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G = (C3×C6).8D12order 432 = 24·33

1st non-split extension by C3×C6 of D12 acting via D12/C3=D4

non-abelian, soluble, monomial

Aliases: C6.7S3≀C2, (C3×C6).8D12, C324(D6⋊C4), C324D62C4, C335(C22⋊C4), (C32×C6).13D4, C2.1(C322D12), C32(S32⋊C4), C3⋊S3.4(C4×S3), (C2×C32⋊C4)⋊2S3, (C6×C32⋊C4)⋊3C2, C339(C2×C4)⋊8C2, (C2×C3⋊S3).13D6, (C3×C3⋊S3).11D4, C3⋊S3.4(C3⋊D4), (C6×C3⋊S3).9C22, (C2×C324D6).2C2, (C3×C3⋊S3).11(C2×C4), SmallGroup(432,586)

Series: Derived Chief Lower central Upper central

C1C32C3×C3⋊S3 — (C3×C6).8D12
C1C3C33C3×C3⋊S3C6×C3⋊S3C2×C324D6 — (C3×C6).8D12
C33C3×C3⋊S3 — (C3×C6).8D12
C1C2

Generators and relations for (C3×C6).8D12
 G = < a,b,c,d | a3=b6=c12=1, d2=b3, ab=ba, cac-1=b2, dad-1=a-1, cbc-1=ab3, bd=db, dcd-1=b3c-1 >

Subgroups: 1024 in 132 conjugacy classes, 21 normal (19 characteristic)
C1, C2, C2 [×4], C3, C3 [×4], C4 [×2], C22 [×5], S3 [×10], C6, C6 [×8], C2×C4 [×2], C23, C32, C32 [×4], Dic3 [×3], C12 [×2], D6 [×12], C2×C6 [×2], C22⋊C4, C3×S3 [×10], C3⋊S3 [×2], C3⋊S3 [×2], C3×C6, C3×C6 [×4], C4×S3, C2×Dic3, C2×C12, C22×S3 [×2], C33, C3×Dic3 [×3], C3⋊Dic3, C32⋊C4, S32 [×7], S3×C6 [×5], C2×C3⋊S3, C2×C3⋊S3, D6⋊C4, C3×C3⋊S3 [×2], C3×C3⋊S3 [×2], C32×C6, S3×Dic3, C6.D6, C2×C32⋊C4, C2×S32 [×2], C3×C3⋊Dic3, C3×C32⋊C4, C324D6 [×2], C324D6, C6×C3⋊S3, C6×C3⋊S3, S32⋊C4, C339(C2×C4), C6×C32⋊C4, C2×C324D6, (C3×C6).8D12
Quotients: C1, C2 [×3], C4 [×2], C22, S3, C2×C4, D4 [×2], D6, C22⋊C4, C4×S3, D12, C3⋊D4, D6⋊C4, S3≀C2, S32⋊C4, C322D12, (C3×C6).8D12

Character table of (C3×C6).8D12

 class 12A2B2C2D2E3A3B3C3D3E4A4B4C4D6A6B6C6D6E6F6G6H6I12A12B12C12D12E12F
 size 1199181824488181818182448818183636181818183636
ρ1111111111111111111111111111111    trivial
ρ211111111111-1-1-1-1111111111-1-1-1-1-1-1    linear of order 2
ρ31111-1-111111-111-11111111-1-11111-1-1    linear of order 2
ρ41111-1-1111111-1-111111111-1-1-1-1-1-111    linear of order 2
ρ51-11-11-111111ii-i-i-1-1-1-1-1-111-1ii-i-i-ii    linear of order 4
ρ61-11-11-111111-i-iii-1-1-1-1-1-111-1-i-iiii-i    linear of order 4
ρ71-11-1-1111111-ii-ii-1-1-1-1-1-11-11ii-i-ii-i    linear of order 4
ρ81-11-1-1111111i-ii-i-1-1-1-1-1-11-11-i-iii-ii    linear of order 4
ρ9222200-122-1-10220-122-1-1-1-100-1-1-1-100    orthogonal lifted from S3
ρ1022-2-20022222000022222-2-200000000    orthogonal lifted from D4
ρ112-2-2200222220000-2-2-2-2-22-200000000    orthogonal lifted from D4
ρ12222200-122-1-10-2-20-122-1-1-1-100111100    orthogonal lifted from D6
ρ1322-2-200-122-1-10000-122-1-111003-3-3300    orthogonal lifted from D12
ρ1422-2-200-122-1-10000-122-1-11100-333-300    orthogonal lifted from D12
ρ152-22-200-122-1-102i-2i01-2-2111-100-i-iii00    complex lifted from C4×S3
ρ162-22-200-122-1-10-2i2i01-2-2111-100ii-i-i00    complex lifted from C4×S3
ρ172-2-2200-122-1-100001-2-211-1100-3--3-3--300    complex lifted from C3⋊D4
ρ182-2-2200-122-1-100001-2-211-1100--3-3--3-300    complex lifted from C3⋊D4
ρ194-4002-24-21-210000-4-12-1200-11000000    orthogonal lifted from S32⋊C4
ρ2044000041-21-220024-21-2100000000-1-1    orthogonal lifted from S3≀C2
ρ214-400-224-21-210000-4-12-12001-1000000    orthogonal lifted from S32⋊C4
ρ224400-2-24-21-21000041-21-20011000000    orthogonal lifted from S3≀C2
ρ234400224-21-21000041-21-200-1-1000000    orthogonal lifted from S3≀C2
ρ2444000041-21-2-200-24-21-210000000011    orthogonal lifted from S3≀C2
ρ254-4000041-21-2-2i002i-42-12-100000000-ii    complex lifted from S32⋊C4
ρ264-4000041-21-22i00-2i-42-12-100000000i-i    complex lifted from S32⋊C4
ρ278-80000-4-422-100004-241-20000000000    orthogonal faithful
ρ28880000-4-422-10000-42-4-120000000000    orthogonal lifted from C322D12
ρ29880000-42-4-120000-4-422-10000000000    orthogonal lifted from C322D12
ρ308-80000-42-4-12000044-2-210000000000    symplectic faithful, Schur index 2

Permutation representations of (C3×C6).8D12
On 24 points - transitive group 24T1305
Generators in S24
(1 9 5)(2 6 10)(3 7 11)(4 12 8)(13 21 17)(14 18 22)(15 19 23)(16 24 20)
(1 21 5 13 9 17)(2 22 6 14 10 18)(3 19 11 15 7 23)(4 20 12 16 8 24)
(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 12 13 24)(2 23 14 11)(3 10 15 22)(4 21 16 9)(5 8 17 20)(6 19 18 7)

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

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

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

G:=TransitiveGroup(24,1305);

Matrix representation of (C3×C6).8D12 in GL6(𝔽13)

100000
010000
0000012
0000120
0001120
0010012
,
1200000
0120000
0012001
0000120
0001120
0012000
,
11110000
290000
0000120
0012000
0000012
0001200
,
1140000
220000
000010
000001
001000
000100

G:=sub<GL(6,GF(13))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,12,12,0,0,0,12,0,0,12],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,12,0,0,0,0,1,0,0,0,0,12,12,0,0,0,1,0,0,0],[11,2,0,0,0,0,11,9,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,12,0,0,12,0,0,0,0,0,0,0,12,0],[11,2,0,0,0,0,4,2,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,1,0,0] >;

(C3×C6).8D12 in GAP, Magma, Sage, TeX

(C_3\times C_6)._8D_{12}
% in TeX

G:=Group("(C3xC6).8D12");
// GroupNames label

G:=SmallGroup(432,586);
// by ID

G=gap.SmallGroup(432,586);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,3,-3,85,92,1684,1691,298,677,348,1027,14118]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^6=c^12=1,d^2=b^3,a*b=b*a,c*a*c^-1=b^2,d*a*d^-1=a^-1,c*b*c^-1=a*b^3,b*d=d*b,d*c*d^-1=b^3*c^-1>;
// generators/relations

Export

Character table of (C3×C6).8D12 in TeX

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