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G = D4.Dic3order 96 = 25·3

The non-split extension by D4 of Dic3 acting through Inn(D4)

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D4.Dic3, Q8.2Dic3, C12.42C23, D4(C3⋊C8), Q8(C3⋊C8), (C3×D4).C4, (C3×Q8).C4, C33(C8○D4), C4○D4.5S3, (C2×C4).58D6, C12.15(C2×C4), C3⋊C8.13C22, C4.Dic38C2, C4.5(C2×Dic3), C4.42(C22×S3), C6.27(C22×C4), (C2×C12).41C22, C2.8(C22×Dic3), C22.1(C2×Dic3), (C2×C3⋊C8)⋊7C2, C4○D4(C3⋊C8), (C2×C6).7(C2×C4), (C3×C4○D4).2C2, SmallGroup(96,155)

Series: Derived Chief Lower central Upper central

C1C6 — D4.Dic3
C1C3C6C12C3⋊C8C2×C3⋊C8 — D4.Dic3
C3C6 — D4.Dic3
C1C4C4○D4

Generators and relations for D4.Dic3
 G = < a,b,c,d | a4=b2=1, c6=a2, d2=a2c3, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c5 >

Subgroups: 90 in 62 conjugacy classes, 45 normal (12 characteristic)
C1, C2, C2 [×3], C3, C4, C4 [×3], C22 [×3], C6, C6 [×3], C8 [×4], C2×C4 [×3], D4 [×3], Q8, C12, C12 [×3], C2×C6 [×3], C2×C8 [×3], M4(2) [×3], C4○D4, C3⋊C8, C3⋊C8 [×3], C2×C12 [×3], C3×D4 [×3], C3×Q8, C8○D4, C2×C3⋊C8 [×3], C4.Dic3 [×3], C3×C4○D4, D4.Dic3
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3, C2×C4 [×6], C23, Dic3 [×4], D6 [×3], C22×C4, C2×Dic3 [×6], C22×S3, C8○D4, C22×Dic3, D4.Dic3

Character table of D4.Dic3

 class 12A2B2C2D34A4B4C4D4E6A6B6C6D8A8B8C8D8E8F8G8H8I8J12A12B12C12D12E
 size 112222112222444333366666622444
ρ1111111111111111111111111111111    trivial
ρ211-1-111111-1-11-1-11-1-1-1-1-1111-11111-1-1    linear of order 2
ρ311-11-1111-11-111-1-1-1-1-1-11-1111-111-1-11    linear of order 2
ρ4111-1-1111-1-111-11-11111-1-111-1-111-11-1    linear of order 2
ρ511-11-1111-11-111-1-11111-11-1-1-1111-1-11    linear of order 2
ρ6111-1-1111-1-111-11-1-1-1-1-111-1-11111-11-1    linear of order 2
ρ7111111111111111-1-1-1-1-1-1-1-1-1-111111    linear of order 2
ρ811-1-111111-1-11-1-1111111-1-1-11-1111-1-1    linear of order 2
ρ91111-11-1-11-1-1111-1-ii-ii-i-ii-iii-1-11-1-1    linear of order 4
ρ1011-1111-1-1-1-1111-11-ii-iii-i-ii-ii-1-1-11-1    linear of order 4
ρ1111-1-1-11-1-11111-1-1-1i-ii-ii-ii-i-ii-1-1111    linear of order 4
ρ12111-111-1-1-11-11-111i-ii-i-i-i-iiii-1-1-1-11    linear of order 4
ρ1311-1111-1-1-1-1111-11i-ii-i-iii-ii-i-1-1-11-1    linear of order 4
ρ141111-11-1-11-1-1111-1i-ii-iii-ii-i-i-1-11-1-1    linear of order 4
ρ15111-111-1-1-11-11-111-ii-iiiii-i-i-i-1-1-1-11    linear of order 4
ρ1611-1-1-11-1-11111-1-1-1-ii-ii-ii-iii-i-1-1111    linear of order 4
ρ1722-22-2-122-22-2-1-1110000000000-1-111-1    orthogonal lifted from D6
ρ1822222-122222-1-1-1-10000000000-1-1-1-1-1    orthogonal lifted from S3
ρ19222-2-2-122-2-22-11-110000000000-1-11-11    orthogonal lifted from D6
ρ2022-2-22-1222-2-2-111-10000000000-1-1-111    orthogonal lifted from D6
ρ21222-22-1-2-2-22-2-11-1-100000000001111-1    symplectic lifted from Dic3, Schur index 2
ρ2222-2-2-2-1-2-2222-1111000000000011-1-1-1    symplectic lifted from Dic3, Schur index 2
ρ2322-222-1-2-2-2-22-1-11-10000000000111-11    symplectic lifted from Dic3, Schur index 2
ρ242222-2-1-2-22-2-2-1-1-11000000000011-111    symplectic lifted from Dic3, Schur index 2
ρ252-200022i-2i000-20008388785000000-2i2i000    complex lifted from C8○D4
ρ262-20002-2i2i000-200085878830000002i-2i000    complex lifted from C8○D4
ρ272-20002-2i2i000-200088385870000002i-2i000    complex lifted from C8○D4
ρ282-200022i-2i000-20008785838000000-2i2i000    complex lifted from C8○D4
ρ294-4000-2-4i4i00020000000000000-2i2i000    complex faithful
ρ304-4000-24i-4i000200000000000002i-2i000    complex faithful

Smallest permutation representation of D4.Dic3
On 48 points
Generators in S48
(1 10 7 4)(2 11 8 5)(3 12 9 6)(13 16 19 22)(14 17 20 23)(15 18 21 24)(25 34 31 28)(26 35 32 29)(27 36 33 30)(37 40 43 46)(38 41 44 47)(39 42 45 48)
(1 24)(2 13)(3 14)(4 15)(5 16)(6 17)(7 18)(8 19)(9 20)(10 21)(11 22)(12 23)(25 37)(26 38)(27 39)(28 40)(29 41)(30 42)(31 43)(32 44)(33 45)(34 46)(35 47)(36 48)
(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 28 10 25 7 34 4 31)(2 33 11 30 8 27 5 36)(3 26 12 35 9 32 6 29)(13 45 22 42 19 39 16 48)(14 38 23 47 20 44 17 41)(15 43 24 40 21 37 18 46)

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

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

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

D4.Dic3 is a maximal subgroup of
M4(2).22D6  C42.196D6  D85Dic3  D84Dic3  M4(2).D6  M4(2).13D6  M4(2).15D6  M4(2).16D6  S3×C8○D4  M4(2)⋊28D6  C12.76C24  D12.32C23  D12.33C23  D12.34C23  D12.35C23  D4.Dic9  SL2(𝔽3).Dic3  D12.2Dic3  D12.Dic3  D4.(C3⋊Dic3)  D20.3Dic3  D20.2Dic3  D4.Dic15  Dic10.Dic3  D20.Dic3
D4.Dic3 is a maximal quotient of
C12.5C42  C42.43D6  C42.187D6  D4×C3⋊C8  C42.47D6  C123M4(2)  Q8×C3⋊C8  C42.210D6  (C6×D4).11C4  D4.Dic9  D12.2Dic3  D12.Dic3  D4.(C3⋊Dic3)  D20.3Dic3  D20.2Dic3  D4.Dic15  Dic10.Dic3  D20.Dic3

Matrix representation of D4.Dic3 in GL4(𝔽5) generated by

0001
1010
0404
4000
,
0004
1010
0101
4000
,
0030
0104
2030
0202
,
3040
0101
2020
0104
G:=sub<GL(4,GF(5))| [0,1,0,4,0,0,4,0,0,1,0,0,1,0,4,0],[0,1,0,4,0,0,1,0,0,1,0,0,4,0,1,0],[0,0,2,0,0,1,0,2,3,0,3,0,0,4,0,2],[3,0,2,0,0,1,0,1,4,0,2,0,0,1,0,4] >;

D4.Dic3 in GAP, Magma, Sage, TeX

D_4.{\rm Dic}_3
% in TeX

G:=Group("D4.Dic3");
// GroupNames label

G:=SmallGroup(96,155);
// by ID

G=gap.SmallGroup(96,155);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-3,48,188,69,2309]);
// Polycyclic

G:=Group<a,b,c,d|a^4=b^2=1,c^6=a^2,d^2=a^2*c^3,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^5>;
// generators/relations

Export

Character table of D4.Dic3 in TeX

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