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G = D21⋊C4order 168 = 23·3·7

The semidirect product of D21 and C4 acting via C4/C2=C2

metabelian, supersoluble, monomial, A-group, 2-hyperelementary

Aliases: D21⋊C4, D42.C2, C14.3D6, C6.3D14, Dic32D7, Dic72S3, C42.3C22, C71(C4×S3), C31(C4×D7), C213(C2×C4), C2.3(S3×D7), (C7×Dic3)⋊2C2, (C3×Dic7)⋊2C2, SmallGroup(168,14)

Series: Derived Chief Lower central Upper central

C1C21 — D21⋊C4
C1C7C21C42C3×Dic7 — D21⋊C4
C21 — D21⋊C4
C1C2

Generators and relations for D21⋊C4
 G = < a,b,c | a21=b2=c4=1, bab=a-1, cac-1=a8, cbc-1=a7b >

21C2
21C2
3C4
7C4
21C22
7S3
7S3
3D7
3D7
21C2×C4
7C12
7D6
3C28
3D14
7C4×S3
3C4×D7

Character table of D21⋊C4

 class 12A2B2C34A4B4C4D67A7B7C12A12B14A14B14C21A21B21C28A28B28C28D28E28F42A42B42C
 size 1121212337722221414222444666666444
ρ1111111111111111111111111111111    trivial
ρ211111-1-1-1-11111-1-1111111-1-1-1-1-1-1111    linear of order 2
ρ311-1-11-1-111111111111111-1-1-1-1-1-1111    linear of order 2
ρ411-1-1111-1-11111-1-1111111111111111    linear of order 2
ρ51-1-111-ii-ii-1111i-i-1-1-1111-iii-i-ii-1-1-1    linear of order 4
ρ61-11-11i-i-ii-1111i-i-1-1-1111i-i-iii-i-1-1-1    linear of order 4
ρ71-1-111i-ii-i-1111-ii-1-1-1111i-i-iii-i-1-1-1    linear of order 4
ρ81-11-11-iii-i-1111-ii-1-1-1111-iii-i-ii-1-1-1    linear of order 4
ρ92200-10022-1222-1-1222-1-1-1000000-1-1-1    orthogonal lifted from S3
ρ102200-100-2-2-122211222-1-1-1000000-1-1-1    orthogonal lifted from D6
ρ1122002-2-2002ζ7473ζ7572ζ76700ζ7473ζ7572ζ767ζ7473ζ767ζ75727677473767757274737572ζ767ζ7473ζ7572    orthogonal lifted from D14
ρ122200222002ζ767ζ7473ζ757200ζ767ζ7473ζ7572ζ767ζ7572ζ7473ζ7572ζ767ζ7572ζ7473ζ767ζ7473ζ7572ζ767ζ7473    orthogonal lifted from D7
ρ132200222002ζ7473ζ7572ζ76700ζ7473ζ7572ζ767ζ7473ζ767ζ7572ζ767ζ7473ζ767ζ7572ζ7473ζ7572ζ767ζ7473ζ7572    orthogonal lifted from D7
ρ1422002-2-2002ζ767ζ7473ζ757200ζ767ζ7473ζ7572ζ767ζ7572ζ74737572767757274737677473ζ7572ζ767ζ7473    orthogonal lifted from D14
ρ1522002-2-2002ζ7572ζ767ζ747300ζ7572ζ767ζ7473ζ7572ζ7473ζ7677473757274737677572767ζ7473ζ7572ζ767    orthogonal lifted from D14
ρ162200222002ζ7572ζ767ζ747300ζ7572ζ767ζ7473ζ7572ζ7473ζ767ζ7473ζ7572ζ7473ζ767ζ7572ζ767ζ7473ζ7572ζ767    orthogonal lifted from D7
ρ172-200-1002i-2i1222i-i-2-2-2-1-1-1000000111    complex lifted from C4×S3
ρ182-200-100-2i2i1222-ii-2-2-2-1-1-1000000111    complex lifted from C4×S3
ρ192-20022i-2i00-2ζ7572ζ767ζ74730075727677473ζ7572ζ7473ζ767ζ4ζ744ζ73ζ43ζ7543ζ72ζ43ζ7443ζ73ζ4ζ764ζ7ζ4ζ754ζ72ζ43ζ7643ζ774737572767    complex lifted from C4×D7
ρ202-2002-2i2i00-2ζ767ζ7473ζ75720076774737572ζ767ζ7572ζ7473ζ43ζ7543ζ72ζ4ζ764ζ7ζ4ζ754ζ72ζ43ζ7443ζ73ζ43ζ7643ζ7ζ4ζ744ζ7375727677473    complex lifted from C4×D7
ρ212-2002-2i2i00-2ζ7572ζ767ζ74730075727677473ζ7572ζ7473ζ767ζ43ζ7443ζ73ζ4ζ754ζ72ζ4ζ744ζ73ζ43ζ7643ζ7ζ43ζ7543ζ72ζ4ζ764ζ774737572767    complex lifted from C4×D7
ρ222-2002-2i2i00-2ζ7473ζ7572ζ7670074737572767ζ7473ζ767ζ7572ζ43ζ7643ζ7ζ4ζ744ζ73ζ4ζ764ζ7ζ43ζ7543ζ72ζ43ζ7443ζ73ζ4ζ754ζ7276774737572    complex lifted from C4×D7
ρ232-20022i-2i00-2ζ767ζ7473ζ75720076774737572ζ767ζ7572ζ7473ζ4ζ754ζ72ζ43ζ7643ζ7ζ43ζ7543ζ72ζ4ζ744ζ73ζ4ζ764ζ7ζ43ζ7443ζ7375727677473    complex lifted from C4×D7
ρ242-20022i-2i00-2ζ7473ζ7572ζ7670074737572767ζ7473ζ767ζ7572ζ4ζ764ζ7ζ43ζ7443ζ73ζ43ζ7643ζ7ζ4ζ754ζ72ζ4ζ744ζ73ζ43ζ7543ζ7276774737572    complex lifted from C4×D7
ρ254-400-20000275+2ζ7276+2ζ774+2ζ7300-2ζ75-2ζ72-2ζ76-2ζ7-2ζ74-2ζ7375727473767000000ζ7473ζ7572ζ767    orthogonal faithful, Schur index 2
ρ264-400-20000276+2ζ774+2ζ7375+2ζ7200-2ζ76-2ζ7-2ζ74-2ζ73-2ζ75-2ζ7276775727473000000ζ7572ζ767ζ7473    orthogonal faithful, Schur index 2
ρ274400-20000-276+2ζ774+2ζ7375+2ζ720076+2ζ774+2ζ7375+2ζ727677572747300000075727677473    orthogonal lifted from S3×D7
ρ284-400-20000274+2ζ7375+2ζ7276+2ζ700-2ζ74-2ζ73-2ζ75-2ζ72-2ζ76-2ζ774737677572000000ζ767ζ7473ζ7572    orthogonal faithful, Schur index 2
ρ294400-20000-274+2ζ7375+2ζ7276+2ζ70074+2ζ7375+2ζ7276+2ζ77473767757200000076774737572    orthogonal lifted from S3×D7
ρ304400-20000-275+2ζ7276+2ζ774+2ζ730075+2ζ7276+2ζ774+2ζ737572747376700000074737572767    orthogonal lifted from S3×D7

Smallest permutation representation of D21⋊C4
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 21)(2 20)(3 19)(4 18)(5 17)(6 16)(7 15)(8 14)(9 13)(10 12)(22 38)(23 37)(24 36)(25 35)(26 34)(27 33)(28 32)(29 31)(39 42)(40 41)(43 58)(44 57)(45 56)(46 55)(47 54)(48 53)(49 52)(50 51)(59 63)(60 62)(64 74)(65 73)(66 72)(67 71)(68 70)(75 84)(76 83)(77 82)(78 81)(79 80)
(1 80 41 51)(2 67 42 59)(3 75 22 46)(4 83 23 54)(5 70 24 62)(6 78 25 49)(7 65 26 57)(8 73 27 44)(9 81 28 52)(10 68 29 60)(11 76 30 47)(12 84 31 55)(13 71 32 63)(14 79 33 50)(15 66 34 58)(16 74 35 45)(17 82 36 53)(18 69 37 61)(19 77 38 48)(20 64 39 56)(21 72 40 43)

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

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

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

D21⋊C4 is a maximal subgroup of   D84⋊C2  D21⋊Q8  D14.D6  C4×S3×D7  Dic7.D6  Dic3.D14  D6⋊D14
D21⋊C4 is a maximal quotient of   D21⋊C8  D42.C4  Dic3×Dic7  D42⋊C4  Dic21⋊C4

Matrix representation of D21⋊C4 in GL4(𝔽337) generated by

0100
33633600
0034194
00178193
,
0100
1000
000228
00340
,
336000
1100
001480
000148
G:=sub<GL(4,GF(337))| [0,336,0,0,1,336,0,0,0,0,34,178,0,0,194,193],[0,1,0,0,1,0,0,0,0,0,0,34,0,0,228,0],[336,1,0,0,0,1,0,0,0,0,148,0,0,0,0,148] >;

D21⋊C4 in GAP, Magma, Sage, TeX

D_{21}\rtimes C_4
% in TeX

G:=Group("D21:C4");
// GroupNames label

G:=SmallGroup(168,14);
// by ID

G=gap.SmallGroup(168,14);
# by ID

G:=PCGroup([5,-2,-2,-2,-3,-7,20,26,168,3604]);
// Polycyclic

G:=Group<a,b,c|a^21=b^2=c^4=1,b*a*b=a^-1,c*a*c^-1=a^8,c*b*c^-1=a^7*b>;
// generators/relations

Export

Subgroup lattice of D21⋊C4 in TeX
Character table of D21⋊C4 in TeX

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