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G = C23.38D4order 64 = 26

9th non-split extension by C23 of D4 acting via D4/C22=C2

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

Aliases: C23.38D4, (C2×Q8)⋊6C4, C4.52(C2×D4), Q8.6(C2×C4), (C2×C4).125D4, C4.6(C22×C4), Q8⋊C415C2, C4⋊C4.45C22, (C2×C4).64C23, (C2×C8).44C22, C22.46(C2×D4), (C22×Q8).5C2, C4.15(C22⋊C4), C42⋊C2.4C2, C2.2(C8.C22), (C2×Q8).43C22, (C2×M4(2)).12C2, (C22×C4).36C22, C22.18(C22⋊C4), (C2×C4).23(C2×C4), C2.22(C2×C22⋊C4), SmallGroup(64,100)

Series: Derived Chief Lower central Upper central Jennings

C1C4 — C23.38D4
C1C2C22C2×C4C22×C4C22×Q8 — C23.38D4
C1C2C4 — C23.38D4
C1C22C22×C4 — C23.38D4
C1C2C2C2×C4 — C23.38D4

Generators and relations for C23.38D4
 G = < a,b,c,d,e | a2=b2=c2=1, d4=c, e2=b, ab=ba, dad-1=eae-1=ac=ca, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede-1=bd3 >

Subgroups: 113 in 75 conjugacy classes, 41 normal (13 characteristic)
C1, C2, C2 [×2], C2 [×2], C4 [×2], C4 [×2], C4 [×6], C22, C22 [×2], C22 [×2], C8 [×2], C2×C4 [×2], C2×C4 [×4], C2×C4 [×8], Q8 [×4], Q8 [×6], C23, C42, C22⋊C4, C4⋊C4 [×2], C2×C8 [×2], M4(2) [×2], C22×C4, C22×C4, C2×Q8 [×6], C2×Q8 [×3], Q8⋊C4 [×4], C42⋊C2, C2×M4(2), C22×Q8, C23.38D4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×4], C23, C22⋊C4 [×4], C22×C4, C2×D4 [×2], C2×C22⋊C4, C8.C22 [×2], C23.38D4

Character table of C23.38D4

 class 12A2B2C2D2E4A4B4C4D4E4F4G4H4I4J4K4L8A8B8C8D
 size 1111222222444444444444
ρ11111111111111111111111    trivial
ρ21111-1-111-1-1-111-1-111-1-11-11    linear of order 2
ρ31111-1-111-1-11-1-111-1-11-11-11    linear of order 2
ρ41111111111-1-1-1-1-1-1-1-11111    linear of order 2
ρ5111111111111-1-111-1-1-1-1-1-1    linear of order 2
ρ61111-1-111-1-1-11-11-11-111-11-1    linear of order 2
ρ71111-1-111-1-11-11-11-11-11-11-1    linear of order 2
ρ81111111111-1-111-1-111-1-1-1-1    linear of order 2
ρ91-11-1-111-1-11ii11-i-i-1-1ii-i-i    linear of order 4
ρ101-11-11-11-11-1-ii1-1i-i-11-iii-i    linear of order 4
ρ111-11-1-111-1-11-i-i-1-1ii11ii-i-i    linear of order 4
ρ121-11-11-11-11-1i-i-11-ii1-1-iii-i    linear of order 4
ρ131-11-1-111-1-11ii-1-1-i-i11-i-iii    linear of order 4
ρ141-11-11-11-11-1-ii-11i-i1-1i-i-ii    linear of order 4
ρ151-11-1-111-1-11-i-i11ii-1-1-i-iii    linear of order 4
ρ161-11-11-11-11-1i-i1-1-ii-11i-i-ii    linear of order 4
ρ172-22-22-2-22-22000000000000    orthogonal lifted from D4
ρ18222222-2-2-2-2000000000000    orthogonal lifted from D4
ρ192222-2-2-2-222000000000000    orthogonal lifted from D4
ρ202-22-2-22-222-2000000000000    orthogonal lifted from D4
ρ2144-4-4000000000000000000    symplectic lifted from C8.C22, Schur index 2
ρ224-4-44000000000000000000    symplectic lifted from C8.C22, Schur index 2

Smallest permutation representation of C23.38D4
On 32 points
Generators in S32
(2 6)(4 8)(10 14)(12 16)(18 22)(20 24)(26 30)(28 32)
(1 25)(2 26)(3 27)(4 28)(5 29)(6 30)(7 31)(8 32)(9 17)(10 18)(11 19)(12 20)(13 21)(14 22)(15 23)(16 24)
(1 5)(2 6)(3 7)(4 8)(9 13)(10 14)(11 15)(12 16)(17 21)(18 22)(19 23)(20 24)(25 29)(26 30)(27 31)(28 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 10 25 18)(2 21 26 13)(3 16 27 24)(4 19 28 11)(5 14 29 22)(6 17 30 9)(7 12 31 20)(8 23 32 15)

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

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

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

C23.38D4 is a maximal subgroup of
(C22×Q8)⋊C4  C42.6D4  M4(2).9D4  C24.98D4  2- 1+44C4  C4×C8.C22  C24.178D4  C24.104D4  (C2×Q8)⋊16D4  Q8.(C2×D4)  C42.445D4  C42.446D4  C42.220D4  C24.118D4  (C2×D4).302D4  C42.241D4  C42.242D4  C23.15S4  (C2×Q8)⋊4F5
 C4⋊C4.D2p: C8.C22⋊C4  C24.23D4  C4⋊Q815C4  C4.10D43C4  M4(2).31D4  M4(2).33D4  C4210D4  C42.130D4 ...
 (C2×C8).D2p: M4(2).46D4  C4.(C4×D4)  M4(2).D4  (C2×C8).6D4  C8.D4⋊C2  C23.51D12  C23.46D20  C23.46D28 ...
 C4p.(C2×D4): M4(2)⋊15D4  (C2×C8)⋊11D4  (C6×Q8)⋊6C4  (Q8×C10)⋊16C4  (Q8×C14)⋊6C4 ...
C23.38D4 is a maximal quotient of
C42.399D4  C42.401D4  Q85M4(2)  C42.54D4  C42.56D4  C24.55D4  C24.57D4  C42.60D4  C42.415D4  C42.418D4  C42.83D4  C42.85D4  C42.86D4  C24.152D4  Q8⋊C42  C24.155D4  C42.101D4  C42.122D4  C42.125D4  (C2×Q8)⋊4F5
 C23.D4p: C23.36D8  C23.51D12  C23.46D20  C23.46D28 ...
 C4.(C2×D4p): C42.414D4  C4⋊C4.237D6  C4.(C2×D20)  (C2×C4).47D28 ...
 (C2×C4p).D4: C24.75D4  C42.111D4  (C6×Q8)⋊6C4  (Q8×C10)⋊16C4  (Q8×C14)⋊6C4 ...
 C4⋊C4.D2p: C24.73D4  C42.117D4  (S3×Q8)⋊C4  (Q8×D5)⋊C4  (Q8×D7)⋊C4 ...

Matrix representation of C23.38D4 in GL6(𝔽17)

1600000
0160000
001000
000100
0000160
0000016
,
1600000
0160000
001000
000100
000010
000001
,
100000
010000
0016000
0001600
0000160
0000016
,
040000
1300000
000004
000040
004000
0001300
,
040000
400000
000010
000001
001000
000100

G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[0,13,0,0,0,0,4,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,13,0,0,0,4,0,0,0,0,4,0,0,0],[0,4,0,0,0,0,4,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,1,0,0] >;

C23.38D4 in GAP, Magma, Sage, TeX

C_2^3._{38}D_4
% in TeX

G:=Group("C2^3.38D4");
// GroupNames label

G:=SmallGroup(64,100);
// by ID

G=gap.SmallGroup(64,100);
# by ID

G:=PCGroup([6,-2,2,2,-2,2,-2,96,121,199,332,158,963,489,117]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^2=c^2=1,d^4=c,e^2=b,a*b=b*a,d*a*d^-1=e*a*e^-1=a*c=c*a,b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=b*d^3>;
// generators/relations

Export

Character table of C23.38D4 in TeX

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