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G = C88D4order 64 = 26

2nd semidirect product of C8 and D4 acting via D4/C22=C2

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

Aliases: C88D4, C221SD16, C23.26D4, C4.Q86C2, (C22×C8)⋊9C2, C4.53(C2×D4), (C2×C4).54D4, C22⋊Q83C2, D4⋊C41C2, Q8⋊C41C2, C4.7(C4○D4), C4⋊D4.3C2, C4⋊C4.5C22, C2.9(C2×SD16), (C2×SD16)⋊13C2, C2.10(C4○D8), (C2×C8).91C22, (C2×C4).93C23, C22.89(C2×D4), C2.17(C4⋊D4), (C2×D4).15C22, (C2×Q8).11C22, (C22×C4).117C22, SmallGroup(64,146)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C88D4
C1C2C22C2×C4C22×C4C22×C8 — C88D4
C1C2C2×C4 — C88D4
C1C22C22×C4 — C88D4
C1C2C2C2×C4 — C88D4

Generators and relations for C88D4
 G = < a,b,c | a8=b4=c2=1, bab-1=cac=a3, cbc=b-1 >

Subgroups: 117 in 62 conjugacy classes, 29 normal (25 characteristic)
C1, C2 [×3], C2 [×3], C4 [×2], C4 [×4], C22, C22 [×2], C22 [×5], C8 [×2], C8, C2×C4 [×2], C2×C4 [×5], D4 [×4], Q8 [×2], C23, C23, C22⋊C4 [×2], C4⋊C4 [×2], C4⋊C4, C2×C8 [×2], C2×C8 [×2], SD16 [×2], C22×C4, C2×D4, C2×D4, C2×Q8, D4⋊C4, Q8⋊C4, C4.Q8, C4⋊D4, C22⋊Q8, C22×C8, C2×SD16, C88D4
Quotients: C1, C2 [×7], C22 [×7], D4 [×4], C23, SD16 [×2], C2×D4 [×2], C4○D4, C4⋊D4, C2×SD16, C4○D8, C88D4

Character table of C88D4

 class 12A2B2C2D2E2F4A4B4C4D4E4F4G8A8B8C8D8E8F8G8H
 size 1111228222288822222222
ρ11111111111111111111111    trivial
ρ2111111-11111-1-1-111111111    linear of order 2
ρ31111-1-11-111-1-11-111-11-11-1-1    linear of order 2
ρ41111-1-1-1-111-11-1111-11-11-1-1    linear of order 2
ρ511111111111-1-11-1-1-1-1-1-1-1-1    linear of order 2
ρ6111111-1111111-1-1-1-1-1-1-1-1-1    linear of order 2
ρ71111-1-11-111-11-1-1-1-11-11-111    linear of order 2
ρ81111-1-1-1-111-1-111-1-11-11-111    linear of order 2
ρ92-2-2200002-2000000-20-2022    orthogonal lifted from D4
ρ102-2-2200002-20000002020-2-2    orthogonal lifted from D4
ρ112222220-2-2-2-200000000000    orthogonal lifted from D4
ρ122222-2-202-2-2200000000000    orthogonal lifted from D4
ρ132-2-220000-220000-2i2i02i0-2i00    complex lifted from C4○D4
ρ142-2-220000-2200002i-2i0-2i02i00    complex lifted from C4○D4
ρ152-22-22-200000000-2--2-2-2--2--2-2--2    complex lifted from SD16
ρ1622-2-20002i00-2i000-2-2-22--22--2-2    complex lifted from C4○D8
ρ1722-2-2000-2i002i00022-2-2--2-2--2-2    complex lifted from C4○D8
ρ182-22-22-200000000--2-2--2--2-2-2--2-2    complex lifted from SD16
ρ192-22-2-2200000000-2--2--2-2-2--2--2-2    complex lifted from SD16
ρ2022-2-20002i00-2i00022--2-2-2-2-2--2    complex lifted from C4○D8
ρ2122-2-2000-2i002i000-2-2--22-22-2--2    complex lifted from C4○D8
ρ222-22-2-2200000000--2-2-2--2--2-2-2--2    complex lifted from SD16

Smallest permutation representation of C88D4
On 32 points
Generators in S32
(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 16 31 21)(2 11 32 24)(3 14 25 19)(4 9 26 22)(5 12 27 17)(6 15 28 20)(7 10 29 23)(8 13 30 18)
(2 4)(3 7)(6 8)(9 24)(10 19)(11 22)(12 17)(13 20)(14 23)(15 18)(16 21)(25 29)(26 32)(28 30)

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

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

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

C88D4 is a maximal subgroup of
M4(2)⋊15D4  (C2×C8)⋊11D4  M4(2)⋊17D4  C42.308D4  C42.258D4  C234SD16  C24.123D4  C4.152+ 1+4  C4.162+ 1+4  C42.296D4  C42.298D4  C82S4
 D2p⋊SD16: D47SD16  D48SD16  D49SD16  D6⋊SD16  D62SD16  C88D12  C2414D4  D10⋊SD16 ...
 C4⋊C4.D2p: C42.385C23  C42.386C23  C42.390C23  C42.391C23  C4.2- 1+4  C42.25C23  C42.27C23  C42.30C23 ...
 C8pD4⋊C2: C24.144D4  M4(2)⋊14D4  (C2×C8)⋊13D4  M4(2)⋊16D4  C42.365D4  C42.257D4  C24.121D4  C24.124D4 ...
 (C2×C2p)⋊SD16: C42.294D4  C2430D4  C4030D4  C5630D4 ...
C88D4 is a maximal quotient of
C24.133D4  C24.135D4  C2.(C88D4)  C2.(C87D4)  C87(C4⋊C4)  (C2×C4)⋊9SD16  C24.89D4  C4.(C4⋊Q8)
 C8⋊D4p: C88D8  C88D12  C88D20  C88D28 ...
 C23.D4p: C23.23D8  C2430D4  C4030D4  C5630D4 ...
 C4⋊C4.D2p: C814SD16  C811SD16  C88Q16  D4.2SD16  Q8.2SD16  D4.3SD16  Q8.3SD16  C24.84D4 ...
 (C2×D4).D2p: C233SD16  (C2×C4)⋊5SD16  C2414D4  C4014D4  C5614D4 ...

Matrix representation of C88D4 in GL4(𝔽17) generated by

51200
5500
00125
001212
,
01300
13000
00160
0001
,
1000
01600
0010
00016
G:=sub<GL(4,GF(17))| [5,5,0,0,12,5,0,0,0,0,12,12,0,0,5,12],[0,13,0,0,13,0,0,0,0,0,16,0,0,0,0,1],[1,0,0,0,0,16,0,0,0,0,1,0,0,0,0,16] >;

C88D4 in GAP, Magma, Sage, TeX

C_8\rtimes_8D_4
% in TeX

G:=Group("C8:8D4");
// GroupNames label

G:=SmallGroup(64,146);
// by ID

G=gap.SmallGroup(64,146);
# by ID

G:=PCGroup([6,-2,2,2,-2,2,-2,121,55,362,963,117]);
// Polycyclic

G:=Group<a,b,c|a^8=b^4=c^2=1,b*a*b^-1=c*a*c=a^3,c*b*c=b^-1>;
// generators/relations

Export

Character table of C88D4 in TeX

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