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

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

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

C23.10D4 is a maximal subgroup of
C4⋊C4.D4  (C2×C4)⋊SD16  C23.288C24  C4215D4  C23.295C24  C42.163D4  C24.243C23  C24.244C23  C23.308C24  C248D4  C23.311C24  C24.249C23  C23.315C24  C23.316C24  C23.318C24  C24.254C23  C23.322C24  C23.324C24  C24.258C23  C24.259C23  C23.327C24  C23.328C24  C24.262C23  C24.264C23  C23.335C24  C24.269C23  C23.344C24  C23.345C24  C24.276C23  C24.278C23  C24.279C23  C23.359C24  C23.360C24  C24.282C23  C24.283C23  C23.364C24  C24.286C23  C23.367C24  C23.368C24  C24.289C23  C23.372C24  C23.374C24  C24.293C23  C23.377C24  C23.379C24  C23.385C24  C23.390C24  C23.391C24  C23.400C24  C23.416C24  C23.418C24  C24.311C23  C23.426C24  C23.431C24  C23.434C24  C42.165D4  C4218D4  C23.443C24  C4221D4  C42.171D4  C24.326C23  C24.327C23  C23.455C24  C23.458C24  C24.331C23  C24.332C23  C42.172D4  C23.472C24  C24.340C23  C23.478C24  C23.491C24  C23.493C24  C24.347C23  C24.348C23  C4222D4  C23.500C24  C4223D4  C23.502C24  C4224D4  C4225D4  C4226D4  C249D4  C24.361C23  C2410D4  C24.587C23  C4227D4  C4228D4  C23.524C24  C23.530C24  C42.189D4  C42.190D4  C23.535C24  C4230D4  C24.374C23  C24.592C23  C42.194D4  C23.548C24  C24.375C23  C23.551C24  C23.553C24  C24.377C23  C4232D4  C24.378C23  C23.568C24  C23.569C24  C23.570C24  C23.571C24  C23.572C24  C23.573C24  C23.574C24  C23.576C24  C23.578C24  C25⋊C22  C23.580C24  C23.581C24  C24.389C23  C23.584C24  C23.585C24  C24.393C23  C24.394C23  C24.395C23  C23.591C24  C24.401C23  C23.595C24  C24.403C23  C23.597C24  C24.406C23  C23.600C24  C24.407C23  C23.602C24  C23.603C24  C23.605C24  C23.606C24  C23.607C24  C23.608C24  C24.411C23  C24.412C23  C23.611C24  C23.612C24  C24.413C23  C23.615C24  C23.618C24  C24.418C23  C23.627C24  C24.420C23  C23.630C24  C23.631C24  C23.633C24  C23.637C24  C23.640C24  C23.641C24  C23.643C24  C24.432C23  C23.649C24  C24.435C23  C23.651C24  C23.652C24  C24.437C23  C23.656C24  C24.438C23  C24.440C23  C23.678C24  C23.679C24  C23.681C24  C23.682C24  C23.685C24  C23.686C24  C23.695C24  C23.696C24  C23.697C24  C23.700C24  C23.701C24  C23.703C24  C23.708C24  C2411D4  C24.459C23  C23.714C24  C23.716C24  C4233D4  C4234D4  C42.199D4  C4235D4  C23.724C24  C23.725C24  C23.726C24  C23.727C24  C23.728C24  C23.729C24  C23.730C24  C23.732C24  C23.737C24
 (C2×C4)⋊D4p: (C2×C4)⋊D8  (C2×C4)⋊3D12  (C2×C4)⋊3D20  (C2×C4)⋊3D28 ...
 C24.D2p: C24.94D4  C24.95D4  C24.97D4  C24.25D6  C24.27D6  C24.31D6  C24.14D10  C24.16D10 ...
 (C2×C4).D4p: C2.C2≀C4  C6.C22≀C2  (C2×Dic5)⋊3D4  (C2×Dic7)⋊3D4 ...
C23.10D4 is a maximal quotient of
C24.631C23  C24.632C23  M4(2).7D4  C4211D4  C4212D4  C4⋊C47D4  C4⋊C4.94D4  C4⋊C4.95D4  M4(2).8D4  M4(2).9D4
 (C2×C4)⋊D4p: (C2×C4)⋊3D8  (C2×C4)⋊3D12  (C2×C4)⋊3D20  (C2×C4)⋊3D28 ...
 C24.D2p: C24.50D4  C24.5Q8  C24.52D4  C24.83D4  C24.84D4  C24.85D4  C24.86D4  M4(2)⋊6D4 ...
 (C2×C4p).D4: C4⋊C4.96D4  C4⋊C4.97D4  C4⋊C4.98D4  C42.131D4  M4(2).10D4  M4(2).11D4  C22⋊C4.7D4  (C2×C4)⋊5SD16 ...

Matrix representation of C23.10D4 in GL6(𝔽5)

400000
010000
000400
004000
000001
000010
,
100000
010000
004000
000400
000040
000004
,
400000
040000
004000
000400
000040
000004
,
010000
400000
000400
004000
000020
000002
,
100000
040000
001000
000400
000010
000004

G:=sub<GL(6,GF(5))| [4,0,0,0,0,0,0,1,0,0,0,0,0,0,0,4,0,0,0,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[0,4,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,4,0,0,0,0,0,0,0,2,0,0,0,0,0,0,2],[1,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,4] >;

C23.10D4 in GAP, Magma, Sage, TeX

C_2^3._{10}D_4
% in TeX

G:=Group("C2^3.10D4");
// GroupNames label

G:=SmallGroup(64,75);
// by ID

G=gap.SmallGroup(64,75);
# by ID

G:=PCGroup([6,-2,2,2,-2,2,2,121,362,332,50]);
// Polycyclic

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

Export

Character table of C23.10D4 in TeX

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