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G = C6.F7order 252 = 22·32·7

The non-split extension by C6 of F7 acting via F7/C7⋊C3=C2

metacyclic, supersoluble, monomial, A-group

Aliases: C6.F7, C211C12, C42.1C6, Dic21⋊C3, C3⋊(C7⋊C12), C7⋊C3⋊Dic3, C7⋊(C3×Dic3), C14.(C3×S3), C2.(C3⋊F7), (C3×C7⋊C3)⋊1C4, (C2×C7⋊C3).S3, (C6×C7⋊C3).1C2, SmallGroup(252,18)

Series: Derived Chief Lower central Upper central

C1C21 — C6.F7
C1C7C21C42C6×C7⋊C3 — C6.F7
C21 — C6.F7
C1C2

Generators and relations for C6.F7
 G = < a,b,c | a6=b7=1, c6=a3, ab=ba, cac-1=a-1, cbc-1=b5 >

7C3
14C3
21C4
7C6
14C6
7C32
2C7⋊C3
7Dic3
21C12
7C3×C6
3Dic7
2C2×C7⋊C3
7C3×Dic3
3C7⋊C12

Character table of C6.F7

 class 123A3B3C3D3E4A4B6A6B6C6D6E712A12B12C12D1421A21B42A42B
 size 1127714142121277141462121212166666
ρ1111111111111111111111111    trivial
ρ21111111-1-1111111-1-1-1-111111    linear of order 2
ρ3111ζ32ζ3ζ3ζ32-1-11ζ3ζ32ζ3ζ321ζ6ζ65ζ65ζ611111    linear of order 6
ρ4111ζ3ζ32ζ32ζ3111ζ32ζ3ζ32ζ31ζ3ζ32ζ32ζ311111    linear of order 3
ρ5111ζ3ζ32ζ32ζ3-1-11ζ32ζ3ζ32ζ31ζ65ζ6ζ6ζ6511111    linear of order 6
ρ6111ζ32ζ3ζ3ζ32111ζ3ζ32ζ3ζ321ζ32ζ3ζ3ζ3211111    linear of order 3
ρ71-111111i-i-1-1-1-1-11i-ii-i-111-1-1    linear of order 4
ρ81-111111-ii-1-1-1-1-11-ii-ii-111-1-1    linear of order 4
ρ91-11ζ3ζ32ζ32ζ3-ii-1ζ6ζ65ζ6ζ651ζ43ζ3ζ4ζ32ζ43ζ32ζ4ζ3-111-1-1    linear of order 12
ρ101-11ζ32ζ3ζ3ζ32i-i-1ζ65ζ6ζ65ζ61ζ4ζ32ζ43ζ3ζ4ζ3ζ43ζ32-111-1-1    linear of order 12
ρ111-11ζ32ζ3ζ3ζ32-ii-1ζ65ζ6ζ65ζ61ζ43ζ32ζ4ζ3ζ43ζ3ζ4ζ32-111-1-1    linear of order 12
ρ121-11ζ3ζ32ζ32ζ3i-i-1ζ6ζ65ζ6ζ651ζ4ζ3ζ43ζ32ζ4ζ32ζ43ζ3-111-1-1    linear of order 12
ρ1322-122-1-100-122-1-1200002-1-1-1-1    orthogonal lifted from S3
ρ142-2-122-1-1001-2-21120000-2-1-111    symplectic lifted from Dic3, Schur index 2
ρ152-2-1-1--3-1+-3ζ65ζ60011--31+-3ζ3ζ3220000-2-1-111    complex lifted from C3×Dic3
ρ1622-1-1+-3-1--3ζ6ζ6500-1-1--3-1+-3ζ6ζ65200002-1-1-1-1    complex lifted from C3×S3
ρ1722-1-1--3-1+-3ζ65ζ600-1-1+-3-1--3ζ65ζ6200002-1-1-1-1    complex lifted from C3×S3
ρ182-2-1-1+-3-1--3ζ6ζ650011+-31--3ζ32ζ320000-2-1-111    complex lifted from C3×Dic3
ρ1966600000060000-10000-1-1-1-1-1    orthogonal lifted from F7
ρ2066-3000000-30000-10000-11-21/21+21/21+21/21-21/2    orthogonal lifted from C3⋊F7
ρ2166-3000000-30000-10000-11+21/21-21/21-21/21+21/2    orthogonal lifted from C3⋊F7
ρ226-6-300000030000-1000011-21/21+21/2-1-21/2-1+21/2    symplectic faithful, Schur index 2
ρ236-6-300000030000-1000011+21/21-21/2-1+21/2-1-21/2    symplectic faithful, Schur index 2
ρ246-66000000-60000-100001-1-111    symplectic lifted from C7⋊C12, Schur index 2

Smallest permutation representation of C6.F7
On 84 points
Generators in S84
(1 3 5 7 9 11)(2 12 10 8 6 4)(13 33 59 19 27 53)(14 54 28 20 60 34)(15 35 49 21 29 55)(16 56 30 22 50 36)(17 25 51 23 31 57)(18 58 32 24 52 26)(37 75 72 43 81 66)(38 67 82 44 61 76)(39 77 62 45 83 68)(40 69 84 46 63 78)(41 79 64 47 73 70)(42 71 74 48 65 80)
(1 43 62 18 79 56 28)(2 57 19 44 29 80 63)(3 81 45 58 64 30 20)(4 31 59 82 21 65 46)(5 66 83 32 47 22 60)(6 23 33 67 49 48 84)(7 37 68 24 73 50 34)(8 51 13 38 35 74 69)(9 75 39 52 70 36 14)(10 25 53 76 15 71 40)(11 72 77 26 41 16 54)(12 17 27 61 55 42 78)
(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)(73 74 75 76 77 78 79 80 81 82 83 84)

G:=sub<Sym(84)| (1,3,5,7,9,11)(2,12,10,8,6,4)(13,33,59,19,27,53)(14,54,28,20,60,34)(15,35,49,21,29,55)(16,56,30,22,50,36)(17,25,51,23,31,57)(18,58,32,24,52,26)(37,75,72,43,81,66)(38,67,82,44,61,76)(39,77,62,45,83,68)(40,69,84,46,63,78)(41,79,64,47,73,70)(42,71,74,48,65,80), (1,43,62,18,79,56,28)(2,57,19,44,29,80,63)(3,81,45,58,64,30,20)(4,31,59,82,21,65,46)(5,66,83,32,47,22,60)(6,23,33,67,49,48,84)(7,37,68,24,73,50,34)(8,51,13,38,35,74,69)(9,75,39,52,70,36,14)(10,25,53,76,15,71,40)(11,72,77,26,41,16,54)(12,17,27,61,55,42,78), (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)(73,74,75,76,77,78,79,80,81,82,83,84)>;

G:=Group( (1,3,5,7,9,11)(2,12,10,8,6,4)(13,33,59,19,27,53)(14,54,28,20,60,34)(15,35,49,21,29,55)(16,56,30,22,50,36)(17,25,51,23,31,57)(18,58,32,24,52,26)(37,75,72,43,81,66)(38,67,82,44,61,76)(39,77,62,45,83,68)(40,69,84,46,63,78)(41,79,64,47,73,70)(42,71,74,48,65,80), (1,43,62,18,79,56,28)(2,57,19,44,29,80,63)(3,81,45,58,64,30,20)(4,31,59,82,21,65,46)(5,66,83,32,47,22,60)(6,23,33,67,49,48,84)(7,37,68,24,73,50,34)(8,51,13,38,35,74,69)(9,75,39,52,70,36,14)(10,25,53,76,15,71,40)(11,72,77,26,41,16,54)(12,17,27,61,55,42,78), (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)(73,74,75,76,77,78,79,80,81,82,83,84) );

G=PermutationGroup([(1,3,5,7,9,11),(2,12,10,8,6,4),(13,33,59,19,27,53),(14,54,28,20,60,34),(15,35,49,21,29,55),(16,56,30,22,50,36),(17,25,51,23,31,57),(18,58,32,24,52,26),(37,75,72,43,81,66),(38,67,82,44,61,76),(39,77,62,45,83,68),(40,69,84,46,63,78),(41,79,64,47,73,70),(42,71,74,48,65,80)], [(1,43,62,18,79,56,28),(2,57,19,44,29,80,63),(3,81,45,58,64,30,20),(4,31,59,82,21,65,46),(5,66,83,32,47,22,60),(6,23,33,67,49,48,84),(7,37,68,24,73,50,34),(8,51,13,38,35,74,69),(9,75,39,52,70,36,14),(10,25,53,76,15,71,40),(11,72,77,26,41,16,54),(12,17,27,61,55,42,78)], [(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),(73,74,75,76,77,78,79,80,81,82,83,84)])

Matrix representation of C6.F7 in GL6(𝔽337)

29224624602460
02922462460246
9191460091
24600292246246
9109191460
0910919146
,
336336336336336336
100000
010000
001000
000100
000010
,
3301481148221168
67134207154323316
7320189182203270
16916218325031753
21881552281757
6714087256249270

G:=sub<GL(6,GF(337))| [292,0,91,246,91,0,246,292,91,0,0,91,246,246,46,0,91,0,0,246,0,292,91,91,246,0,0,246,46,91,0,246,91,246,0,46],[336,1,0,0,0,0,336,0,1,0,0,0,336,0,0,1,0,0,336,0,0,0,1,0,336,0,0,0,0,1,336,0,0,0,0,0],[330,67,73,169,21,67,14,134,20,162,88,140,81,207,189,183,155,87,148,154,182,250,228,256,221,323,203,317,175,249,168,316,270,53,7,270] >;

C6.F7 in GAP, Magma, Sage, TeX

C_6.F_7
% in TeX

G:=Group("C6.F7");
// GroupNames label

G:=SmallGroup(252,18);
// by ID

G=gap.SmallGroup(252,18);
# by ID

G:=PCGroup([5,-2,-3,-2,-3,-7,30,483,5404,909]);
// Polycyclic

G:=Group<a,b,c|a^6=b^7=1,c^6=a^3,a*b=b*a,c*a*c^-1=a^-1,c*b*c^-1=b^5>;
// generators/relations

Export

Subgroup lattice of C6.F7 in TeX
Character table of C6.F7 in TeX

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