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G = D6.D6order 144 = 24·32

1st non-split extension by D6 of D6 acting via D6/C6=C2

metabelian, supersoluble, monomial

Aliases: D6.5D6, C12.32D6, Dic3.6D6, C4.17S32, (C4×S3)⋊4S3, (S3×C12)⋊1C2, C3⋊D127C2, D6⋊S36C2, C32(C4○D12), C322Q86C2, C323(C4○D4), C6.5(C22×S3), (C3×C6).5C23, (S3×C6).6C22, (C3×C12).31C22, C3⋊Dic3.13C22, (C3×Dic3).6C22, C2.8(C2×S32), (C4×C3⋊S3)⋊6C2, (C2×C3⋊S3).13C22, SmallGroup(144,141)

Series: Derived Chief Lower central Upper central

C1C3×C6 — D6.D6
C1C3C32C3×C6S3×C6D6⋊S3 — D6.D6
C32C3×C6 — D6.D6
C1C4

Generators and relations for D6.D6
 G = < a,b,c,d | a6=b2=1, c6=d2=a3, bab=a-1, ac=ca, ad=da, bc=cb, dbd-1=a3b, dcd-1=c5 >

Subgroups: 288 in 88 conjugacy classes, 32 normal (12 characteristic)
C1, C2, C2 [×3], C3 [×2], C3, C4, C4 [×3], C22 [×3], S3 [×6], C6 [×2], C6 [×3], C2×C4 [×3], D4 [×3], Q8, C32, Dic3 [×2], Dic3 [×3], C12 [×2], C12 [×3], D6 [×2], D6 [×3], C2×C6 [×2], C4○D4, C3×S3 [×2], C3⋊S3, C3×C6, Dic6 [×2], C4×S3 [×2], C4×S3 [×3], D12 [×2], C3⋊D4 [×4], C2×C12 [×2], C3×Dic3 [×2], C3⋊Dic3, C3×C12, S3×C6 [×2], C2×C3⋊S3, C4○D12 [×2], D6⋊S3, C3⋊D12 [×2], C322Q8, S3×C12 [×2], C4×C3⋊S3, D6.D6
Quotients: C1, C2 [×7], C22 [×7], S3 [×2], C23, D6 [×6], C4○D4, C22×S3 [×2], S32, C4○D12 [×2], C2×S32, D6.D6

Character table of D6.D6

 class 12A2B2C2D3A3B3C4A4B4C4D4E6A6B6C6D6E6F6G12A12B12C12D12E12F12G12H12I12J
 size 11661822411661822466662222446666
ρ1111111111111111111111111111111    trivial
ρ211-111111-1-11-1-1111-1-111-1-1-1-1-1-1-1-111    linear of order 2
ρ31111-1111-1-1-1-111111111-1-1-1-1-1-1-1-1-1-1    linear of order 2
ρ411-11-111111-11-1111-1-11111111111-1-1    linear of order 2
ρ511-1-1-1111-1-1111111-1-1-1-1-1-1-1-1-1-11111    linear of order 2
ρ6111-1-1111111-1-111111-1-1111111-1-111    linear of order 2
ρ711-1-1111111-1-11111-1-1-1-1111111-1-1-1-1    linear of order 2
ρ8111-11111-1-1-11-111111-1-1-1-1-1-1-1-111-1-1    linear of order 2
ρ922-2002-1-1-2-2200-12-11100-2-2111100-1-1    orthogonal lifted from D6
ρ1022-2002-1-122-200-12-1110022-1-1-1-10011    orthogonal lifted from D6
ρ11222002-1-1-2-2-200-12-1-1-100-2-211110011    orthogonal lifted from D6
ρ12220-20-12-1220-202-1-10011-1-122-1-11100    orthogonal lifted from D6
ρ13220-20-12-1-2-20202-1-1001111-2-211-1-100    orthogonal lifted from D6
ρ1422020-12-1-2-20-202-1-100-1-111-2-2111100    orthogonal lifted from D6
ρ1522020-12-1220202-1-100-1-1-1-122-1-1-1-100    orthogonal lifted from S3
ρ16222002-1-122200-12-1-1-10022-1-1-1-100-1-1    orthogonal lifted from S3
ρ172-20002222i-2i000-2-2-20000-2i2i2i-2i2i-2i0000    complex lifted from C4○D4
ρ182-2000222-2i2i000-2-2-200002i-2i-2i2i-2i2i0000    complex lifted from C4○D4
ρ192-2000-12-12i-2i000-21100--3-3i-i2i-2i-ii3-300    complex lifted from C4○D12
ρ202-2000-12-1-2i2i000-21100-3--3-ii-2i2ii-i3-300    complex lifted from C4○D12
ρ212-20002-1-12i-2i0001-21-3--300-2i2i-ii-ii00-33    complex lifted from C4○D12
ρ222-2000-12-12i-2i000-21100-3--3i-i2i-2i-ii-3300    complex lifted from C4○D12
ρ232-2000-12-1-2i2i000-21100--3-3-ii-2i2ii-i-3300    complex lifted from C4○D12
ρ242-20002-1-1-2i2i0001-21-3--3002i-2ii-ii-i003-3    complex lifted from C4○D12
ρ252-20002-1-12i-2i0001-21--3-300-2i2i-ii-ii003-3    complex lifted from C4○D12
ρ262-20002-1-1-2i2i0001-21--3-3002i-2ii-ii-i00-33    complex lifted from C4○D12
ρ2744000-2-21-4-4000-2-2100002222-1-10000    orthogonal lifted from C2×S32
ρ2844000-2-2144000-2-210000-2-2-2-2110000    orthogonal lifted from S32
ρ294-4000-2-214i-4i00022-100002i-2i-2i2ii-i0000    complex faithful
ρ304-4000-2-21-4i4i00022-10000-2i2i2i-2i-ii0000    complex faithful

Permutation representations of D6.D6
On 24 points - transitive group 24T228
Generators in S24
(1 3 5 7 9 11)(2 4 6 8 10 12)(13 23 21 19 17 15)(14 24 22 20 18 16)
(1 18)(2 19)(3 20)(4 21)(5 22)(6 23)(7 24)(8 13)(9 14)(10 15)(11 16)(12 17)
(1 2 3 4 5 6 7 8 9 10 11 12)(13 14 15 16 17 18 19 20 21 22 23 24)
(1 24 7 18)(2 17 8 23)(3 22 9 16)(4 15 10 21)(5 20 11 14)(6 13 12 19)

G:=sub<Sym(24)| (1,3,5,7,9,11)(2,4,6,8,10,12)(13,23,21,19,17,15)(14,24,22,20,18,16), (1,18)(2,19)(3,20)(4,21)(5,22)(6,23)(7,24)(8,13)(9,14)(10,15)(11,16)(12,17), (1,2,3,4,5,6,7,8,9,10,11,12)(13,14,15,16,17,18,19,20,21,22,23,24), (1,24,7,18)(2,17,8,23)(3,22,9,16)(4,15,10,21)(5,20,11,14)(6,13,12,19)>;

G:=Group( (1,3,5,7,9,11)(2,4,6,8,10,12)(13,23,21,19,17,15)(14,24,22,20,18,16), (1,18)(2,19)(3,20)(4,21)(5,22)(6,23)(7,24)(8,13)(9,14)(10,15)(11,16)(12,17), (1,2,3,4,5,6,7,8,9,10,11,12)(13,14,15,16,17,18,19,20,21,22,23,24), (1,24,7,18)(2,17,8,23)(3,22,9,16)(4,15,10,21)(5,20,11,14)(6,13,12,19) );

G=PermutationGroup([(1,3,5,7,9,11),(2,4,6,8,10,12),(13,23,21,19,17,15),(14,24,22,20,18,16)], [(1,18),(2,19),(3,20),(4,21),(5,22),(6,23),(7,24),(8,13),(9,14),(10,15),(11,16),(12,17)], [(1,2,3,4,5,6,7,8,9,10,11,12),(13,14,15,16,17,18,19,20,21,22,23,24)], [(1,24,7,18),(2,17,8,23),(3,22,9,16),(4,15,10,21),(5,20,11,14),(6,13,12,19)])

G:=TransitiveGroup(24,228);

D6.D6 is a maximal subgroup of
C32⋊C4≀C2  C24.63D6  C24.64D6  C24.D6  C32⋊D85C2  C32⋊D8⋊C2  C32⋊Q16⋊C2  S3×C4○D12  D1223D6  Dic6.24D6  D1212D6  D1213D6  D12.25D6  Dic6.26D6  D1216D6  D6.D18  C12.91S32  (S3×C6).D6  D6.S32  Dic3.S32  C12.73S32  C12.95S32
D6.D6 is a maximal quotient of
C62.20C23  D6⋊Dic6  Dic3.D12  C62.23C23  C62.25C23  C62.29C23  C62.32C23  C62.35C23  C62.37C23  C62.38C23  C62.40C23  C62.44C23  C4×D6⋊S3  C4×C3⋊D12  C62.75C23  D6⋊D12  C62.82C23  C62.85C23  C4×C322Q8  D6.D18  C12.91S32  (S3×C6).D6  D6.S32  Dic3.S32  C12.73S32  C12.95S32

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

0030
0101
3010
0400
,
0101
0030
0200
1020
,
3010
0303
1000
0200
,
0004
0020
0200
1000
G:=sub<GL(4,GF(5))| [0,0,3,0,0,1,0,4,3,0,1,0,0,1,0,0],[0,0,0,1,1,0,2,0,0,3,0,2,1,0,0,0],[3,0,1,0,0,3,0,2,1,0,0,0,0,3,0,0],[0,0,0,1,0,0,2,0,0,2,0,0,4,0,0,0] >;

D6.D6 in GAP, Magma, Sage, TeX

D_6.D_6
% in TeX

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

G:=SmallGroup(144,141);
// by ID

G=gap.SmallGroup(144,141);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-3,-3,121,50,490,3461]);
// Polycyclic

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

Export

Character table of D6.D6 in TeX

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