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G = C21⋊D4order 168 = 23·3·7

1st semidirect product of C21 and D4 acting via D4/C2=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C211D4, D61D7, D141S3, C6.4D14, C14.4D6, Dic214C2, C42.4C22, (C6×D7)⋊1C2, C72(C3⋊D4), C32(C7⋊D4), C2.4(S3×D7), (S3×C14)⋊1C2, SmallGroup(168,15)

Series: Derived Chief Lower central Upper central

C1C42 — C21⋊D4
C1C7C21C42C6×D7 — C21⋊D4
C21C42 — C21⋊D4
C1C2

Generators and relations for C21⋊D4
 G = < a,b,c | a21=b4=c2=1, bab-1=a-1, cac=a13, cbc=b-1 >

6C2
14C2
3C22
7C22
21C4
2S3
14C6
2D7
6C14
21D4
7C2×C6
7Dic3
3Dic7
3C2×C14
2S3×C7
2C3×D7
7C3⋊D4
3C7⋊D4

Character table of C21⋊D4

 class 12A2B2C346A6B6C7A7B7C14A14B14C14D14E14F14G14H14I21A21B21C42A42B42C
 size 1161424221414222222666666444444
ρ1111111111111111111111111111    trivial
ρ211-111-1111111111-1-1-1-1-1-1111111    linear of order 2
ρ311-1-1111-1-1111111-1-1-1-1-1-1111111    linear of order 2
ρ4111-11-11-1-1111111111111111111    linear of order 2
ρ5220-2-10-111222222000000-1-1-1-1-1-1    orthogonal lifted from D6
ρ62202-10-1-1-1222222000000-1-1-1-1-1-1    orthogonal lifted from S3
ρ72-20020-200222-2-2-2000000222-2-2-2    orthogonal lifted from D4
ρ822-2020200ζ7473ζ767ζ7572ζ767ζ7572ζ74737572757276774737473767ζ767ζ7572ζ7473ζ7572ζ7473ζ767    orthogonal lifted from D14
ρ9222020200ζ767ζ7572ζ7473ζ7572ζ7473ζ767ζ7473ζ7473ζ7572ζ767ζ767ζ7572ζ7572ζ7473ζ767ζ7473ζ767ζ7572    orthogonal lifted from D7
ρ10222020200ζ7473ζ767ζ7572ζ767ζ7572ζ7473ζ7572ζ7572ζ767ζ7473ζ7473ζ767ζ767ζ7572ζ7473ζ7572ζ7473ζ767    orthogonal lifted from D7
ρ1122-2020200ζ7572ζ7473ζ767ζ7473ζ767ζ75727677677473757275727473ζ7473ζ767ζ7572ζ767ζ7572ζ7473    orthogonal lifted from D14
ρ1222-2020200ζ767ζ7572ζ7473ζ7572ζ7473ζ7677473747375727677677572ζ7572ζ7473ζ767ζ7473ζ767ζ7572    orthogonal lifted from D14
ρ13222020200ζ7572ζ7473ζ767ζ7473ζ767ζ7572ζ767ζ767ζ7473ζ7572ζ7572ζ7473ζ7473ζ767ζ7572ζ767ζ7572ζ7473    orthogonal lifted from D7
ρ142-200-101-3--3222-2-2-2000000-1-1-1111    complex lifted from C3⋊D4
ρ152-20020-200ζ7473ζ767ζ7572767757274737572ζ7572767ζ74737473ζ767ζ767ζ7572ζ747375727473767    complex lifted from C7⋊D4
ρ162-20020-200ζ7473ζ767ζ757276775727473ζ75727572ζ7677473ζ7473767ζ767ζ7572ζ747375727473767    complex lifted from C7⋊D4
ρ172-20020-200ζ767ζ7572ζ7473757274737677473ζ7473ζ7572ζ7677677572ζ7572ζ7473ζ76774737677572    complex lifted from C7⋊D4
ρ182-20020-200ζ7572ζ7473ζ76774737677572767ζ767ζ74737572ζ75727473ζ7473ζ767ζ757276775727473    complex lifted from C7⋊D4
ρ192-20020-200ζ767ζ7572ζ747375727473767ζ747374737572767ζ767ζ7572ζ7572ζ7473ζ76774737677572    complex lifted from C7⋊D4
ρ202-200-101--3-3222-2-2-2000000-1-1-1111    complex lifted from C3⋊D4
ρ212-20020-200ζ7572ζ7473ζ76774737677572ζ7677677473ζ75727572ζ7473ζ7473ζ767ζ757276775727473    complex lifted from C7⋊D4
ρ224400-20-20075+2ζ7274+2ζ7376+2ζ774+2ζ7376+2ζ775+2ζ720000007473767757276775727473    orthogonal lifted from S3×D7
ρ234400-20-20074+2ζ7376+2ζ775+2ζ7276+2ζ775+2ζ7274+2ζ730000007677572747375727473767    orthogonal lifted from S3×D7
ρ244400-20-20076+2ζ775+2ζ7274+2ζ7375+2ζ7274+2ζ7376+2ζ70000007572747376774737677572    orthogonal lifted from S3×D7
ρ254-400-2020075+2ζ7274+2ζ7376+2ζ7-2ζ74-2ζ73-2ζ76-2ζ7-2ζ75-2ζ7200000074737677572ζ767ζ7572ζ7473    symplectic faithful, Schur index 2
ρ264-400-2020074+2ζ7376+2ζ775+2ζ72-2ζ76-2ζ7-2ζ75-2ζ72-2ζ74-2ζ7300000076775727473ζ7572ζ7473ζ767    symplectic faithful, Schur index 2
ρ274-400-2020076+2ζ775+2ζ7274+2ζ73-2ζ75-2ζ72-2ζ74-2ζ73-2ζ76-2ζ700000075727473767ζ7473ζ767ζ7572    symplectic faithful, Schur index 2

Smallest permutation representation of C21⋊D4
On 84 points
Generators in S84
(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 43 40 84)(2 63 41 83)(3 62 42 82)(4 61 22 81)(5 60 23 80)(6 59 24 79)(7 58 25 78)(8 57 26 77)(9 56 27 76)(10 55 28 75)(11 54 29 74)(12 53 30 73)(13 52 31 72)(14 51 32 71)(15 50 33 70)(16 49 34 69)(17 48 35 68)(18 47 36 67)(19 46 37 66)(20 45 38 65)(21 44 39 64)
(2 14)(3 6)(4 19)(5 11)(7 16)(9 21)(10 13)(12 18)(17 20)(22 37)(23 29)(24 42)(25 34)(27 39)(28 31)(30 36)(32 41)(35 38)(43 84)(44 76)(45 68)(46 81)(47 73)(48 65)(49 78)(50 70)(51 83)(52 75)(53 67)(54 80)(55 72)(56 64)(57 77)(58 69)(59 82)(60 74)(61 66)(62 79)(63 71)

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

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

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

C21⋊D4 is a maximal subgroup of   D285S3  D6.D14  D125D7  C28⋊D6  C42.C23  D7×C3⋊D4  S3×C7⋊D4
C21⋊D4 is a maximal quotient of   C21⋊D8  C28.D6  C42.D4  C21⋊Q16  D14⋊Dic3  D6⋊Dic7  Dic21⋊C4

Matrix representation of C21⋊D4 in GL4(𝔽337) generated by

019400
3322700
002080
0084128
,
019400
304000
00206205
0033131
,
019400
304000
0010
00238336
G:=sub<GL(4,GF(337))| [0,33,0,0,194,227,0,0,0,0,208,84,0,0,0,128],[0,304,0,0,194,0,0,0,0,0,206,33,0,0,205,131],[0,304,0,0,194,0,0,0,0,0,1,238,0,0,0,336] >;

C21⋊D4 in GAP, Magma, Sage, TeX

C_{21}\rtimes D_4
% in TeX

G:=Group("C21:D4");
// GroupNames label

G:=SmallGroup(168,15);
// by ID

G=gap.SmallGroup(168,15);
# by ID

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

G:=Group<a,b,c|a^21=b^4=c^2=1,b*a*b^-1=a^-1,c*a*c=a^13,c*b*c=b^-1>;
// generators/relations

Export

Subgroup lattice of C21⋊D4 in TeX
Character table of C21⋊D4 in TeX

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