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G = C12.D6order 144 = 24·32

24th non-split extension by C12 of D6 acting via D6/C3=C22

metabelian, supersoluble, monomial

Aliases: C12.24D6, C62.14C22, (C3×D4)⋊3S3, (C2×C6).6D6, D42(C3⋊S3), (D4×C32)⋊6C2, C327D44C2, C35(D42S3), C324Q86C2, C3210(C4○D4), C6.35(C22×S3), (C3×C6).34C23, (C3×C12).24C22, C3⋊Dic3.18C22, (C4×C3⋊S3)⋊4C2, C4.5(C2×C3⋊S3), (C2×C3⋊Dic3)⋊7C2, C22.1(C2×C3⋊S3), C2.7(C22×C3⋊S3), (C2×C3⋊S3).18C22, SmallGroup(144,173)

Series: Derived Chief Lower central Upper central

C1C3×C6 — C12.D6
C1C3C32C3×C6C2×C3⋊S3C4×C3⋊S3 — C12.D6
C32C3×C6 — C12.D6
C1C2D4

Generators and relations for C12.D6
 G = < a,b,c | a12=b6=1, c2=a6, bab-1=a7, cac-1=a-1, cbc-1=a6b-1 >

Subgroups: 346 in 120 conjugacy classes, 47 normal (13 characteristic)
C1, C2, C2 [×3], C3 [×4], C4, C4 [×3], C22 [×2], C22, S3 [×4], C6 [×4], C6 [×8], C2×C4 [×3], D4, D4 [×2], Q8, C32, Dic3 [×12], C12 [×4], D6 [×4], C2×C6 [×8], C4○D4, C3⋊S3, C3×C6, C3×C6 [×2], Dic6 [×4], C4×S3 [×4], C2×Dic3 [×8], C3⋊D4 [×8], C3×D4 [×4], C3⋊Dic3, C3⋊Dic3 [×2], C3×C12, C2×C3⋊S3, C62 [×2], D42S3 [×4], C324Q8, C4×C3⋊S3, C2×C3⋊Dic3 [×2], C327D4 [×2], D4×C32, C12.D6
Quotients: C1, C2 [×7], C22 [×7], S3 [×4], C23, D6 [×12], C4○D4, C3⋊S3, C22×S3 [×4], C2×C3⋊S3 [×3], D42S3 [×4], C22×C3⋊S3, C12.D6

Character table of C12.D6

 class 12A2B2C2D3A3B3C3D4A4B4C4D4E6A6B6C6D6E6F6G6H6I6J6K6L12A12B12C12D
 size 112218222229918182222444444444444
ρ1111111111111111111111111111111    trivial
ρ211-11-11111-111-111111-1-11111-1-1-1-1-1-1    linear of order 2
ρ311-1111111-1-1-11-11111-1-11111-1-1-1-1-1-1    linear of order 2
ρ41111-111111-1-1-1-11111111111111111    linear of order 2
ρ511-1-111111111-1-11111-1-1-1-1-1-1-1-11111    linear of order 2
ρ6111-1-11111-1111-1111111-1-1-1-111-1-1-1-1    linear of order 2
ρ7111-111111-1-1-1-11111111-1-1-1-111-1-1-1-1    linear of order 2
ρ811-1-1-111111-1-1111111-1-1-1-1-1-1-1-11111    linear of order 2
ρ922-2-20-1-12-1200002-1-1-11111-21-212-1-1-1    orthogonal lifted from D6
ρ10222-20-12-1-1-20000-1-1-12-121-211-1-1111-2    orthogonal lifted from D6
ρ1122-220-12-1-1-20000-1-1-121-2-12-1-111111-2    orthogonal lifted from D6
ρ1222220-1-1-1220000-1-12-1-1-1-1-1-12-12-12-1-1    orthogonal lifted from S3
ρ1322-2202-1-1-1-20000-12-1-1-212-1-1-11111-21    orthogonal lifted from D6
ρ1422-2-20-1-1-1220000-1-12-111111-21-2-12-1-1    orthogonal lifted from D6
ρ15222-202-1-1-1-20000-12-1-12-1-2111-1-111-21    orthogonal lifted from D6
ρ1622-220-1-12-1-200002-1-1-111-1-12-1-21-2111    orthogonal lifted from D6
ρ1722220-1-12-1200002-1-1-1-1-1-1-12-12-12-1-1-1    orthogonal lifted from S3
ρ1822-2-202-1-1-120000-12-1-1-21-211111-1-12-1    orthogonal lifted from D6
ρ1922-220-1-1-12-20000-1-12-111-1-1-121-21-211    orthogonal lifted from D6
ρ20222202-1-1-120000-12-1-12-12-1-1-1-1-1-1-12-1    orthogonal lifted from S3
ρ21222-20-1-1-12-20000-1-12-1-1-1111-2-121-211    orthogonal lifted from D6
ρ2222220-12-1-120000-1-1-12-12-12-1-1-1-1-1-1-12    orthogonal lifted from S3
ρ23222-20-1-12-1-200002-1-1-1-1-111-212-1-2111    orthogonal lifted from D6
ρ2422-2-20-12-1-120000-1-1-121-21-21111-1-1-12    orthogonal lifted from D6
ρ252-200022220-2i2i00-2-2-2-2000000000000    complex lifted from C4○D4
ρ262-2000222202i-2i00-2-2-2-2000000000000    complex lifted from C4○D4
ρ274-4000-2-24-200000-4222000000000000    symplectic lifted from D42S3, Schur index 2
ρ284-40004-2-2-2000002-422000000000000    symplectic lifted from D42S3, Schur index 2
ρ294-4000-24-2-200000222-4000000000000    symplectic lifted from D42S3, Schur index 2
ρ304-4000-2-2-240000022-42000000000000    symplectic lifted from D42S3, Schur index 2

Smallest permutation representation of C12.D6
On 72 points
Generators in S72
(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)(49 50 51 52 53 54 55 56 57 58 59 60)(61 62 63 64 65 66 67 68 69 70 71 72)
(1 29 54 66 23 40)(2 36 55 61 24 47)(3 31 56 68 13 42)(4 26 57 63 14 37)(5 33 58 70 15 44)(6 28 59 65 16 39)(7 35 60 72 17 46)(8 30 49 67 18 41)(9 25 50 62 19 48)(10 32 51 69 20 43)(11 27 52 64 21 38)(12 34 53 71 22 45)
(1 40 7 46)(2 39 8 45)(3 38 9 44)(4 37 10 43)(5 48 11 42)(6 47 12 41)(13 27 19 33)(14 26 20 32)(15 25 21 31)(16 36 22 30)(17 35 23 29)(18 34 24 28)(49 71 55 65)(50 70 56 64)(51 69 57 63)(52 68 58 62)(53 67 59 61)(54 66 60 72)

G:=sub<Sym(72)| (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)(49,50,51,52,53,54,55,56,57,58,59,60)(61,62,63,64,65,66,67,68,69,70,71,72), (1,29,54,66,23,40)(2,36,55,61,24,47)(3,31,56,68,13,42)(4,26,57,63,14,37)(5,33,58,70,15,44)(6,28,59,65,16,39)(7,35,60,72,17,46)(8,30,49,67,18,41)(9,25,50,62,19,48)(10,32,51,69,20,43)(11,27,52,64,21,38)(12,34,53,71,22,45), (1,40,7,46)(2,39,8,45)(3,38,9,44)(4,37,10,43)(5,48,11,42)(6,47,12,41)(13,27,19,33)(14,26,20,32)(15,25,21,31)(16,36,22,30)(17,35,23,29)(18,34,24,28)(49,71,55,65)(50,70,56,64)(51,69,57,63)(52,68,58,62)(53,67,59,61)(54,66,60,72)>;

G:=Group( (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)(49,50,51,52,53,54,55,56,57,58,59,60)(61,62,63,64,65,66,67,68,69,70,71,72), (1,29,54,66,23,40)(2,36,55,61,24,47)(3,31,56,68,13,42)(4,26,57,63,14,37)(5,33,58,70,15,44)(6,28,59,65,16,39)(7,35,60,72,17,46)(8,30,49,67,18,41)(9,25,50,62,19,48)(10,32,51,69,20,43)(11,27,52,64,21,38)(12,34,53,71,22,45), (1,40,7,46)(2,39,8,45)(3,38,9,44)(4,37,10,43)(5,48,11,42)(6,47,12,41)(13,27,19,33)(14,26,20,32)(15,25,21,31)(16,36,22,30)(17,35,23,29)(18,34,24,28)(49,71,55,65)(50,70,56,64)(51,69,57,63)(52,68,58,62)(53,67,59,61)(54,66,60,72) );

G=PermutationGroup([(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),(49,50,51,52,53,54,55,56,57,58,59,60),(61,62,63,64,65,66,67,68,69,70,71,72)], [(1,29,54,66,23,40),(2,36,55,61,24,47),(3,31,56,68,13,42),(4,26,57,63,14,37),(5,33,58,70,15,44),(6,28,59,65,16,39),(7,35,60,72,17,46),(8,30,49,67,18,41),(9,25,50,62,19,48),(10,32,51,69,20,43),(11,27,52,64,21,38),(12,34,53,71,22,45)], [(1,40,7,46),(2,39,8,45),(3,38,9,44),(4,37,10,43),(5,48,11,42),(6,47,12,41),(13,27,19,33),(14,26,20,32),(15,25,21,31),(16,36,22,30),(17,35,23,29),(18,34,24,28),(49,71,55,65),(50,70,56,64),(51,69,57,63),(52,68,58,62),(53,67,59,61),(54,66,60,72)])

C12.D6 is a maximal subgroup of
C326C4≀C2  D12.D6  Dic6.D6  D12.8D6  C248D6  C24.26D6  C24.32D6  C24.40D6  C62.(C2×C4)  Dic6.24D6  S3×D42S3  D1212D6  C3282+ 1+4  C4○D4×C3⋊S3  C3292- 1+4  C62.13D6  C36.27D6  (C3×D12)⋊S3  D12⋊(C3⋊S3)  C62.90D6  C62.91D6  C62.100D6
C12.D6 is a maximal quotient of
C62.221C23  C626Q8  C62.223C23  C62.225C23  C62.227C23  C62.229C23  C62.69D4  C62.231C23  C62.233C23  C62.234C23  C62.236C23  C12.31D12  C62.242C23  D4×C3⋊Dic3  C62.72D4  C62.254C23  C62.256C23  C6214D4  C36.27D6  C62.16D6  (C3×D12)⋊S3  D12⋊(C3⋊S3)  C62.90D6  C62.91D6  C62.100D6

Matrix representation of C12.D6 in GL6(𝔽13)

100000
010000
0001200
001100
000050
000008
,
12120000
100000
001100
0012000
000001
000010
,
12120000
010000
000100
001000
0000012
000010

G:=sub<GL(6,GF(13))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,12,1,0,0,0,0,0,0,5,0,0,0,0,0,0,8],[12,1,0,0,0,0,12,0,0,0,0,0,0,0,1,12,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[12,0,0,0,0,0,12,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,12,0] >;

C12.D6 in GAP, Magma, Sage, TeX

C_{12}.D_6
% in TeX

G:=Group("C12.D6");
// GroupNames label

G:=SmallGroup(144,173);
// by ID

G=gap.SmallGroup(144,173);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-3,-3,55,218,116,964,3461]);
// Polycyclic

G:=Group<a,b,c|a^12=b^6=1,c^2=a^6,b*a*b^-1=a^7,c*a*c^-1=a^-1,c*b*c^-1=a^6*b^-1>;
// generators/relations

Export

Character table of C12.D6 in TeX

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