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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 [×3], C4 [×2], C4 [×5], C22, C22 [×2], C8 [×2], C8 [×2], C2×C4, C2×C4 [×8], D4, D4 [×5], Q8, Q8 [×8], C16 [×2], C2×C8, C2×C8, M4(2), M4(2) [×2], D8, SD16 [×4], Q16, Q16 [×2], Q16 [×6], C2×Q8 [×5], C4○D4, C4○D4 [×6], C4.10D4, C8.C4, C2×C16, M5(2), SD32, Q32 [×3], C8○D4, C2×Q16 [×2], C2×Q16, C4○D8, C4○D8, C8.C22 [×4], 2- 1+4, D4.C8, D8.C4, C8.17D4, D4.5D4, C2×Q32, Q32⋊C2, Q8○D8, D8.12D4
Quotients: C1, C2 [×7], C22 [×7], D4 [×6], C23, D8 [×2], C2×D4 [×3], 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 22 9 30)(2 31 10 23)(3 24 11 32)(4 17 12 25)(5 26 13 18)(6 19 14 27)(7 28 15 20)(8 21 16 29)(33 52 41 60)(34 61 42 53)(35 54 43 62)(36 63 44 55)(37 56 45 64)(38 49 46 57)(39 58 47 50)(40 51 48 59)
(1 35)(2 63)(3 37)(4 49)(5 39)(6 51)(7 41)(8 53)(9 43)(10 55)(11 45)(12 57)(13 47)(14 59)(15 33)(16 61)(17 38)(18 58)(19 40)(20 60)(21 42)(22 62)(23 44)(24 64)(25 46)(26 50)(27 48)(28 52)(29 34)(30 54)(31 36)(32 56)
(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 26 25 18)(19 24 27 32)(20 31 28 23)(21 22 29 30)(33 36 41 44)(34 43 42 35)(37 48 45 40)(38 39 46 47)(49 50 57 58)(51 64 59 56)(52 55 60 63)(53 62 61 54)

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

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

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

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

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