Copied to
clipboard

G = D8.12D4order 128 = 27

4th non-split extension by D8 of D4 acting via D4/C22=C2

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

Aliases: D8.12D4, D4.11D8, Q16.3D4, Q8.11D8, M4(2).19D4, M5(2).2C22, Q8○D8.C2, D4.C84C2, (C2×Q32)⋊3C2, C8.69(C2×D4), C4.40(C2×D8), C4○D4.12D4, Q32⋊C23C2, C4.27C22≀C2, D4.5D42C2, D8.C42C2, C8.17D42C2, (C2×C16).3C22, C4○D8.8C22, C8○D4.4C22, (C2×C8).234C23, C2.35(C22⋊D8), C22.7(C8⋊C22), C8.C4.4C22, (C2×Q16).45C22, (C2×C4).42(C2×D4), SmallGroup(128,927)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C8 — D8.12D4
C1C2C4C2×C4C2×C8C8○D4Q8○D8 — D8.12D4
C1C2C4C2×C8 — D8.12D4
C1C2C2×C4C8○D4 — D8.12D4
C1C2C2C2C2C4C4C2×C8 — D8.12D4

Generators and relations for D8.12D4
 G = < a,b,c,d | a4=b2=1, c8=d2=a2, bab=cac-1=dad-1=a-1, cbc-1=ab, dbd-1=a-1b, dcd-1=c7 >

Subgroups: 240 in 103 conjugacy classes, 32 normal (30 characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, C16, C2×C8, C2×C8, M4(2), M4(2), D8, SD16, Q16, Q16, Q16, C2×Q8, C4○D4, C4○D4, C4.10D4, C8.C4, C2×C16, M5(2), SD32, Q32, C8○D4, C2×Q16, C2×Q16, C4○D8, C4○D8, C8.C22, 2- 1+4, D4.C8, D8.C4, C8.17D4, D4.5D4, C2×Q32, Q32⋊C2, Q8○D8, D8.12D4
Quotients: C1, C2, C22, D4, C23, D8, C2×D4, C22≀C2, C2×D8, C8⋊C22, C22⋊D8, D8.12D4

Character table of D8.12D4

 class 12A2B2C2D4A4B4C4D4E4F4G8A8B8C8D8E16A16B16C16D16E16F
 size 1124822488816224816444488
ρ111111111111111111111111    trivial
ρ2111-1111-1-11-1-1111-111111-1-1    linear of order 2
ρ31111-1111-1-1-1-11111-1111111    linear of order 2
ρ4111-1-111-11-111111-1-11111-1-1    linear of order 2
ρ5111-1-111-11-11-1111-11-1-1-1-111    linear of order 2
ρ61111-1111-1-1-1111111-1-1-1-1-1-1    linear of order 2
ρ7111-1111-1-11-11111-1-1-1-1-1-111    linear of order 2
ρ811111111111-11111-1-1-1-1-1-1-1    linear of order 2
ρ922-2022-200-20022-200000000    orthogonal lifted from D4
ρ10222202220000-2-2-2-20000000    orthogonal lifted from D4
ρ1122-2002-2020-20-2-2200000000    orthogonal lifted from D4
ρ1222-20-22-20020022-200000000    orthogonal lifted from D4
ρ13222-2022-20000-2-2-220000000    orthogonal lifted from D4
ρ1422-2002-20-2020-2-2200000000    orthogonal lifted from D4
ρ1522-2-20-222000000000-22-222-2    orthogonal lifted from D8
ρ1622-220-22-2000000000-22-22-22    orthogonal lifted from D8
ρ1722-2-20-2220000000002-22-2-22    orthogonal lifted from D8
ρ1822-220-22-20000000002-22-22-2    orthogonal lifted from D8
ρ1944400-4-40000000000000000    orthogonal lifted from C8⋊C22
ρ204-40000000000-222200016716516316ζ1615169165163ζ16716516316161516916516300    symplectic faithful, Schur index 2
ρ214-40000000000-2222000ζ16716516316161516916516316716516316ζ161516916516300    symplectic faithful, Schur index 2
ρ224-4000000000022-22000161516916516316716516316ζ1615169165163ζ1671651631600    symplectic faithful, Schur index 2
ρ234-4000000000022-22000ζ1615169165163ζ1671651631616151691651631671651631600    symplectic faithful, Schur index 2

Smallest permutation representation of D8.12D4
On 64 points
Generators in S64
(1 60 9 52)(2 53 10 61)(3 62 11 54)(4 55 12 63)(5 64 13 56)(6 57 14 49)(7 50 15 58)(8 59 16 51)(17 47 25 39)(18 40 26 48)(19 33 27 41)(20 42 28 34)(21 35 29 43)(22 44 30 36)(23 37 31 45)(24 46 32 38)
(1 42)(2 21)(3 44)(4 23)(5 46)(6 25)(7 48)(8 27)(9 34)(10 29)(11 36)(12 31)(13 38)(14 17)(15 40)(16 19)(18 58)(20 60)(22 62)(24 64)(26 50)(28 52)(30 54)(32 56)(33 59)(35 61)(37 63)(39 49)(41 51)(43 53)(45 55)(47 57)
(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 8 9 16)(2 15 10 7)(3 6 11 14)(4 13 12 5)(17 30 25 22)(18 21 26 29)(19 28 27 20)(23 24 31 32)(33 42 41 34)(35 40 43 48)(36 47 44 39)(37 38 45 46)(49 54 57 62)(50 61 58 53)(51 52 59 60)(55 64 63 56)

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

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

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

Matrix representation of D8.12D4 in GL4(𝔽17) generated by

0010
16161615
16000
1101
,
11112
7010
109103
16766
,
131531
0101512
014212
4729
,
0101512
131531
1315108
4729
G:=sub<GL(4,GF(17))| [0,16,16,1,0,16,0,1,1,16,0,0,0,15,0,1],[1,7,10,16,11,0,9,7,1,1,10,6,2,0,3,6],[13,0,0,4,15,10,14,7,3,15,2,2,1,12,12,9],[0,13,13,4,10,15,15,7,15,3,10,2,12,1,8,9] >;

D8.12D4 in GAP, Magma, Sage, TeX

D_8._{12}D_4
% in TeX

G:=Group("D8.12D4");
// GroupNames label

G:=SmallGroup(128,927);
// by ID

G=gap.SmallGroup(128,927);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,-2,448,141,422,352,1123,570,360,2804,718,172,4037,2028,124]);
// Polycyclic

G:=Group<a,b,c,d|a^4=b^2=1,c^8=d^2=a^2,b*a*b=c*a*c^-1=d*a*d^-1=a^-1,c*b*c^-1=a*b,d*b*d^-1=a^-1*b,d*c*d^-1=c^7>;
// generators/relations

Export

Character table of D8.12D4 in TeX

׿
×
𝔽