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G = C6.D12order 144 = 24·32

6th non-split extension by C6 of D12 acting via D12/D6=C2

metabelian, supersoluble, monomial

Aliases: C6.17D12, C62.3C22, C6.3(C4×S3), C22.5S32, C31(D6⋊C4), (C3×C6).14D4, (C2×C6).10D6, (C2×Dic3)⋊2S3, (C6×Dic3)⋊2C2, C6.5(C3⋊D4), C324(C22⋊C4), C2.2(C3⋊D12), C2.4(C6.D6), (C2×C3⋊S3)⋊1C4, (C3×C6).14(C2×C4), (C22×C3⋊S3).1C2, SmallGroup(144,65)

Series: Derived Chief Lower central Upper central

C1C3×C6 — C6.D12
C1C3C32C3×C6C62C6×Dic3 — C6.D12
C32C3×C6 — C6.D12
C1C22

Generators and relations for C6.D12
 G = < a,b,c | a6=b12=c2=1, bab-1=cac=a-1, cbc=a3b-1 >

Subgroups: 336 in 84 conjugacy classes, 28 normal (8 characteristic)
C1, C2, C2 [×2], C2 [×2], C3 [×2], C3, C4 [×2], C22, C22 [×4], S3 [×8], C6 [×6], C6 [×3], C2×C4 [×2], C23, C32, Dic3 [×2], C12 [×2], D6 [×14], C2×C6 [×2], C2×C6, C22⋊C4, C3⋊S3 [×2], C3×C6, C3×C6 [×2], C2×Dic3 [×2], C2×C12 [×2], C22×S3 [×3], C3×Dic3 [×2], C2×C3⋊S3 [×2], C2×C3⋊S3 [×2], C62, D6⋊C4 [×2], C6×Dic3 [×2], C22×C3⋊S3, C6.D12
Quotients: C1, C2 [×3], C4 [×2], C22, S3 [×2], C2×C4, D4 [×2], D6 [×2], C22⋊C4, C4×S3 [×2], D12 [×2], C3⋊D4 [×2], S32, D6⋊C4 [×2], C6.D6, C3⋊D12 [×2], C6.D12

Character table of C6.D12

 class 12A2B2C2D2E3A3B3C4A4B4C4D6A6B6C6D6E6F6G6H6I12A12B12C12D12E12F12G12H
 size 11111818224666622222244466666666
ρ1111111111111111111111111111111    trivial
ρ21111-1-11111-11-1111111111-111-111-1-1    linear of order 2
ρ31111-1-1111-11-111111111111-1-11-1-111    linear of order 2
ρ4111111111-1-1-1-1111111111-1-1-1-1-1-1-1-1    linear of order 2
ρ511-1-11-1111-i-iii1-1-1-1-111-1-1iii-i-i-ii-i    linear of order 4
ρ611-1-1-11111-iii-i1-1-1-1-111-1-1-iiii-i-i-ii    linear of order 4
ρ711-1-11-1111ii-i-i1-1-1-1-111-1-1-i-i-iiii-ii    linear of order 4
ρ811-1-1-11111i-i-ii1-1-1-1-111-1-1i-i-i-iiii-i    linear of order 4
ρ9222200-12-10-20-22-1-122-1-1-1-110010011    orthogonal lifted from D6
ρ102-22-2002220000-22-2-22-2-2-2200000000    orthogonal lifted from D4
ρ112222002-1-1-20-20-122-1-12-1-1-101101100    orthogonal lifted from D6
ρ122222002-1-12020-122-1-12-1-1-10-1-10-1-100    orthogonal lifted from S3
ρ132-2-22002220000-2-222-2-2-22-200000000    orthogonal lifted from D4
ρ14222200-12-102022-1-122-1-1-1-1-100-100-1-1    orthogonal lifted from S3
ρ152-22-200-12-10000-2-11-22111-1300300-3-3    orthogonal lifted from D12
ρ162-2-22002-1-100001-22-11-21-110-3303-300    orthogonal lifted from D12
ρ172-22-200-12-10000-2-11-22111-1-300-30033    orthogonal lifted from D12
ρ182-2-22002-1-100001-22-11-21-1103-30-3300    orthogonal lifted from D12
ρ192-2-2200-12-10000-21-12-211-11--300-300-3--3    complex lifted from C3⋊D4
ρ202-22-2002-1-1000012-21-1-211-10--3-30--3-300    complex lifted from C3⋊D4
ρ212-2-2200-12-10000-21-12-211-11-300--300--3-3    complex lifted from C3⋊D4
ρ222-22-2002-1-1000012-21-1-211-10-3--30-3--300    complex lifted from C3⋊D4
ρ2322-2-2002-1-1-2i02i0-1-2-2112-1110-i-i0ii00    complex lifted from C4×S3
ρ2422-2-2002-1-12i0-2i0-1-2-2112-1110ii0-i-i00    complex lifted from C4×S3
ρ2522-2-200-12-10-2i02i211-2-2-1-111-i00i00-ii    complex lifted from C4×S3
ρ2622-2-200-12-102i0-2i211-2-2-1-111i00-i00i-i    complex lifted from C4×S3
ρ2744-4-400-2-210000-22222-21-1-100000000    orthogonal lifted from C6.D6
ρ284-44-400-2-2100002-222-22-1-1100000000    orthogonal lifted from C3⋊D12
ρ29444400-2-210000-2-2-2-2-2-211100000000    orthogonal lifted from S32
ρ304-4-4400-2-21000022-2-222-11-100000000    orthogonal lifted from C3⋊D12

Permutation representations of C6.D12
On 24 points - transitive group 24T235
Generators in S24
(1 17 5 21 9 13)(2 14 10 22 6 18)(3 19 7 23 11 15)(4 16 12 24 8 20)
(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 9)(2 16)(3 7)(4 14)(6 24)(8 22)(10 20)(12 18)(15 23)(17 21)

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

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

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

G:=TransitiveGroup(24,235);

C6.D12 is a maximal subgroup of
C62.2D4  C62.6C23  C62.18C23  C62.19C23  C62.20C23  Dic3.D12  C62.23C23  C12.28D12  C62.35C23  C62.38C23  C12.30D12  C62.44C23  C62.51C23  C62.58C23  D6.D12  Dic35D12  C62.65C23  C62.67C23  C4×C3⋊D12  C62.77C23  C127D12  Dic33D12  S3×D6⋊C4  D65D12  C62.94C23  C62.95C23  C62.60D4  C62.113C23  C62.116C23  C62.117C23  C626D4  C628D4  C62.125C23  C6.18D36  C62.4D6  C62.79D6  C62.84D6
C6.D12 is a maximal quotient of
C12.78D12  C12.70D12  C12.71D12  C6.17D24  C12.73D12  C12.80D12  C62.6Q8  C62.32D4  C6.18D36  C62.5D6  C62.79D6  C62.84D6

Matrix representation of C6.D12 in GL6(𝔽13)

1200000
0120000
0011200
001000
000010
000001
,
010000
100000
000800
008000
0000121
0000120
,
100000
0120000
0001200
0012000
0000120
0000121

G:=sub<GL(6,GF(13))| [12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,1,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,8,0,0,0,0,8,0,0,0,0,0,0,0,12,12,0,0,0,0,1,0],[1,0,0,0,0,0,0,12,0,0,0,0,0,0,0,12,0,0,0,0,12,0,0,0,0,0,0,0,12,12,0,0,0,0,0,1] >;

C6.D12 in GAP, Magma, Sage, TeX

C_6.D_{12}
% in TeX

G:=Group("C6.D12");
// GroupNames label

G:=SmallGroup(144,65);
// by ID

G=gap.SmallGroup(144,65);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-3,-3,24,121,79,490,3461]);
// Polycyclic

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

Export

Character table of C6.D12 in TeX

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