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G = D4.D6order 96 = 25·3

4th non-split extension by D4 of D6 acting via D6/S3=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C8.2D6, D4.4D6, D6.8D4, Q8.6D6, SD162S3, Dic126C2, C12.6C23, C24.9C22, Dic3.10D4, Dic6.2C22, (S3×Q8)⋊2C2, C8⋊S32C2, D42S3.C2, D4.S34C2, C3⋊Q161C2, C6.32(C2×D4), C2.20(S3×D4), C3⋊C8.1C22, (C3×SD16)⋊2C2, C32(C8.C22), C4.6(C22×S3), (C4×S3).3C22, (C3×D4).4C22, (C3×Q8).1C22, SmallGroup(96,122)

Series: Derived Chief Lower central Upper central

C1C12 — D4.D6
C1C3C6C12C4×S3S3×Q8 — D4.D6
C3C6C12 — D4.D6
C1C2C4SD16

Generators and relations for D4.D6
 G = < a,b,c,d | a4=b2=1, c6=d2=a2, bab=cac-1=dad-1=a-1, cbc-1=ab, dbd-1=a-1b, dcd-1=c5 >

Subgroups: 138 in 60 conjugacy classes, 27 normal (all characteristic)
C1, C2, C2 [×2], C3, C4, C4 [×4], C22 [×2], S3, C6, C6, C8, C8, C2×C4 [×3], D4, D4, Q8, Q8 [×3], Dic3, Dic3 [×2], C12, C12, D6, C2×C6, M4(2), SD16, SD16, Q16 [×2], C2×Q8, C4○D4, C3⋊C8, C24, Dic6 [×2], Dic6, C4×S3, C4×S3, C2×Dic3, C3⋊D4, C3×D4, C3×Q8, C8.C22, C8⋊S3, Dic12, D4.S3, C3⋊Q16, C3×SD16, D42S3, S3×Q8, D4.D6
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], C23, D6 [×3], C2×D4, C22×S3, C8.C22, S3×D4, D4.D6

Character table of D4.D6

 class 12A2B2C34A4B4C4D4E6A6B8A8B12A12B24A24B
 size 114622461212284124844
ρ1111111111111111111    trivial
ρ211-1111-11-1-11-1111-111    linear of order 2
ρ3111-1111-1-1-1111-11111    linear of order 2
ρ4111-111-1-11-111-111-1-1-1    linear of order 2
ρ511-1111111-11-1-1-111-1-1    linear of order 2
ρ611-1-111-1-1111-11-11-111    linear of order 2
ρ711-1-1111-1-111-1-1111-1-1    linear of order 2
ρ8111111-11-1111-1-11-1-1-1    linear of order 2
ρ922022-20-2002000-2000    orthogonal lifted from D4
ρ1022-20-122000-11-20-1-111    orthogonal lifted from D6
ρ11220-22-202002000-2000    orthogonal lifted from D4
ρ122220-122000-1-120-1-1-1-1    orthogonal lifted from S3
ρ132220-12-2000-1-1-20-1111    orthogonal lifted from D6
ρ1422-20-12-2000-1120-11-1-1    orthogonal lifted from D6
ρ154400-2-40000-20002000    orthogonal lifted from S3×D4
ρ164-400400000-40000000    symplectic lifted from C8.C22, Schur index 2
ρ174-400-2000002000006-6    symplectic faithful, Schur index 2
ρ184-400-200000200000-66    symplectic faithful, Schur index 2

Smallest permutation representation of D4.D6
On 48 points
Generators in S48
(1 15 7 21)(2 22 8 16)(3 17 9 23)(4 24 10 18)(5 19 11 13)(6 14 12 20)(25 48 31 42)(26 43 32 37)(27 38 33 44)(28 45 34 39)(29 40 35 46)(30 47 36 41)
(1 33)(2 39)(3 35)(4 41)(5 25)(6 43)(7 27)(8 45)(9 29)(10 47)(11 31)(12 37)(13 48)(14 26)(15 38)(16 28)(17 40)(18 30)(19 42)(20 32)(21 44)(22 34)(23 46)(24 36)
(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)
(1 6 7 12)(2 11 8 5)(3 4 9 10)(13 22 19 16)(14 15 20 21)(17 18 23 24)(25 28 31 34)(26 33 32 27)(29 36 35 30)(37 44 43 38)(39 42 45 48)(40 47 46 41)

G:=sub<Sym(48)| (1,15,7,21)(2,22,8,16)(3,17,9,23)(4,24,10,18)(5,19,11,13)(6,14,12,20)(25,48,31,42)(26,43,32,37)(27,38,33,44)(28,45,34,39)(29,40,35,46)(30,47,36,41), (1,33)(2,39)(3,35)(4,41)(5,25)(6,43)(7,27)(8,45)(9,29)(10,47)(11,31)(12,37)(13,48)(14,26)(15,38)(16,28)(17,40)(18,30)(19,42)(20,32)(21,44)(22,34)(23,46)(24,36), (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), (1,6,7,12)(2,11,8,5)(3,4,9,10)(13,22,19,16)(14,15,20,21)(17,18,23,24)(25,28,31,34)(26,33,32,27)(29,36,35,30)(37,44,43,38)(39,42,45,48)(40,47,46,41)>;

G:=Group( (1,15,7,21)(2,22,8,16)(3,17,9,23)(4,24,10,18)(5,19,11,13)(6,14,12,20)(25,48,31,42)(26,43,32,37)(27,38,33,44)(28,45,34,39)(29,40,35,46)(30,47,36,41), (1,33)(2,39)(3,35)(4,41)(5,25)(6,43)(7,27)(8,45)(9,29)(10,47)(11,31)(12,37)(13,48)(14,26)(15,38)(16,28)(17,40)(18,30)(19,42)(20,32)(21,44)(22,34)(23,46)(24,36), (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), (1,6,7,12)(2,11,8,5)(3,4,9,10)(13,22,19,16)(14,15,20,21)(17,18,23,24)(25,28,31,34)(26,33,32,27)(29,36,35,30)(37,44,43,38)(39,42,45,48)(40,47,46,41) );

G=PermutationGroup([(1,15,7,21),(2,22,8,16),(3,17,9,23),(4,24,10,18),(5,19,11,13),(6,14,12,20),(25,48,31,42),(26,43,32,37),(27,38,33,44),(28,45,34,39),(29,40,35,46),(30,47,36,41)], [(1,33),(2,39),(3,35),(4,41),(5,25),(6,43),(7,27),(8,45),(9,29),(10,47),(11,31),(12,37),(13,48),(14,26),(15,38),(16,28),(17,40),(18,30),(19,42),(20,32),(21,44),(22,34),(23,46),(24,36)], [(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)], [(1,6,7,12),(2,11,8,5),(3,4,9,10),(13,22,19,16),(14,15,20,21),(17,18,23,24),(25,28,31,34),(26,33,32,27),(29,36,35,30),(37,44,43,38),(39,42,45,48),(40,47,46,41)])

D4.D6 is a maximal subgroup of
SD1613D6  SD16⋊D6  D8.10D6  D84D6  D86D6  S3×C8.C22  SD16.D6  SD16⋊D9  C24.3D6  D12.4D6  Dic6.19D6  Dic6.D6  D12.24D6  D12.15D6  C24.32D6  D6.S4  C40.2D6  D30.3D4  C60.10C23  D30.9D4  D20.28D6  D20.17D6  SD16⋊D15
D4.D6 is a maximal quotient of
D4.S3⋊C4  Dic3.D8  C12⋊Q8⋊C2  Dic6.D4  D4⋊(C4×S3)  D6.D8  C3⋊C81D4  D4.D12  C3⋊Q16⋊C4  Dic3⋊Q16  Q8.3Dic6  (C2×Q8).36D6  (S3×Q8)⋊C4  D6⋊Q16  D6⋊C8.C2  C3⋊C8.D4  Dic129C4  Dic6⋊Q8  C243Q8  Dic6.Q8  C8⋊(C4×S3)  D6.2SD16  C8.2D12  C6.(C4○D8)  Dic33SD16  SD16⋊Dic3  (C3×Q8).D4  C24.31D4  D68SD16  Dic6.16D4  C248D4  SD16⋊D9  C24.3D6  D12.4D6  Dic6.19D6  Dic6.D6  D12.24D6  D12.15D6  C24.32D6  C40.2D6  D30.3D4  C60.10C23  D30.9D4  D20.28D6  D20.17D6  SD16⋊D15

Matrix representation of D4.D6 in GL4(𝔽5) generated by

0003
4022
2204
3000
,
1013
4421
1244
3101
,
4340
3231
1223
0432
,
2000
3044
1102
0003
G:=sub<GL(4,GF(5))| [0,4,2,3,0,0,2,0,0,2,0,0,3,2,4,0],[1,4,1,3,0,4,2,1,1,2,4,0,3,1,4,1],[4,3,1,0,3,2,2,4,4,3,2,3,0,1,3,2],[2,3,1,0,0,0,1,0,0,4,0,0,0,4,2,3] >;

D4.D6 in GAP, Magma, Sage, TeX

D_4.D_6
% in TeX

G:=Group("D4.D6");
// GroupNames label

G:=SmallGroup(96,122);
// by ID

G=gap.SmallGroup(96,122);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-3,103,362,116,297,159,69,2309]);
// Polycyclic

G:=Group<a,b,c,d|a^4=b^2=1,c^6=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^5>;
// generators/relations

Export

Character table of D4.D6 in TeX

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