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G = C7⋊D12order 168 = 23·3·7

The semidirect product of C7 and D12 acting via D12/D6=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C213D4, C72D12, D62D7, Dic7⋊S3, D424C2, C6.6D14, C14.6D6, C42.6C22, C31(C7⋊D4), C2.6(S3×D7), (S3×C14)⋊2C2, (C3×Dic7)⋊3C2, SmallGroup(168,17)

Series: Derived Chief Lower central Upper central

C1C42 — C7⋊D12
C1C7C21C42C3×Dic7 — C7⋊D12
C21C42 — C7⋊D12
C1C2

Generators and relations for C7⋊D12
 G = < a,b,c | a7=b12=c2=1, bab-1=cac=a-1, cbc=b-1 >

6C2
42C2
3C22
7C4
21C22
2S3
14S3
6D7
6C14
21D4
7C12
7D6
3D14
3C2×C14
2S3×C7
2D21
7D12
3C7⋊D4

Character table of C7⋊D12

 class 12A2B2C3467A7B7C12A12B14A14B14C14D14E14F14G14H14I21A21B21C42A42B42C
 size 1164221422221414222666666444444
ρ1111111111111111111111111111    trivial
ρ211-111-11111-1-1111-1-1-1-1-1-1111111    linear of order 2
ρ311-1-111111111111-1-1-1-1-1-1111111    linear of order 2
ρ4111-11-11111-1-1111111111111111    linear of order 2
ρ52200-1-2-122211222000000-1-1-1-1-1-1    orthogonal lifted from D6
ρ62200-12-1222-1-1222000000-1-1-1-1-1-1    orthogonal lifted from S3
ρ72-20020-222200-2-2-2000000222-2-2-2    orthogonal lifted from D4
ρ82220202ζ7572ζ7473ζ76700ζ767ζ7572ζ7473ζ7473ζ7473ζ767ζ7572ζ7572ζ767ζ767ζ7572ζ7473ζ7572ζ7473ζ767    orthogonal lifted from D7
ρ92220202ζ7473ζ767ζ757200ζ7572ζ7473ζ767ζ767ζ767ζ7572ζ7473ζ7473ζ7572ζ7572ζ7473ζ767ζ7473ζ767ζ7572    orthogonal lifted from D7
ρ1022-20202ζ7572ζ7473ζ76700ζ767ζ7572ζ74737473747376775727572767ζ767ζ7572ζ7473ζ7572ζ7473ζ767    orthogonal lifted from D14
ρ1122-20202ζ767ζ7572ζ747300ζ7473ζ767ζ75727572757274737677677473ζ7473ζ767ζ7572ζ767ζ7572ζ7473    orthogonal lifted from D14
ρ122-200-1012223-3-2-2-2000000-1-1-1111    orthogonal lifted from D12
ρ1322-20202ζ7473ζ767ζ757200ζ7572ζ7473ζ7677677677572747374737572ζ7572ζ7473ζ767ζ7473ζ767ζ7572    orthogonal lifted from D14
ρ142220202ζ767ζ7572ζ747300ζ7473ζ767ζ7572ζ7572ζ7572ζ7473ζ767ζ767ζ7473ζ7473ζ767ζ7572ζ767ζ7572ζ7473    orthogonal lifted from D7
ρ152-200-101222-33-2-2-2000000-1-1-1111    orthogonal lifted from D12
ρ162-20020-2ζ7572ζ7473ζ7670076775727473ζ74737473ζ7677572ζ7572767ζ767ζ7572ζ747375727473767    complex lifted from C7⋊D4
ρ172-20020-2ζ767ζ7572ζ747300747376775727572ζ75727473767ζ767ζ7473ζ7473ζ767ζ757276775727473    complex lifted from C7⋊D4
ρ182-20020-2ζ7473ζ767ζ75720075727473767767ζ767ζ7572ζ747374737572ζ7572ζ7473ζ76774737677572    complex lifted from C7⋊D4
ρ192-20020-2ζ767ζ7572ζ74730074737677572ζ75727572ζ7473ζ7677677473ζ7473ζ767ζ757276775727473    complex lifted from C7⋊D4
ρ202-20020-2ζ7572ζ7473ζ76700767757274737473ζ7473767ζ75727572ζ767ζ767ζ7572ζ747375727473767    complex lifted from C7⋊D4
ρ212-20020-2ζ7473ζ767ζ75720075727473767ζ76776775727473ζ7473ζ7572ζ7572ζ7473ζ76774737677572    complex lifted from C7⋊D4
ρ224400-20-275+2ζ7274+2ζ7376+2ζ70076+2ζ775+2ζ7274+2ζ730000007677572747375727473767    orthogonal lifted from S3×D7
ρ234-400-20275+2ζ7274+2ζ7376+2ζ700-2ζ76-2ζ7-2ζ75-2ζ72-2ζ74-2ζ7300000076775727473ζ7572ζ7473ζ767    orthogonal faithful, Schur index 2
ρ244-400-20276+2ζ775+2ζ7274+2ζ7300-2ζ74-2ζ73-2ζ76-2ζ7-2ζ75-2ζ7200000074737677572ζ767ζ7572ζ7473    orthogonal faithful, Schur index 2
ρ254400-20-274+2ζ7376+2ζ775+2ζ720075+2ζ7274+2ζ7376+2ζ70000007572747376774737677572    orthogonal lifted from S3×D7
ρ264400-20-276+2ζ775+2ζ7274+2ζ730074+2ζ7376+2ζ775+2ζ720000007473767757276775727473    orthogonal lifted from S3×D7
ρ274-400-20274+2ζ7376+2ζ775+2ζ7200-2ζ75-2ζ72-2ζ74-2ζ73-2ζ76-2ζ700000075727473767ζ7473ζ767ζ7572    orthogonal faithful, Schur index 2

Smallest permutation representation of C7⋊D12
On 84 points
Generators in S84
(1 18 80 54 71 39 33)(2 34 40 72 55 81 19)(3 20 82 56 61 41 35)(4 36 42 62 57 83 21)(5 22 84 58 63 43 25)(6 26 44 64 59 73 23)(7 24 74 60 65 45 27)(8 28 46 66 49 75 13)(9 14 76 50 67 47 29)(10 30 48 68 51 77 15)(11 16 78 52 69 37 31)(12 32 38 70 53 79 17)
(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)
(1 9)(2 8)(3 7)(4 6)(10 12)(13 34)(14 33)(15 32)(16 31)(17 30)(18 29)(19 28)(20 27)(21 26)(22 25)(23 36)(24 35)(37 78)(38 77)(39 76)(40 75)(41 74)(42 73)(43 84)(44 83)(45 82)(46 81)(47 80)(48 79)(49 72)(50 71)(51 70)(52 69)(53 68)(54 67)(55 66)(56 65)(57 64)(58 63)(59 62)(60 61)

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

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

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

C7⋊D12 is a maximal subgroup of   D12⋊D7  D84⋊C2  D6.D14  D7×D12  Dic3.D14  S3×C7⋊D4  D6⋊D14
C7⋊D12 is a maximal quotient of   C7⋊D24  D12.D7  Dic6⋊D7  C7⋊Dic12  D6⋊Dic7  D42⋊C4  C42.Q8

Matrix representation of C7⋊D12 in GL4(𝔽337) generated by

0100
33630300
0010
0001
,
1222700
26021500
00292306
00870
,
1000
30333600
00336290
0001
G:=sub<GL(4,GF(337))| [0,336,0,0,1,303,0,0,0,0,1,0,0,0,0,1],[122,260,0,0,27,215,0,0,0,0,292,87,0,0,306,0],[1,303,0,0,0,336,0,0,0,0,336,0,0,0,290,1] >;

C7⋊D12 in GAP, Magma, Sage, TeX

C_7\rtimes D_{12}
% in TeX

G:=Group("C7:D12");
// GroupNames label

G:=SmallGroup(168,17);
// by ID

G=gap.SmallGroup(168,17);
# by ID

G:=PCGroup([5,-2,-2,-2,-3,-7,20,61,168,3604]);
// Polycyclic

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

Export

Subgroup lattice of C7⋊D12 in TeX
Character table of C7⋊D12 in TeX

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