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G = C6.11D12order 144 = 24·32

11st non-split extension by C6 of D12 acting via D12/C12=C2

metabelian, supersoluble, monomial

Aliases: C6.11D12, C62.24C22, (C6×C12)⋊2C2, (C2×C12)⋊2S3, C32(D6⋊C4), C6.15(C4×S3), (C2×C6).33D6, (C3×C6).26D4, C6.21(C3⋊D4), C326(C22⋊C4), C2.2(C12⋊S3), C2.2(C327D4), (C2×C3⋊S3)⋊2C4, C2.5(C4×C3⋊S3), (C2×C4)⋊1(C3⋊S3), (C3×C6).26(C2×C4), (C2×C3⋊Dic3)⋊2C2, C22.6(C2×C3⋊S3), (C22×C3⋊S3).2C2, SmallGroup(144,95)

Series: Derived Chief Lower central Upper central

C1C3×C6 — C6.11D12
C1C3C32C3×C6C62C22×C3⋊S3 — C6.11D12
C32C3×C6 — C6.11D12
C1C22C2×C4

Generators and relations for C6.11D12
 G = < a,b,c | a6=b12=1, c2=a3, ab=ba, cac-1=a-1, cbc-1=a3b-1 >

Subgroups: 378 in 102 conjugacy classes, 41 normal (15 characteristic)
C1, C2 [×3], C2 [×2], C3 [×4], C4 [×2], C22, C22 [×4], S3 [×8], C6 [×12], C2×C4, C2×C4, C23, C32, Dic3 [×4], C12 [×4], D6 [×16], C2×C6 [×4], C22⋊C4, C3⋊S3 [×2], C3×C6 [×3], C2×Dic3 [×4], C2×C12 [×4], C22×S3 [×4], C3⋊Dic3, C3×C12, C2×C3⋊S3 [×2], C2×C3⋊S3 [×2], C62, D6⋊C4 [×4], C2×C3⋊Dic3, C6×C12, C22×C3⋊S3, C6.11D12
Quotients: C1, C2 [×3], C4 [×2], C22, S3 [×4], C2×C4, D4 [×2], D6 [×4], C22⋊C4, C3⋊S3, C4×S3 [×4], D12 [×4], C3⋊D4 [×4], C2×C3⋊S3, D6⋊C4 [×4], C4×C3⋊S3, C12⋊S3, C327D4, C6.11D12

Smallest permutation representation of C6.11D12
On 72 points
Generators in S72
(1 27 64 20 42 52)(2 28 65 21 43 53)(3 29 66 22 44 54)(4 30 67 23 45 55)(5 31 68 24 46 56)(6 32 69 13 47 57)(7 33 70 14 48 58)(8 34 71 15 37 59)(9 35 72 16 38 60)(10 36 61 17 39 49)(11 25 62 18 40 50)(12 26 63 19 41 51)
(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 19 20 12)(2 11 21 18)(3 17 22 10)(4 9 23 16)(5 15 24 8)(6 7 13 14)(25 65 40 53)(26 52 41 64)(27 63 42 51)(28 50 43 62)(29 61 44 49)(30 60 45 72)(31 71 46 59)(32 58 47 70)(33 69 48 57)(34 56 37 68)(35 67 38 55)(36 54 39 66)

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

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

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

C6.11D12 is a maximal subgroup of
C62.6C23  C62.18C23  C62.20C23  Dic3.D12  C62.24C23  C62.38C23  Dic34D12  Dic3⋊D12  C62.58C23  D6.D12  C62.65C23  C62.74C23  D6⋊D12  C62.77C23  S3×D6⋊C4  D64D12  C12216C2  C4×C12⋊S3  C1226C2  C1222C2  C22⋊C4×C3⋊S3  C62.225C23  C6212D4  C62.227C23  C62.228C23  C62.229C23  C62.69D4  C62.236C23  C62.237C23  C62.238C23  C123D12  C62.240C23  C12.31D12  C62.242C23  C4×C327D4  C62.129D4  C6219D4  C6213D4  C6214D4  C62.261C23  C62.262C23  C62.21D6  C6.11D36  C62.78D6  C62.79D6  C62.148D6
C6.11D12 is a maximal quotient of
C122⋊C2  C62.110D4  C62.113D4  C62.114D4  C6.4Dic12  C12.60D12  C62.84D4  C12.19D12  C12.20D12  C62.37D4  C62.15Q8  C6.11D36  C62.31D6  C62.78D6  C62.79D6  C62.148D6

42 conjugacy classes

class 1 2A2B2C2D2E3A3B3C3D4A4B4C4D6A···6L12A···12P
order122222333344446···612···12
size1111181822222218182···22···2

42 irreducible representations

dim11111222222
type++++++++
imageC1C2C2C2C4S3D4D6C4×S3D12C3⋊D4
kernelC6.11D12C2×C3⋊Dic3C6×C12C22×C3⋊S3C2×C3⋊S3C2×C12C3×C6C2×C6C6C6C6
# reps11114424888

Matrix representation of C6.11D12 in GL4(𝔽13) generated by

0100
12100
00120
00012
,
2900
41100
0008
0058
,
41100
2900
0058
0008
G:=sub<GL(4,GF(13))| [0,12,0,0,1,1,0,0,0,0,12,0,0,0,0,12],[2,4,0,0,9,11,0,0,0,0,0,5,0,0,8,8],[4,2,0,0,11,9,0,0,0,0,5,0,0,0,8,8] >;

C6.11D12 in GAP, Magma, Sage, TeX

C_6._{11}D_{12}
% in TeX

G:=Group("C6.11D12");
// GroupNames label

G:=SmallGroup(144,95);
// by ID

G=gap.SmallGroup(144,95);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-3,-3,121,31,964,3461]);
// Polycyclic

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

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