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G = D88D4order 128 = 27

2nd semidirect product of D8 and D4 acting via D4/C22=C2

p-group, metabelian, nilpotent (class 4), monomial

Aliases: D88D4, Q168D4, C23.14D8, (C2×D16)⋊2C2, C8.62(C2×D4), C2.D169C2, C87D416C2, C22⋊C167C2, (C2×SD32)⋊8C2, (C2×C4).112D8, (C2×C8).172D4, C2.Q324C2, C4.18C22≀C2, C2.6(C4○D16), (C2×C16).1C22, C22.96(C2×D8), C4.10(C8⋊C22), C2.7(C16⋊C22), (C2×C8).510C23, C2.D8.2C22, (C22×C4).345D4, C2.26(C22⋊D8), (C2×D8).107C22, (C22×C8).172C22, (C2×Q16).107C22, (C2×C4○D8)⋊1C2, (C2×C4).778(C2×D4), SmallGroup(128,918)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C8 — D88D4
C1C2C4C2×C4C2×C8C22×C8C2×C4○D8 — D88D4
C1C2C4C2×C8 — D88D4
C1C22C22×C4C22×C8 — D88D4
C1C2C2C2C2C4C4C2×C8 — D88D4

Generators and relations for D88D4
 G = < a,b,c,d | a8=b2=c4=d2=1, bab=cac-1=dad=a-1, cbc-1=ab, dbd=a5b, dcd=c-1 >

Subgroups: 324 in 115 conjugacy classes, 34 normal (30 characteristic)
C1, C2 [×3], C2 [×4], C4 [×2], C4 [×4], C22, C22 [×10], C8 [×2], C8, C2×C4 [×2], C2×C4 [×8], D4 [×11], Q8 [×3], C23, C23 [×2], C16 [×2], C22⋊C4, C4⋊C4, C2×C8 [×2], C2×C8 [×2], D8 [×2], D8 [×3], SD16 [×4], Q16 [×2], Q16, C22×C4, C22×C4, C2×D4 [×4], C2×Q8, C4○D4 [×6], D4⋊C4, C2.D8, C2×C16 [×2], D16 [×2], SD32 [×2], C4⋊D4, C22×C8, C2×D8 [×2], C2×SD16, C2×Q16, C4○D8 [×4], C2×C4○D4, C22⋊C16, C2.D16, C2.Q32, C87D4, C2×D16, C2×SD32, C2×C4○D8, D88D4
Quotients: C1, C2 [×7], C22 [×7], D4 [×6], C23, D8 [×2], C2×D4 [×3], C22≀C2, C2×D8, C8⋊C22, C22⋊D8, C4○D16, C16⋊C22, D88D4

Character table of D88D4

 class 12A2B2C2D2E2F2G4A4B4C4D4E4F4G8A8B8C8D8E8F16A16B16C16D16E16F16G16H
 size 1111488162222881622224444444444
ρ111111111111111111111111111111    trivial
ρ211111-1-111111-1-11111111-1-1-1-1-1-1-1-1    linear of order 2
ρ31111-1-1-111-1-1111-11111-1-111-11-1-1-11    linear of order 2
ρ41111-11111-1-11-1-1-11111-1-1-1-11-1111-1    linear of order 2
ρ51111-111-11-1-11-1-111111-1-111-11-1-1-11    linear of order 2
ρ61111-1-1-1-11-1-111111111-1-1-1-11-1111-1    linear of order 2
ρ711111-1-1-11111-1-1-111111111111111    linear of order 2
ρ81111111-1111111-1111111-1-1-1-1-1-1-1-1    linear of order 2
ρ92-2-220-220200-2000-222-20000000000    orthogonal lifted from D4
ρ102-2-220000200-22-202-2-220000000000    orthogonal lifted from D4
ρ112-2-220000200-2-2202-2-220000000000    orthogonal lifted from D4
ρ122-2-2202-20200-2000-222-20000000000    orthogonal lifted from D4
ρ132222-20002-2-22000-2-2-2-22200000000    orthogonal lifted from D4
ρ14222220002222000-2-2-2-2-2-200000000    orthogonal lifted from D4
ρ152222-2000-222-2000000000-2-222-2-222    orthogonal lifted from D8
ρ162222-2000-222-200000000022-2-222-2-2    orthogonal lifted from D8
ρ1722222000-2-2-2-2000000000222-2-2-22-2    orthogonal lifted from D8
ρ1822222000-2-2-2-2000000000-2-2-2222-22    orthogonal lifted from D8
ρ192-22-2000002i-2i000022-2-2--2-2ζ16131611ζ165163165163ζ167161615169ζ1615169ζ165163ζ1615169    complex lifted from C4○D16
ρ202-22-200000-2i2i000022-2-2-2--2ζ165163ζ16131611165163ζ16151691615169ζ1615169ζ165163ζ16716    complex lifted from C4○D16
ρ212-22-2000002i-2i0000-2-222-2--2ζ16716ζ1615169ζ1615169ζ165163165163ζ1651631615169ζ16131611    complex lifted from C4○D16
ρ222-22-200000-2i2i0000-2-222--2-2ζ1615169ζ16716ζ1615169ζ16131611165163ζ1651631615169ζ165163    complex lifted from C4○D16
ρ232-22-200000-2i2i0000-2-222--2-2ζ16716ζ16151691615169ζ165163ζ165163165163ζ1615169ζ16131611    complex lifted from C4○D16
ρ242-22-2000002i-2i000022-2-2--2-2ζ165163ζ16131611ζ165163ζ1615169ζ16151691615169165163ζ16716    complex lifted from C4○D16
ρ252-22-200000-2i2i000022-2-2-2--2ζ16131611ζ165163ζ165163ζ16716ζ16151691615169165163ζ1615169    complex lifted from C4○D16
ρ262-22-2000002i-2i0000-2-222-2--2ζ1615169ζ167161615169ζ16131611ζ165163165163ζ1615169ζ165163    complex lifted from C4○D16
ρ274-4-440000-400400000000000000000    orthogonal lifted from C8⋊C22
ρ2844-4-40000000000022-2222-220000000000    orthogonal lifted from C16⋊C22
ρ2944-4-400000000000-2222-22220000000000    orthogonal lifted from C16⋊C22

Smallest permutation representation of D88D4
On 64 points
Generators in S64
(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)(33 34 35 36 37 38 39 40)(41 42 43 44 45 46 47 48)(49 50 51 52 53 54 55 56)(57 58 59 60 61 62 63 64)
(1 50)(2 49)(3 56)(4 55)(5 54)(6 53)(7 52)(8 51)(9 26)(10 25)(11 32)(12 31)(13 30)(14 29)(15 28)(16 27)(17 45)(18 44)(19 43)(20 42)(21 41)(22 48)(23 47)(24 46)(33 59)(34 58)(35 57)(36 64)(37 63)(38 62)(39 61)(40 60)
(1 30 23 39)(2 29 24 38)(3 28 17 37)(4 27 18 36)(5 26 19 35)(6 25 20 34)(7 32 21 33)(8 31 22 40)(9 42 57 53)(10 41 58 52)(11 48 59 51)(12 47 60 50)(13 46 61 49)(14 45 62 56)(15 44 63 55)(16 43 64 54)
(2 8)(3 7)(4 6)(9 60)(10 59)(11 58)(12 57)(13 64)(14 63)(15 62)(16 61)(17 21)(18 20)(22 24)(25 36)(26 35)(27 34)(28 33)(29 40)(30 39)(31 38)(32 37)(41 48)(42 47)(43 46)(44 45)(49 54)(50 53)(51 52)(55 56)

G:=sub<Sym(64)| (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)(33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56)(57,58,59,60,61,62,63,64), (1,50)(2,49)(3,56)(4,55)(5,54)(6,53)(7,52)(8,51)(9,26)(10,25)(11,32)(12,31)(13,30)(14,29)(15,28)(16,27)(17,45)(18,44)(19,43)(20,42)(21,41)(22,48)(23,47)(24,46)(33,59)(34,58)(35,57)(36,64)(37,63)(38,62)(39,61)(40,60), (1,30,23,39)(2,29,24,38)(3,28,17,37)(4,27,18,36)(5,26,19,35)(6,25,20,34)(7,32,21,33)(8,31,22,40)(9,42,57,53)(10,41,58,52)(11,48,59,51)(12,47,60,50)(13,46,61,49)(14,45,62,56)(15,44,63,55)(16,43,64,54), (2,8)(3,7)(4,6)(9,60)(10,59)(11,58)(12,57)(13,64)(14,63)(15,62)(16,61)(17,21)(18,20)(22,24)(25,36)(26,35)(27,34)(28,33)(29,40)(30,39)(31,38)(32,37)(41,48)(42,47)(43,46)(44,45)(49,54)(50,53)(51,52)(55,56)>;

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)(33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56)(57,58,59,60,61,62,63,64), (1,50)(2,49)(3,56)(4,55)(5,54)(6,53)(7,52)(8,51)(9,26)(10,25)(11,32)(12,31)(13,30)(14,29)(15,28)(16,27)(17,45)(18,44)(19,43)(20,42)(21,41)(22,48)(23,47)(24,46)(33,59)(34,58)(35,57)(36,64)(37,63)(38,62)(39,61)(40,60), (1,30,23,39)(2,29,24,38)(3,28,17,37)(4,27,18,36)(5,26,19,35)(6,25,20,34)(7,32,21,33)(8,31,22,40)(9,42,57,53)(10,41,58,52)(11,48,59,51)(12,47,60,50)(13,46,61,49)(14,45,62,56)(15,44,63,55)(16,43,64,54), (2,8)(3,7)(4,6)(9,60)(10,59)(11,58)(12,57)(13,64)(14,63)(15,62)(16,61)(17,21)(18,20)(22,24)(25,36)(26,35)(27,34)(28,33)(29,40)(30,39)(31,38)(32,37)(41,48)(42,47)(43,46)(44,45)(49,54)(50,53)(51,52)(55,56) );

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),(33,34,35,36,37,38,39,40),(41,42,43,44,45,46,47,48),(49,50,51,52,53,54,55,56),(57,58,59,60,61,62,63,64)], [(1,50),(2,49),(3,56),(4,55),(5,54),(6,53),(7,52),(8,51),(9,26),(10,25),(11,32),(12,31),(13,30),(14,29),(15,28),(16,27),(17,45),(18,44),(19,43),(20,42),(21,41),(22,48),(23,47),(24,46),(33,59),(34,58),(35,57),(36,64),(37,63),(38,62),(39,61),(40,60)], [(1,30,23,39),(2,29,24,38),(3,28,17,37),(4,27,18,36),(5,26,19,35),(6,25,20,34),(7,32,21,33),(8,31,22,40),(9,42,57,53),(10,41,58,52),(11,48,59,51),(12,47,60,50),(13,46,61,49),(14,45,62,56),(15,44,63,55),(16,43,64,54)], [(2,8),(3,7),(4,6),(9,60),(10,59),(11,58),(12,57),(13,64),(14,63),(15,62),(16,61),(17,21),(18,20),(22,24),(25,36),(26,35),(27,34),(28,33),(29,40),(30,39),(31,38),(32,37),(41,48),(42,47),(43,46),(44,45),(49,54),(50,53),(51,52),(55,56)])

Matrix representation of D88D4 in GL4(𝔽17) generated by

16000
01600
00143
001414
,
01600
16000
00114
0046
,
0100
16000
0004
0040
,
16000
0100
0010
00016
G:=sub<GL(4,GF(17))| [16,0,0,0,0,16,0,0,0,0,14,14,0,0,3,14],[0,16,0,0,16,0,0,0,0,0,11,4,0,0,4,6],[0,16,0,0,1,0,0,0,0,0,0,4,0,0,4,0],[16,0,0,0,0,1,0,0,0,0,1,0,0,0,0,16] >;

D88D4 in GAP, Magma, Sage, TeX

D_8\rtimes_8D_4
% in TeX

G:=Group("D8:8D4");
// GroupNames label

G:=SmallGroup(128,918);
// by ID

G=gap.SmallGroup(128,918);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,-2,141,422,352,1123,570,360,4037,2028,124]);
// Polycyclic

G:=Group<a,b,c,d|a^8=b^2=c^4=d^2=1,b*a*b=c*a*c^-1=d*a*d=a^-1,c*b*c^-1=a*b,d*b*d=a^5*b,d*c*d=c^-1>;
// generators/relations

Export

Character table of D88D4 in TeX

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