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G = C23.36D4order 64 = 26

7th non-split extension by C23 of D4 acting via D4/C22=C2

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

Aliases: C23.36D4, C4○D42C4, D44(C2×C4), Q84(C2×C4), C4.50(C2×D4), (C2×C4).123D4, C4.4(C22×C4), D4⋊C414C2, Q8⋊C414C2, C4⋊C4.44C22, C2.1(C8⋊C22), (C2×C8).43C22, (C2×C4).62C23, C22.44(C2×D4), C4.24(C22⋊C4), (C2×M4(2))⋊10C2, (C2×D4).48C22, C2.1(C8.C22), (C2×Q8).42C22, C22.4(C22⋊C4), (C22×C4).34C22, (C2×C4⋊C4)⋊10C2, (C2×C4).21(C2×C4), (C2×C4○D4).5C2, C2.20(C2×C22⋊C4), SmallGroup(64,98)

Series: Derived Chief Lower central Upper central Jennings

C1C4 — C23.36D4
C1C2C22C2×C4C22×C4C2×C4○D4 — C23.36D4
C1C2C4 — C23.36D4
C1C22C22×C4 — C23.36D4
C1C2C2C2×C4 — C23.36D4

Generators and relations for C23.36D4
 G = < a,b,c,d,e | a2=b2=c2=1, d4=c, e2=b, ab=ba, dad-1=ac=ca, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede-1=bcd3 >

Subgroups: 137 in 81 conjugacy classes, 41 normal (17 characteristic)
C1, C2 [×3], C2 [×4], C4 [×2], C4 [×2], C4 [×4], C22, C22 [×2], C22 [×6], C8 [×2], C2×C4 [×2], C2×C4 [×4], C2×C4 [×9], D4 [×2], D4 [×5], Q8 [×2], Q8, C23, C23, C4⋊C4 [×2], C4⋊C4, C2×C8 [×2], M4(2) [×2], C22×C4, C22×C4 [×2], C2×D4, C2×D4, C2×Q8, C4○D4 [×4], C4○D4 [×2], D4⋊C4 [×2], Q8⋊C4 [×2], C2×C4⋊C4, C2×M4(2), C2×C4○D4, C23.36D4
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, C8.C22, C23.36D4

Character table of C23.36D4

 class 12A2B2C2D2E2F2G4A4B4C4D4E4F4G4H4I4J8A8B8C8D
 size 1111224422224444444444
ρ11111111111111111111111    trivial
ρ21111-1-1111-1-111-11-1-1-11-11-1    linear of order 2
ρ31111-1-1-1-11-1-11111-1-11-11-11    linear of order 2
ρ4111111-1-111111-1111-1-1-1-1-1    linear of order 2
ρ5111111-1-11111-1-1-1-1-1-11111    linear of order 2
ρ61111-1-1-1-11-1-11-11-11111-11-1    linear of order 2
ρ71111-1-1111-1-11-1-1-111-1-11-11    linear of order 2
ρ8111111111111-11-1-1-11-1-1-1-1    linear of order 2
ρ91-11-11-1-111-11-1-i-1i-ii1-iii-i    linear of order 4
ρ101-11-1-11-1111-1-1-i1ii-i-1-i-iii    linear of order 4
ρ111-11-1-111-111-1-1-i-1ii-i1ii-i-i    linear of order 4
ρ121-11-11-11-11-11-1-i1i-ii-1i-i-ii    linear of order 4
ρ131-11-11-11-11-11-1i1-ii-i-1-iii-i    linear of order 4
ρ141-11-1-111-111-1-1i-1-i-ii1-i-iii    linear of order 4
ρ151-11-1-11-1111-1-1i1-i-ii-1ii-i-i    linear of order 4
ρ161-11-11-1-111-11-1i-1-ii-i1i-i-ii    linear of order 4
ρ172222-2-200-222-20000000000    orthogonal lifted from D4
ρ182-22-2-2200-2-2220000000000    orthogonal lifted from D4
ρ1922222200-2-2-2-20000000000    orthogonal lifted from D4
ρ202-22-22-200-22-220000000000    orthogonal lifted from D4
ρ214-4-44000000000000000000    orthogonal lifted from C8⋊C22
ρ2244-4-4000000000000000000    symplectic lifted from C8.C22, Schur index 2

Smallest permutation representation of C23.36D4
On 32 points
Generators in S32
(1 19)(2 24)(3 21)(4 18)(5 23)(6 20)(7 17)(8 22)(9 25)(10 30)(11 27)(12 32)(13 29)(14 26)(15 31)(16 28)
(1 29)(2 30)(3 31)(4 32)(5 25)(6 26)(7 27)(8 28)(9 23)(10 24)(11 17)(12 18)(13 19)(14 20)(15 21)(16 22)
(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 28 29 8)(2 7 30 27)(3 26 31 6)(4 5 32 25)(9 18 23 12)(10 11 24 17)(13 22 19 16)(14 15 20 21)

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

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

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

C23.36D4 is a maximal subgroup of
2+ 1+43C4  2- 1+42C4  C4211D4  C4212D4  C24.98D4  2+ 1+45C4  2- 1+44C4  C4×C8.C22  C24.105D4  C24.106D4  (C2×D4)⋊21D4  (C2×Q8)⋊17D4  C42.211D4  C42.212D4  M4(2)⋊14D4  M4(2)⋊15D4  (C2×C8)⋊13D4  (C2×C8)⋊14D4  C42.448D4  C42.449D4  C24.183D4  C24.116D4  (C2×D4).303D4  (C2×D4).304D4  C42.243D4  C42.244D4  (C2×C4).S4
 C4○D4p⋊C4: C42.280C23  C42.281C23  C4○D12⋊C4  C23.54D12  C4○D209C4  C23.49D20  C4○D4⋊F5  C4○D28⋊C4 ...
 D4p⋊(C2×C4): C4×C8⋊C22  Q87(C4×S3)  Q8⋊(C4×D5)  Q8⋊(C4×D7) ...
 (C2×C4p).D4: C4○D4.4Q8  C4○D4.5Q8  M4(2).48D4  M4(2).49D4  M4(2).10D4  M4(2).11D4  (C2×C8).55D4  (C2×C8).165D4 ...
 C4⋊C4.D2p: C4≀C2⋊C4  C429(C2×C4)  M4(2)⋊6D4  M4(2).7D4  C42.18C23  C42.19C23  C42.22C23  C42.23C23 ...
C23.36D4 is a maximal quotient of
C42.397D4  C42.47D4  D44M4(2)  C42.52D4  C24.150D4  C42.57D4  C42.58D4  C24.58D4  C42.79D4  C42.80D4  C42.81D4  C42.84D4  C42.86D4  C42.87D4  C42.88D4  C24.152D4  D4⋊C42  Q8⋊C42  C24.65D4  C42.99D4  C24.69D4  C24.76D4  C42.123D4  C42.124D4  C4○D4⋊F5
 C23.D4p: C23.38D8  C23.54D12  C23.49D20  C23.49D28 ...
 (C2×C4).D4p: C42.78D4  C42.98D4  C4○D12⋊C4  C4○D209C4  C4○D28⋊C4 ...
 (C2×C4p).D4: C24.75D4  C42.110D4  C4○D43Dic3  C4○D4⋊Dic5  C4○D4⋊Dic7 ...
 C4⋊C4.D2p: C24.160D4  C42.119D4  D4⋊(C4×S3)  Q87(C4×S3)  D4⋊(C4×D5)  Q8⋊(C4×D5)  D4⋊(C4×D7)  Q8⋊(C4×D7) ...

Matrix representation of C23.36D4 in GL6(𝔽17)

1600000
0160000
0001300
004000
004448
001301313
,
1600000
0160000
001000
000100
000010
000001
,
100000
010000
0016000
0001600
0000160
0000016
,
16150000
010000
000010
001112
0001600
00001616
,
16150000
110000
000010
0016161615
001000
000001

G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,0,4,4,13,0,0,13,0,4,0,0,0,0,0,4,13,0,0,0,0,8,13],[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],[16,0,0,0,0,0,15,1,0,0,0,0,0,0,0,1,0,0,0,0,0,1,16,0,0,0,1,1,0,16,0,0,0,2,0,16],[16,1,0,0,0,0,15,1,0,0,0,0,0,0,0,16,1,0,0,0,0,16,0,0,0,0,1,16,0,0,0,0,0,15,0,1] >;

C23.36D4 in GAP, Magma, Sage, TeX

C_2^3._{36}D_4
% in TeX

G:=Group("C2^3.36D4");
// GroupNames label

G:=SmallGroup(64,98);
// by ID

G=gap.SmallGroup(64,98);
# by ID

G:=PCGroup([6,-2,2,2,-2,2,-2,96,121,332,963,489,117]);
// 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=a*c=c*a,a*e=e*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*c*d^3>;
// generators/relations

Export

Character table of C23.36D4 in TeX

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