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G = Q167D4order 128 = 27

1st semidirect product of Q16 and D4 acting via D4/C22=C2

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

Aliases: Q167D4, C223SD32, C23.45D8, (C2×C8).62D4, (C2×C4).31D8, C8.61(C2×D4), (C2×SD32)⋊7C2, C87D4.3C2, C2.5(C2×SD32), C22⋊C1610C2, C2.Q328C2, C4.17C22≀C2, C4.9(C8⋊C22), (C22×Q16)⋊4C2, (C2×D8).1C22, C22.95(C2×D8), (C2×C16).36C22, (C2×C8).509C23, C2.D8.1C22, C2.6(Q32⋊C2), (C22×C4).344D4, C2.25(C22⋊D8), (C22×C8).126C22, (C2×Q16).106C22, (C2×C4).777(C2×D4), SmallGroup(128,917)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C8 — Q167D4
C1C2C4C2×C4C2×C8C22×C8C22×Q16 — Q167D4
C1C2C4C2×C8 — Q167D4
C1C22C22×C4C22×C8 — Q167D4
C1C2C2C2C2C4C4C2×C8 — Q167D4

Generators and relations for Q167D4
 G = < a,b,c,d | a8=c4=d2=1, b2=a4, bab-1=cac-1=dad=a-1, cbc-1=dbd=a-1b, dcd=c-1 >

Subgroups: 292 in 113 conjugacy classes, 36 normal (20 characteristic)
C1, C2 [×3], C2 [×3], C4 [×2], C4 [×6], C22, C22 [×2], C22 [×5], C8 [×2], C8, C2×C4 [×2], C2×C4 [×9], D4 [×4], Q8 [×10], C23, C23, C16 [×2], C22⋊C4, C4⋊C4, C2×C8 [×2], C2×C8 [×2], D8 [×2], Q16 [×4], Q16 [×6], C22×C4, C22×C4, C2×D4 [×2], C2×Q8 [×9], D4⋊C4, C2.D8, C2×C16 [×2], SD32 [×4], C4⋊D4, C22×C8, C2×D8, C2×Q16 [×2], C2×Q16 [×5], C22×Q8, C22⋊C16, C2.Q32 [×2], C87D4, C2×SD32 [×2], C22×Q16, Q167D4
Quotients: C1, C2 [×7], C22 [×7], D4 [×6], C23, D8 [×2], C2×D4 [×3], SD32 [×2], C22≀C2, C2×D8, C8⋊C22, C22⋊D8, C2×SD32, Q32⋊C2, Q167D4

Character table of Q167D4

 class 12A2B2C2D2E2F4A4B4C4D4E4F4G4H8A8B8C8D8E8F16A16B16C16D16E16F16G16H
 size 1111221622488881622224444444444
ρ111111111111111111111111111111    trivial
ρ2111111-11111111-1111111-1-1-1-1-1-1-1-1    linear of order 2
ρ31111-1-1-111-111-1-111111-1-1-1-1-111-111    linear of order 2
ρ41111-1-1111-111-1-1-11111-1-1111-1-11-1-1    linear of order 2
ρ51111111111-1-1-1-11111111-1-1-1-1-1-1-1-1    linear of order 2
ρ6111111-1111-1-1-1-1-111111111111111    linear of order 2
ρ71111-1-1-111-1-1-11111111-1-1111-1-11-1-1    linear of order 2
ρ81111-1-1111-1-1-111-11111-1-1-1-1-111-111    linear of order 2
ρ92-2-220002-20-22000-22-220000000000    orthogonal lifted from D4
ρ102222-2-2022-200000-2-2-2-22200000000    orthogonal lifted from D4
ρ11222222022200000-2-2-2-2-2-200000000    orthogonal lifted from D4
ρ122-2-220002-202-2000-22-220000000000    orthogonal lifted from D4
ρ132-2-220002-2000-2202-22-20000000000    orthogonal lifted from D4
ρ142-2-220002-20002-202-22-20000000000    orthogonal lifted from D4
ρ152222220-2-2-2000000000002-222-2-22-2    orthogonal lifted from D8
ρ162222-2-20-2-22000000000002-22-22-2-22    orthogonal lifted from D8
ρ172222220-2-2-200000000000-22-2-222-22    orthogonal lifted from D8
ρ182222-2-20-2-2200000000000-22-22-222-2    orthogonal lifted from D8
ρ192-22-22-200000000022-2-22-2ζ16716ζ16131611ζ1615169ζ1615169ζ165163ζ165163ζ16716ζ16131611    complex lifted from SD32
ρ202-22-22-200000000022-2-22-2ζ1615169ζ165163ζ16716ζ16716ζ16131611ζ16131611ζ1615169ζ165163    complex lifted from SD32
ρ212-22-22-2000000000-2-222-22ζ16131611ζ1615169ζ165163ζ165163ζ16716ζ16716ζ16131611ζ1615169    complex lifted from SD32
ρ222-22-22-2000000000-2-222-22ζ165163ζ16716ζ16131611ζ16131611ζ1615169ζ1615169ζ165163ζ16716    complex lifted from SD32
ρ232-22-2-2200000000022-2-2-22ζ1615169ζ165163ζ16716ζ1615169ζ165163ζ16131611ζ16716ζ16131611    complex lifted from SD32
ρ242-22-2-2200000000022-2-2-22ζ16716ζ16131611ζ1615169ζ16716ζ16131611ζ165163ζ1615169ζ165163    complex lifted from SD32
ρ252-22-2-22000000000-2-2222-2ζ165163ζ16716ζ16131611ζ165163ζ16716ζ1615169ζ16131611ζ1615169    complex lifted from SD32
ρ262-22-2-22000000000-2-2222-2ζ16131611ζ1615169ζ165163ζ16131611ζ1615169ζ16716ζ165163ζ16716    complex lifted from SD32
ρ274-4-44000-4400000000000000000000    orthogonal lifted from C8⋊C22
ρ2844-4-40000000000022-22-22220000000000    symplectic lifted from Q32⋊C2, Schur index 2
ρ2944-4-400000000000-222222-220000000000    symplectic lifted from Q32⋊C2, Schur index 2

Smallest permutation representation of Q167D4
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 58 5 62)(2 57 6 61)(3 64 7 60)(4 63 8 59)(9 22 13 18)(10 21 14 17)(11 20 15 24)(12 19 16 23)(25 52 29 56)(26 51 30 55)(27 50 31 54)(28 49 32 53)(33 46 37 42)(34 45 38 41)(35 44 39 48)(36 43 40 47)
(1 37 19 26)(2 36 20 25)(3 35 21 32)(4 34 22 31)(5 33 23 30)(6 40 24 29)(7 39 17 28)(8 38 18 27)(9 51 59 42)(10 50 60 41)(11 49 61 48)(12 56 62 47)(13 55 63 46)(14 54 64 45)(15 53 57 44)(16 52 58 43)
(2 8)(3 7)(4 6)(9 16)(10 15)(11 14)(12 13)(17 21)(18 20)(22 24)(25 38)(26 37)(27 36)(28 35)(29 34)(30 33)(31 40)(32 39)(41 53)(42 52)(43 51)(44 50)(45 49)(46 56)(47 55)(48 54)(57 60)(58 59)(61 64)(62 63)

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

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

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

Matrix representation of Q167D4 in GL4(𝔽17) generated by

31400
3300
00160
00016
,
71600
161000
0042
00113
,
1000
01600
001315
0004
,
1000
01600
0010
001316
G:=sub<GL(4,GF(17))| [3,3,0,0,14,3,0,0,0,0,16,0,0,0,0,16],[7,16,0,0,16,10,0,0,0,0,4,1,0,0,2,13],[1,0,0,0,0,16,0,0,0,0,13,0,0,0,15,4],[1,0,0,0,0,16,0,0,0,0,1,13,0,0,0,16] >;

Q167D4 in GAP, Magma, Sage, TeX

Q_{16}\rtimes_7D_4
% in TeX

G:=Group("Q16:7D4");
// GroupNames label

G:=SmallGroup(128,917);
// by ID

G=gap.SmallGroup(128,917);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,-2,141,456,422,1123,570,360,4037,2028,124]);
// Polycyclic

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

Export

Character table of Q167D4 in TeX

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