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G = C87D4order 64 = 26

1st semidirect product of C8 and D4 acting via D4/C22=C2

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

Aliases: C87D4, C221D8, C23.27D4, (C2×D8)⋊4C2, C2.D82C2, C2.6(C2×D8), C4⋊D43C2, (C22×C8)⋊6C2, C4.54(C2×D4), (C2×C4).55D4, D4⋊C42C2, C4.8(C4○D4), C4⋊C4.6C22, C2.11(C4○D8), (C2×C4).94C23, (C2×C8).76C22, C22.90(C2×D4), C2.18(C4⋊D4), (C2×D4).16C22, (C22×C4).118C22, SmallGroup(64,147)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C87D4
C1C2C22C2×C4C22×C4C22×C8 — C87D4
C1C2C2×C4 — C87D4
C1C22C22×C4 — C87D4
C1C2C2C2×C4 — C87D4

Generators and relations for C87D4
 G = < a,b,c | a8=b4=c2=1, bab-1=cac=a-1, cbc=b-1 >

Subgroups: 141 in 67 conjugacy classes, 29 normal (17 characteristic)
C1, C2 [×3], C2 [×4], C4 [×2], C4 [×3], C22, C22 [×2], C22 [×8], C8 [×2], C8, C2×C4 [×2], C2×C4 [×4], D4 [×8], C23, C23 [×2], C22⋊C4 [×2], C4⋊C4 [×2], C2×C8 [×2], C2×C8 [×2], D8 [×2], C22×C4, C2×D4 [×2], C2×D4 [×2], D4⋊C4 [×2], C2.D8, C4⋊D4 [×2], C22×C8, C2×D8, C87D4
Quotients: C1, C2 [×7], C22 [×7], D4 [×4], C23, D8 [×2], C2×D4 [×2], C4○D4, C4⋊D4, C2×D8, C4○D8, C87D4

Character table of C87D4

 class 12A2B2C2D2E2F2G4A4B4C4D4E4F8A8B8C8D8E8F8G8H
 size 1111228822228822222222
ρ11111111111111111111111    trivial
ρ21111-1-1-11-111-11-1-11-111-1-11    linear of order 2
ρ31111-1-11-1-111-1-11-11-111-1-11    linear of order 2
ρ4111111-1-11111-1-111111111    linear of order 2
ρ5111111-111111-11-1-1-1-1-1-1-1-1    linear of order 2
ρ61111-1-111-111-1-1-11-11-1-111-1    linear of order 2
ρ71111-1-1-1-1-111-1111-11-1-111-1    linear of order 2
ρ81111111-111111-1-1-1-1-1-1-1-1-1    linear of order 2
ρ92222-2-2002-2-220000000000    orthogonal lifted from D4
ρ102-2-22000002-2000-202002-20    orthogonal lifted from D4
ρ1122222200-2-2-2-20000000000    orthogonal lifted from D4
ρ122-2-22000002-200020-200-220    orthogonal lifted from D4
ρ132-22-22-200000000-2-222-2-222    orthogonal lifted from D8
ρ142-22-22-20000000022-2-222-2-2    orthogonal lifted from D8
ρ152-22-2-2200000000-222-22-22-2    orthogonal lifted from D8
ρ162-22-2-22000000002-2-22-22-22    orthogonal lifted from D8
ρ172-2-2200000-220000-2i02i2i00-2i    complex lifted from C4○D4
ρ1822-2-20000-2i002i00-2--2-2--2-222-2    complex lifted from C4○D8
ρ1922-2-200002i00-2i002--22--2-2-2-2-2    complex lifted from C4○D8
ρ2022-2-200002i00-2i00-2-2-2-2--222--2    complex lifted from C4○D8
ρ2122-2-20000-2i002i002-22-2--2-2-2--2    complex lifted from C4○D8
ρ222-2-2200000-2200002i0-2i-2i002i    complex lifted from C4○D4

Smallest permutation representation of C87D4
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 18 9 29)(2 17 10 28)(3 24 11 27)(4 23 12 26)(5 22 13 25)(6 21 14 32)(7 20 15 31)(8 19 16 30)
(2 8)(3 7)(4 6)(10 16)(11 15)(12 14)(17 30)(18 29)(19 28)(20 27)(21 26)(22 25)(23 32)(24 31)

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

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

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

C87D4 is a maximal subgroup of
C22.SD32  C23.12SD16  D87D4  Q167D4  D88D4  C22.D16  C23.49D8  C23.19D8  C24.144D4  M4(2)⋊14D4  (C2×C8)⋊12D4  (C2×C8)⋊14D4  M4(2)⋊16D4  C42.308D4  C42.387C23  C42.388C23  C42.391C23  C42.257D4  C233D8  C24.121D4  C24.124D4  C4.2+ 1+4  C4.142+ 1+4  C4.182+ 1+4  C42.295D4  C42.298D4  C4.2- 1+4  C42.26C23  C42.29C23  SD16⋊D4  SD167D4  D4×D8  D812D4  Q1612D4  Q1613D4  C42.462C23  C42.466C23  C42.470C23  C42.43C23  C42.44C23  C42.53C23  C42.54C23  C42.488C23  C42.490C23  C42.59C23  C42.61C23  A4⋊D8
 D2p⋊D8: D44D8  D45D8  D6⋊D8  D62D8  D63D8  D10⋊D8  C87D20  C406D4 ...
 C8p⋊D4: C167D4  C168D4  C16⋊D4  C162D4  C2429D4  C4029D4  C5629D4 ...
 (C2×C2p)⋊D8: C42.366D4  C42.293D4  C3⋊C822D4  (C2×C10)⋊D8  C7⋊C822D4 ...
C87D4 is a maximal quotient of
C813SD16  C23.22D8  C23.23D8  C2.(C87D4)  C85(C4⋊C4)  C232D8  (C2×C4).23D8  C23.12D8  (C2×C4).26D8  C16.19D4  C16.D4  D4.3D8  D4.4D8  D4.5D8
 C8p⋊D4: C167D4  C168D4  C16⋊D4  C162D4  D62D8  C2429D4  D63D8  C87D20 ...
 D2p⋊D8: C87D8  D6⋊D8  D10⋊D8  D14⋊D8 ...
 (C2×C2p)⋊D8: (C2×C4)⋊6D8  (C2×C4)⋊3D8  C3⋊C822D4  (C2×C10)⋊D8  C7⋊C822D4 ...
 C2.(D4.pD4): C810SD16  C87Q16  D4.2D8  Q8.2D8  C24.83D4 ...

Matrix representation of C87D4 in GL4(𝔽17) generated by

31400
3300
0049
00013
,
16000
0100
00162
00161
,
1000
01600
0010
00116
G:=sub<GL(4,GF(17))| [3,3,0,0,14,3,0,0,0,0,4,0,0,0,9,13],[16,0,0,0,0,1,0,0,0,0,16,16,0,0,2,1],[1,0,0,0,0,16,0,0,0,0,1,1,0,0,0,16] >;

C87D4 in GAP, Magma, Sage, TeX

C_8\rtimes_7D_4
% in TeX

G:=Group("C8:7D4");
// GroupNames label

G:=SmallGroup(64,147);
// by ID

G=gap.SmallGroup(64,147);
# by ID

G:=PCGroup([6,-2,2,2,-2,2,-2,121,247,362,963,117]);
// Polycyclic

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

Export

Character table of C87D4 in TeX

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