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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, (C3xD4):3S3, (C2xC6).6D6, D4:2(C3:S3), (D4xC32):6C2, C32:7D4:4C2, C3:5(D4:2S3), C32:4Q8:6C2, C32:10(C4oD4), C6.35(C22xS3), (C3xC6).34C23, (C3xC12).24C22, C3:Dic3.18C22, (C4xC3:S3):4C2, C4.5(C2xC3:S3), (C2xC3:Dic3):7C2, C22.1(C2xC3:S3), C2.7(C22xC3:S3), (C2xC3:S3).18C22, SmallGroup(144,173)

Series: Derived Chief Lower central Upper central

C1C3xC6 — C12.D6
C1C3C32C3xC6C2xC3:S3C4xC3:S3 — C12.D6
C32C3xC6 — 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, C3, C4, C4, C22, C22, S3, C6, C6, C2xC4, D4, D4, Q8, C32, Dic3, C12, D6, C2xC6, C4oD4, C3:S3, C3xC6, C3xC6, Dic6, C4xS3, C2xDic3, C3:D4, C3xD4, C3:Dic3, C3:Dic3, C3xC12, C2xC3:S3, C62, D4:2S3, C32:4Q8, C4xC3:S3, C2xC3:Dic3, C32:7D4, D4xC32, C12.D6
Quotients: C1, C2, C22, S3, C23, D6, C4oD4, C3:S3, C22xS3, C2xC3:S3, D4:2S3, C22xC3: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 C4oD4
ρ262-2000222202i-2i00-2-2-2-2000000000000    complex lifted from C4oD4
ρ274-4000-2-24-200000-4222000000000000    symplectic lifted from D4:2S3, Schur index 2
ρ284-40004-2-2-2000002-422000000000000    symplectic lifted from D4:2S3, Schur index 2
ρ294-4000-24-2-200000222-4000000000000    symplectic lifted from D4:2S3, Schur index 2
ρ304-4000-2-2-240000022-42000000000000    symplectic lifted from D4:2S3, 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 26 52 48 17 64)(2 33 53 43 18 71)(3 28 54 38 19 66)(4 35 55 45 20 61)(5 30 56 40 21 68)(6 25 57 47 22 63)(7 32 58 42 23 70)(8 27 59 37 24 65)(9 34 60 44 13 72)(10 29 49 39 14 67)(11 36 50 46 15 62)(12 31 51 41 16 69)
(1 64 7 70)(2 63 8 69)(3 62 9 68)(4 61 10 67)(5 72 11 66)(6 71 12 65)(13 30 19 36)(14 29 20 35)(15 28 21 34)(16 27 22 33)(17 26 23 32)(18 25 24 31)(37 57 43 51)(38 56 44 50)(39 55 45 49)(40 54 46 60)(41 53 47 59)(42 52 48 58)

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

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

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

C12.D6 is a maximal subgroup of
C32:6C4wrC2  D12.D6  Dic6.D6  D12.8D6  C24:8D6  C24.26D6  C24.32D6  C24.40D6  C62.(C2xC4)  Dic6.24D6  S3xD4:2S3  D12:12D6  C32:82+ 1+4  C4oD4xC3:S3  C32:92- 1+4  C62.13D6  C36.27D6  (C3xD12):S3  D12:(C3:S3)  C62.90D6  C62.91D6  C62.100D6
C12.D6 is a maximal quotient of
C62.221C23  C62:6Q8  C62.223C23  C62.225C23  C62.227C23  C62.229C23  C62.69D4  C62.231C23  C62.233C23  C62.234C23  C62.236C23  C12.31D12  C62.242C23  D4xC3:Dic3  C62.72D4  C62.254C23  C62.256C23  C62:14D4  C36.27D6  C62.16D6  (C3xD12):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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