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G = D4.2SD16order 128 = 27

2nd non-split extension by D4 of SD16 acting via SD16/C8=C2

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

Aliases: D4.2SD16, C42.228C23, C82C83C2, (C8×D4)⋊29C2, C4⋊C4.203D4, C4⋊D8.5C2, (C2×C8).311D4, D42Q833C2, (C2×D4).192D4, C4.68(C4○D8), C4.D815C2, C4.4D824C2, C4.6Q164C2, C4.40(C2×SD16), C4⋊Q8.51C22, C2.12(C88D4), C4⋊C8.179C22, C4.89(C8⋊C22), (C4×C8).253C22, C2.8(D4.2D4), (C4×D4).280C22, C41D4.28C22, C2.10(D4.4D4), C22.189(C4⋊D4), (C2×C4).13(C4○D4), (C2×C4).1263(C2×D4), SmallGroup(128,409)

Series: Derived Chief Lower central Upper central Jennings

C1C42 — D4.2SD16
C1C2C22C2×C4C42C4×D4C8×D4 — D4.2SD16
C1C22C42 — D4.2SD16
C1C22C42 — D4.2SD16
C1C22C22C42 — D4.2SD16

Generators and relations for D4.2SD16
 G = < a,b,c,d | a4=b2=c8=d2=1, bab=dad=a-1, ac=ca, bc=cb, dbd=ab, dcd=a2c3 >

Subgroups: 232 in 88 conjugacy classes, 34 normal (32 characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C2×C4, C2×C4, D4, D4, Q8, C23, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, D8, C22×C4, C2×D4, C2×D4, C2×Q8, C4×C8, C22⋊C8, D4⋊C4, C4⋊C8, C4.Q8, C4×D4, C41D4, C4⋊Q8, C22×C8, C2×D8, C4.D8, C4.6Q16, C82C8, C8×D4, C4⋊D8, D42Q8, C4.4D8, D4.2SD16
Quotients: C1, C2, C22, D4, C23, SD16, C2×D4, C4○D4, C4⋊D4, C2×SD16, C4○D8, C8⋊C22, D4.2D4, C88D4, D4.4D4, D4.2SD16

Character table of D4.2SD16

 class 12A2B2C2D2E2F4A4B4C4D4E4F4G4H8A8B8C8D8E8F8G8H8I8J8K8L8M8N
 size 1111441622224441622224444448888
ρ111111111111111111111111111111    trivial
ρ21111-1-1-11111-1-111-1-1-1-1111-1-11-11-11    linear of order 2
ρ31111-1-111111-1-1111111-1-1-111-1-1-1-1-1    linear of order 2
ρ4111111-111111111-1-1-1-1-1-1-1-1-1-11-11-1    linear of order 2
ρ51111-1-1-11111-1-11-11111-1-1-111-11111    linear of order 2
ρ611111111111111-1-1-1-1-1-1-1-1-1-1-1-11-11    linear of order 2
ρ7111111-11111111-11111111111-1-1-1-1    linear of order 2
ρ81111-1-111111-1-11-1-1-1-1-1111-1-111-11-1    linear of order 2
ρ92222000-22-2200-202222000-2-200000    orthogonal lifted from D4
ρ102222000-22-2200-20-2-2-2-20002200000    orthogonal lifted from D4
ρ112222-2-202-22-222-2000000000000000    orthogonal lifted from D4
ρ1222222202-22-2-2-2-2000000000000000    orthogonal lifted from D4
ρ132222000-2-2-2-2002000002i2i-2i00-2i0000    complex lifted from C4○D4
ρ142222000-2-2-2-200200000-2i-2i2i002i0000    complex lifted from C4○D4
ρ1522-2-2000-20202i-2i00--2-2--2-2-22-2--2-220000    complex lifted from C4○D8
ρ1622-2-2-22020-200000-2--2-2--2-2--2--2--2-2-20000    complex lifted from SD16
ρ1722-2-2000-2020-2i2i00--2-2--2-22-22--2-2-20000    complex lifted from C4○D8
ρ1822-2-22-2020-200000-2--2-2--2--2-2-2--2-2--20000    complex lifted from SD16
ρ1922-2-2-22020-200000--2-2--2-2--2-2-2-2--2--20000    complex lifted from SD16
ρ2022-2-2000-2020-2i2i00-2--2-2--2-22-2-2--220000    complex lifted from C4○D8
ρ2122-2-22-2020-200000--2-2--2-2-2--2--2-2--2-20000    complex lifted from SD16
ρ222-22-2000020-20000-2i2i2i-2i000000-2-2--22    complex lifted from C4○D8
ρ232-22-2000020-20000-2i2i2i-2i000000--22-2-2    complex lifted from C4○D8
ρ2422-2-2000-20202i-2i00-2--2-2--22-22-2--2-20000    complex lifted from C4○D8
ρ252-22-2000020-200002i-2i-2i2i000000-22--2-2    complex lifted from C4○D8
ρ262-22-2000020-200002i-2i-2i2i000000--2-2-22    complex lifted from C4○D8
ρ274-44-40000-404000000000000000000    orthogonal lifted from C8⋊C22
ρ284-4-44000000000002222-22-220000000000    orthogonal lifted from D4.4D4
ρ294-4-4400000000000-22-2222220000000000    orthogonal lifted from D4.4D4

Smallest permutation representation of D4.2SD16
On 64 points
Generators in S64
(1 61 22 34)(2 62 23 35)(3 63 24 36)(4 64 17 37)(5 57 18 38)(6 58 19 39)(7 59 20 40)(8 60 21 33)(9 49 43 31)(10 50 44 32)(11 51 45 25)(12 52 46 26)(13 53 47 27)(14 54 48 28)(15 55 41 29)(16 56 42 30)
(1 13)(2 14)(3 15)(4 16)(5 9)(6 10)(7 11)(8 12)(17 42)(18 43)(19 44)(20 45)(21 46)(22 47)(23 48)(24 41)(25 59)(26 60)(27 61)(28 62)(29 63)(30 64)(31 57)(32 58)(33 52)(34 53)(35 54)(36 55)(37 56)(38 49)(39 50)(40 51)
(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)
(2 17)(3 7)(4 23)(6 21)(8 19)(9 31)(10 52)(11 29)(12 50)(13 27)(14 56)(15 25)(16 54)(20 24)(26 44)(28 42)(30 48)(32 46)(33 39)(34 61)(35 37)(36 59)(38 57)(40 63)(41 51)(43 49)(45 55)(47 53)(58 60)(62 64)

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

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

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

Matrix representation of D4.2SD16 in GL4(𝔽17) generated by

0100
16000
0010
0001
,
31400
141400
0010
0001
,
13000
01300
00010
00127
,
1000
01600
0010
001616
G:=sub<GL(4,GF(17))| [0,16,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[3,14,0,0,14,14,0,0,0,0,1,0,0,0,0,1],[13,0,0,0,0,13,0,0,0,0,0,12,0,0,10,7],[1,0,0,0,0,16,0,0,0,0,1,16,0,0,0,16] >;

D4.2SD16 in GAP, Magma, Sage, TeX

D_4._2{\rm SD}_{16}
% in TeX

G:=Group("D4.2SD16");
// GroupNames label

G:=SmallGroup(128,409);
// by ID

G=gap.SmallGroup(128,409);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,2,141,512,422,1123,136,2804,718,172]);
// Polycyclic

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

Export

Character table of D4.2SD16 in TeX

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