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G = C12.26D6order 144 = 24·32

26th non-split extension by C12 of D6 acting via D6/C3=C22

metabelian, supersoluble, monomial

Aliases: C12.26D6, (C3×Q8)⋊5S3, Q83(C3⋊S3), C12⋊S37C2, (Q8×C32)⋊6C2, C33(Q83S3), C3212(C4○D4), C6.37(C22×S3), (C3×C6).36C23, (C3×C12).26C22, C3⋊Dic3.20C22, (C4×C3⋊S3)⋊5C2, C4.7(C2×C3⋊S3), C2.9(C22×C3⋊S3), (C2×C3⋊S3).19C22, SmallGroup(144,175)

Series: Derived Chief Lower central Upper central

C1C3×C6 — C12.26D6
C1C3C32C3×C6C2×C3⋊S3C4×C3⋊S3 — C12.26D6
C32C3×C6 — C12.26D6
C1C2Q8

Generators and relations for C12.26D6
 G = < a,b,c | a12=1, b6=c2=a6, bab-1=a7, cac-1=a5, cbc-1=b5 >

Subgroups: 410 in 120 conjugacy classes, 47 normal (8 characteristic)
C1, C2, C2 [×3], C3 [×4], C4 [×3], C4, C22 [×3], S3 [×12], C6 [×4], C2×C4 [×3], D4 [×3], Q8, C32, Dic3 [×4], C12 [×12], D6 [×12], C4○D4, C3⋊S3 [×3], C3×C6, C4×S3 [×12], D12 [×12], C3×Q8 [×4], C3⋊Dic3, C3×C12 [×3], C2×C3⋊S3 [×3], Q83S3 [×4], C4×C3⋊S3 [×3], C12⋊S3 [×3], Q8×C32, C12.26D6
Quotients: C1, C2 [×7], C22 [×7], S3 [×4], C23, D6 [×12], C4○D4, C3⋊S3, C22×S3 [×4], C2×C3⋊S3 [×3], Q83S3 [×4], C22×C3⋊S3, C12.26D6

Character table of C12.26D6

 class 12A2B2C2D3A3B3C3D4A4B4C4D4E6A6B6C6D12A12B12C12D12E12F12G12H12I12J12K12L
 size 111818182222222992222444444444444
ρ1111111111111111111111111111111    trivial
ρ211-11-11111-1-11111111-1-1-11111-1-1-1-1-1    linear of order 2
ρ3111-111111-1-11-1-11111-1-1-11111-1-1-1-1-1    linear of order 2
ρ411-1-1-11111111-1-11111111111111111    linear of order 2
ρ51111-11111-11-1-1-1111111-1-1-1-1-1-1-1-111    linear of order 2
ρ611-11111111-1-1-1-11111-1-11-1-1-1-1111-1-1    linear of order 2
ρ7111-1-111111-1-1111111-1-11-1-1-1-1111-1-1    linear of order 2
ρ811-1-111111-11-111111111-1-1-1-1-1-1-1-111    linear of order 2
ρ922000-1-12-1-22-200-1-1-122-1111-21-211-1-1    orthogonal lifted from D6
ρ1022000-12-1-122200-1-12-1-1-12-12-1-1-1-1-1-12    orthogonal lifted from S3
ρ1122000-1-1-122-2-2002-1-1-11-2-1111-2-12-111    orthogonal lifted from D6
ρ12220002-1-1-122200-12-1-1-1-1-12-1-1-1-1-122-1    orthogonal lifted from S3
ρ13220002-1-1-12-2-200-12-1-111-1-2111-1-12-21    orthogonal lifted from D6
ρ1422000-1-12-1-2-2200-1-1-12-211-1-12-1-21111    orthogonal lifted from D6
ρ1522000-12-1-1-22-200-1-12-1-1-1-21-211111-12    orthogonal lifted from D6
ρ1622000-1-1-12222002-1-1-1-12-1-1-1-12-12-1-1-1    orthogonal lifted from S3
ρ1722000-1-12-12-2-200-1-1-12-21-111-212-1-111    orthogonal lifted from D6
ρ1822000-1-1-12-22-2002-1-1-1-121111-21-21-1-1    orthogonal lifted from D6
ρ1922000-12-1-1-2-2200-1-12-111-2-12-1-11111-2    orthogonal lifted from D6
ρ20220002-1-1-1-2-2200-12-1-11112-1-1-111-2-21    orthogonal lifted from D6
ρ2122000-1-12-122200-1-1-122-1-1-1-12-12-1-1-1-1    orthogonal lifted from S3
ρ22220002-1-1-1-22-200-12-1-1-1-11-211111-22-1    orthogonal lifted from D6
ρ2322000-1-1-12-2-22002-1-1-11-21-1-1-121-2111    orthogonal lifted from D6
ρ2422000-12-1-12-2-200-1-12-11121-211-1-1-11-2    orthogonal lifted from D6
ρ252-20002222000-2i2i-2-2-2-2000000000000    complex lifted from C4○D4
ρ262-200022220002i-2i-2-2-2-2000000000000    complex lifted from C4○D4
ρ274-4000-24-2-20000022-42000000000000    orthogonal lifted from Q83S3, Schur index 2
ρ284-40004-2-2-2000002-422000000000000    orthogonal lifted from Q83S3, Schur index 2
ρ294-4000-2-24-200000222-4000000000000    orthogonal lifted from Q83S3, Schur index 2
ρ304-4000-2-2-2400000-4222000000000000    orthogonal lifted from Q83S3, Schur index 2

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

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

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

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

C12.26D6 is a maximal subgroup of
C327C4≀C2  D12.10D6  Dic6.10D6  D12.14D6  C247D6  C24.40D6  C24.35D6  C24.28D6  C12⋊S3.C4  Dic6.26D6  S3×Q83S3  D1216D6  C3272- 1+4  C4○D4×C3⋊S3  C62.154C23  C6.(S3×A4)  (Q8×He3)⋊C2  C36.29D6  C3⋊Dic3.2A4  C12.39S32  C12.40S32  (Q8×C33)⋊C2
C12.26D6 is a maximal quotient of
C62.234C23  C62.236C23  C62.237C23  C62.238C23  C123D12  C62.242C23  Q8×C3⋊Dic3  C62.261C23  C62.262C23  C36.29D6  He35D4⋊C2  C12.39S32  C12.40S32  (Q8×C33)⋊C2

Matrix representation of C12.26D6 in GL6(𝔽13)

0120000
110000
0001200
001100
000001
0000120
,
110000
1200000
0001200
001100
000050
000008
,
110000
0120000
001100
0001200
000050
000005

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

C12.26D6 in GAP, Magma, Sage, TeX

C_{12}._{26}D_6
% in TeX

G:=Group("C12.26D6");
// GroupNames label

G:=SmallGroup(144,175);
// by ID

G=gap.SmallGroup(144,175);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-3,-3,55,218,116,50,964,3461]);
// Polycyclic

G:=Group<a,b,c|a^12=1,b^6=c^2=a^6,b*a*b^-1=a^7,c*a*c^-1=a^5,c*b*c^-1=b^5>;
// generators/relations

Export

Character table of C12.26D6 in TeX

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