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G = Q8.D21order 336 = 24·3·7

The non-split extension by Q8 of D21 acting via D21/C7=S3

non-abelian, soluble

Aliases: Q8.D21, C14.1S4, C7⋊CSU2(𝔽3), SL2(𝔽3).D7, C2.2(C7⋊S4), (C7×Q8).1S3, (C7×SL2(𝔽3)).1C2, SmallGroup(336,118)

Series: Derived Chief Lower central Upper central

C1C2Q8C7×SL2(𝔽3) — Q8.D21
C1C2Q8C7×Q8C7×SL2(𝔽3) — Q8.D21
C7×SL2(𝔽3) — Q8.D21
C1C2

Generators and relations for Q8.D21
 G = < a,b,c,d | a4=c21=1, b2=d2=a2, bab-1=a-1, cac-1=b, dad-1=a-1b, cbc-1=ab, dbd-1=a2b, dcd-1=c-1 >

4C3
3C4
42C4
4C6
4C21
21Q8
21C8
28Dic3
3C28
6Dic7
4C42
21Q16
3Dic14
3C7⋊C8
4Dic21
7CSU2(𝔽3)
3C7⋊Q16

Character table of Q8.D21

 class 1234A4B67A7B7C8A8B14A14B14C21A21B21C21D21E21F28A28B28C42A42B42C42D42E42F
 size 11868482224242222888888121212888888
ρ111111111111111111111111111111    trivial
ρ21111-11111-1-1111111111111111111    linear of order 2
ρ322-120-122200222-1-1-1-1-1-1222-1-1-1-1-1-1    orthogonal lifted from S3
ρ4222202ζ7473ζ7572ζ76700ζ7572ζ7473ζ767ζ7572ζ767ζ7572ζ767ζ7473ζ7473ζ7572ζ767ζ7473ζ7572ζ7473ζ7473ζ7572ζ767ζ767    orthogonal lifted from D7
ρ5222202ζ767ζ7473ζ757200ζ7473ζ767ζ7572ζ7473ζ7572ζ7473ζ7572ζ767ζ767ζ7473ζ7572ζ767ζ7473ζ767ζ767ζ7473ζ7572ζ7572    orthogonal lifted from D7
ρ6222202ζ7572ζ767ζ747300ζ767ζ7572ζ7473ζ767ζ7473ζ767ζ7473ζ7572ζ7572ζ767ζ7473ζ7572ζ767ζ7572ζ7572ζ767ζ7473ζ7473    orthogonal lifted from D7
ρ722-120-1ζ767ζ7473ζ757200ζ7473ζ767ζ7572ζ32ζ7432ζ737332ζ7532ζ727532ζ7432ζ73743ζ753ζ72753ζ763ζ776ζ3ζ763ζ77ζ7473ζ7572ζ76732ζ7432ζ7374ζ3ζ763ζ773ζ763ζ776ζ32ζ7432ζ737332ζ7532ζ72753ζ753ζ7275    orthogonal lifted from D21
ρ822-120-1ζ7572ζ767ζ747300ζ767ζ7572ζ74733ζ763ζ77632ζ7432ζ7374ζ3ζ763ζ77ζ32ζ7432ζ737332ζ7532ζ72753ζ753ζ7275ζ767ζ7473ζ7572ζ3ζ763ζ773ζ753ζ727532ζ7532ζ72753ζ763ζ77632ζ7432ζ7374ζ32ζ7432ζ7373    orthogonal lifted from D21
ρ922-120-1ζ7473ζ7572ζ76700ζ7572ζ7473ζ7673ζ753ζ72753ζ763ζ77632ζ7532ζ7275ζ3ζ763ζ77ζ32ζ7432ζ737332ζ7432ζ7374ζ7572ζ767ζ747332ζ7532ζ727532ζ7432ζ7374ζ32ζ7432ζ73733ζ753ζ72753ζ763ζ776ζ3ζ763ζ77    orthogonal lifted from D21
ρ1022-120-1ζ767ζ7473ζ757200ζ7473ζ767ζ757232ζ7432ζ73743ζ753ζ7275ζ32ζ7432ζ737332ζ7532ζ7275ζ3ζ763ζ773ζ763ζ776ζ7473ζ7572ζ767ζ32ζ7432ζ73733ζ763ζ776ζ3ζ763ζ7732ζ7432ζ73743ζ753ζ727532ζ7532ζ7275    orthogonal lifted from D21
ρ1122-120-1ζ7572ζ767ζ747300ζ767ζ7572ζ7473ζ3ζ763ζ77ζ32ζ7432ζ73733ζ763ζ77632ζ7432ζ73743ζ753ζ727532ζ7532ζ7275ζ767ζ7473ζ75723ζ763ζ77632ζ7532ζ72753ζ753ζ7275ζ3ζ763ζ77ζ32ζ7432ζ737332ζ7432ζ7374    orthogonal lifted from D21
ρ1222-120-1ζ7473ζ7572ζ76700ζ7572ζ7473ζ76732ζ7532ζ7275ζ3ζ763ζ773ζ753ζ72753ζ763ζ77632ζ7432ζ7374ζ32ζ7432ζ7373ζ7572ζ767ζ74733ζ753ζ7275ζ32ζ7432ζ737332ζ7432ζ737432ζ7532ζ7275ζ3ζ763ζ773ζ763ζ776    orthogonal lifted from D21
ρ132-2-10012222-2-2-2-2-1-1-1-1-1-1000111111    symplectic lifted from CSU2(𝔽3), Schur index 2
ρ142-2-1001222-22-2-2-2-1-1-1-1-1-1000111111    symplectic lifted from CSU2(𝔽3), Schur index 2
ρ15330-1-1033311333000000-1-1-1000000    orthogonal lifted from S4
ρ16330-110333-1-1333000000-1-1-1000000    orthogonal lifted from S4
ρ174-4100-144400-4-4-4111111000-1-1-1-1-1-1    symplectic lifted from CSU2(𝔽3), Schur index 2
ρ184-4-200275+2ζ7276+2ζ774+2ζ7300-2ζ76-2ζ7-2ζ75-2ζ72-2ζ74-2ζ737677473767747375727572000ζ767ζ7572ζ7572ζ767ζ7473ζ7473    symplectic faithful, Schur index 2
ρ194-4-200276+2ζ774+2ζ7375+2ζ7200-2ζ74-2ζ73-2ζ76-2ζ7-2ζ75-2ζ727473757274737572767767000ζ7473ζ767ζ767ζ7473ζ7572ζ7572    symplectic faithful, Schur index 2
ρ204-4-200274+2ζ7375+2ζ7276+2ζ700-2ζ75-2ζ72-2ζ74-2ζ73-2ζ76-2ζ77572767757276774737473000ζ7572ζ7473ζ7473ζ7572ζ767ζ767    symplectic faithful, Schur index 2
ρ214-4100-174+2ζ7375+2ζ7276+2ζ700-2ζ75-2ζ72-2ζ74-2ζ73-2ζ76-2ζ73ζ753ζ72723ζ763ζ7732ζ7532ζ7272ζ3ζ763ζ776ζ32ζ7432ζ737432ζ7432ζ73730003ζ753ζ7275ζ32ζ7432ζ737332ζ7432ζ737432ζ7532ζ7275ζ3ζ763ζ773ζ763ζ776    symplectic faithful, Schur index 2
ρ224-4100-176+2ζ774+2ζ7375+2ζ7200-2ζ74-2ζ73-2ζ76-2ζ7-2ζ75-2ζ72ζ32ζ7432ζ737432ζ7532ζ727232ζ7432ζ73733ζ753ζ72723ζ763ζ77ζ3ζ763ζ776000ζ32ζ7432ζ73733ζ763ζ776ζ3ζ763ζ7732ζ7432ζ73743ζ753ζ727532ζ7532ζ7275    symplectic faithful, Schur index 2
ρ234-4100-175+2ζ7276+2ζ774+2ζ7300-2ζ76-2ζ7-2ζ75-2ζ72-2ζ74-2ζ73ζ3ζ763ζ776ζ32ζ7432ζ73743ζ763ζ7732ζ7432ζ73733ζ753ζ727232ζ7532ζ7272000ζ3ζ763ζ773ζ753ζ727532ζ7532ζ72753ζ763ζ77632ζ7432ζ7374ζ32ζ7432ζ7373    symplectic faithful, Schur index 2
ρ244-4100-174+2ζ7375+2ζ7276+2ζ700-2ζ75-2ζ72-2ζ74-2ζ73-2ζ76-2ζ732ζ7532ζ7272ζ3ζ763ζ7763ζ753ζ72723ζ763ζ7732ζ7432ζ7373ζ32ζ7432ζ737400032ζ7532ζ727532ζ7432ζ7374ζ32ζ7432ζ73733ζ753ζ72753ζ763ζ776ζ3ζ763ζ77    symplectic faithful, Schur index 2
ρ254-4100-176+2ζ774+2ζ7375+2ζ7200-2ζ74-2ζ73-2ζ76-2ζ7-2ζ75-2ζ7232ζ7432ζ73733ζ753ζ7272ζ32ζ7432ζ737432ζ7532ζ7272ζ3ζ763ζ7763ζ763ζ7700032ζ7432ζ7374ζ3ζ763ζ773ζ763ζ776ζ32ζ7432ζ737332ζ7532ζ72753ζ753ζ7275    symplectic faithful, Schur index 2
ρ264-4100-175+2ζ7276+2ζ774+2ζ7300-2ζ76-2ζ7-2ζ75-2ζ72-2ζ74-2ζ733ζ763ζ7732ζ7432ζ7373ζ3ζ763ζ776ζ32ζ7432ζ737432ζ7532ζ72723ζ753ζ72720003ζ763ζ77632ζ7532ζ72753ζ753ζ7275ζ3ζ763ζ77ζ32ζ7432ζ737332ζ7432ζ7374    symplectic faithful, Schur index 2
ρ27660-20075+3ζ7276+3ζ774+3ζ730076+3ζ775+3ζ7274+3ζ7300000076774737572000000    orthogonal lifted from C7⋊S4
ρ28660-20076+3ζ774+3ζ7375+3ζ720074+3ζ7376+3ζ775+3ζ7200000074737572767000000    orthogonal lifted from C7⋊S4
ρ29660-20074+3ζ7375+3ζ7276+3ζ70075+3ζ7274+3ζ7376+3ζ700000075727677473000000    orthogonal lifted from C7⋊S4

Smallest permutation representation of Q8.D21
On 112 points
Generators in S112
(1 83 21 57)(2 77 15 51)(3 71 16 66)(4 86 17 60)(5 80 18 54)(6 74 19 69)(7 89 20 63)(8 31 25 94)(9 46 26 109)(10 40 27 103)(11 34 28 97)(12 49 22 112)(13 43 23 106)(14 37 24 100)(29 36 92 99)(30 107 93 44)(32 39 95 102)(33 110 96 47)(35 42 98 105)(38 45 101 108)(41 48 104 111)(50 90 76 64)(52 59 78 85)(53 72 79 67)(55 62 81 88)(56 75 82 70)(58 65 84 91)(61 68 87 73)
(1 76 21 50)(2 91 15 65)(3 85 16 59)(4 79 17 53)(5 73 18 68)(6 88 19 62)(7 82 20 56)(8 45 25 108)(9 39 26 102)(10 33 27 96)(11 48 28 111)(12 42 22 105)(13 36 23 99)(14 30 24 93)(29 106 92 43)(31 38 94 101)(32 109 95 46)(34 41 97 104)(35 112 98 49)(37 44 100 107)(40 47 103 110)(51 58 77 84)(52 71 78 66)(54 61 80 87)(55 74 81 69)(57 64 83 90)(60 67 86 72)(63 70 89 75)
(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 85 86 87 88 89 90 91)(92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112)
(1 27 21 10)(2 26 15 9)(3 25 16 8)(4 24 17 14)(5 23 18 13)(6 22 19 12)(7 28 20 11)(29 54 92 80)(30 53 93 79)(31 52 94 78)(32 51 95 77)(33 50 96 76)(34 70 97 75)(35 69 98 74)(36 68 99 73)(37 67 100 72)(38 66 101 71)(39 65 102 91)(40 64 103 90)(41 63 104 89)(42 62 105 88)(43 61 106 87)(44 60 107 86)(45 59 108 85)(46 58 109 84)(47 57 110 83)(48 56 111 82)(49 55 112 81)

G:=sub<Sym(112)| (1,83,21,57)(2,77,15,51)(3,71,16,66)(4,86,17,60)(5,80,18,54)(6,74,19,69)(7,89,20,63)(8,31,25,94)(9,46,26,109)(10,40,27,103)(11,34,28,97)(12,49,22,112)(13,43,23,106)(14,37,24,100)(29,36,92,99)(30,107,93,44)(32,39,95,102)(33,110,96,47)(35,42,98,105)(38,45,101,108)(41,48,104,111)(50,90,76,64)(52,59,78,85)(53,72,79,67)(55,62,81,88)(56,75,82,70)(58,65,84,91)(61,68,87,73), (1,76,21,50)(2,91,15,65)(3,85,16,59)(4,79,17,53)(5,73,18,68)(6,88,19,62)(7,82,20,56)(8,45,25,108)(9,39,26,102)(10,33,27,96)(11,48,28,111)(12,42,22,105)(13,36,23,99)(14,30,24,93)(29,106,92,43)(31,38,94,101)(32,109,95,46)(34,41,97,104)(35,112,98,49)(37,44,100,107)(40,47,103,110)(51,58,77,84)(52,71,78,66)(54,61,80,87)(55,74,81,69)(57,64,83,90)(60,67,86,72)(63,70,89,75), (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)(71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91)(92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112), (1,27,21,10)(2,26,15,9)(3,25,16,8)(4,24,17,14)(5,23,18,13)(6,22,19,12)(7,28,20,11)(29,54,92,80)(30,53,93,79)(31,52,94,78)(32,51,95,77)(33,50,96,76)(34,70,97,75)(35,69,98,74)(36,68,99,73)(37,67,100,72)(38,66,101,71)(39,65,102,91)(40,64,103,90)(41,63,104,89)(42,62,105,88)(43,61,106,87)(44,60,107,86)(45,59,108,85)(46,58,109,84)(47,57,110,83)(48,56,111,82)(49,55,112,81)>;

G:=Group( (1,83,21,57)(2,77,15,51)(3,71,16,66)(4,86,17,60)(5,80,18,54)(6,74,19,69)(7,89,20,63)(8,31,25,94)(9,46,26,109)(10,40,27,103)(11,34,28,97)(12,49,22,112)(13,43,23,106)(14,37,24,100)(29,36,92,99)(30,107,93,44)(32,39,95,102)(33,110,96,47)(35,42,98,105)(38,45,101,108)(41,48,104,111)(50,90,76,64)(52,59,78,85)(53,72,79,67)(55,62,81,88)(56,75,82,70)(58,65,84,91)(61,68,87,73), (1,76,21,50)(2,91,15,65)(3,85,16,59)(4,79,17,53)(5,73,18,68)(6,88,19,62)(7,82,20,56)(8,45,25,108)(9,39,26,102)(10,33,27,96)(11,48,28,111)(12,42,22,105)(13,36,23,99)(14,30,24,93)(29,106,92,43)(31,38,94,101)(32,109,95,46)(34,41,97,104)(35,112,98,49)(37,44,100,107)(40,47,103,110)(51,58,77,84)(52,71,78,66)(54,61,80,87)(55,74,81,69)(57,64,83,90)(60,67,86,72)(63,70,89,75), (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)(71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91)(92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112), (1,27,21,10)(2,26,15,9)(3,25,16,8)(4,24,17,14)(5,23,18,13)(6,22,19,12)(7,28,20,11)(29,54,92,80)(30,53,93,79)(31,52,94,78)(32,51,95,77)(33,50,96,76)(34,70,97,75)(35,69,98,74)(36,68,99,73)(37,67,100,72)(38,66,101,71)(39,65,102,91)(40,64,103,90)(41,63,104,89)(42,62,105,88)(43,61,106,87)(44,60,107,86)(45,59,108,85)(46,58,109,84)(47,57,110,83)(48,56,111,82)(49,55,112,81) );

G=PermutationGroup([(1,83,21,57),(2,77,15,51),(3,71,16,66),(4,86,17,60),(5,80,18,54),(6,74,19,69),(7,89,20,63),(8,31,25,94),(9,46,26,109),(10,40,27,103),(11,34,28,97),(12,49,22,112),(13,43,23,106),(14,37,24,100),(29,36,92,99),(30,107,93,44),(32,39,95,102),(33,110,96,47),(35,42,98,105),(38,45,101,108),(41,48,104,111),(50,90,76,64),(52,59,78,85),(53,72,79,67),(55,62,81,88),(56,75,82,70),(58,65,84,91),(61,68,87,73)], [(1,76,21,50),(2,91,15,65),(3,85,16,59),(4,79,17,53),(5,73,18,68),(6,88,19,62),(7,82,20,56),(8,45,25,108),(9,39,26,102),(10,33,27,96),(11,48,28,111),(12,42,22,105),(13,36,23,99),(14,30,24,93),(29,106,92,43),(31,38,94,101),(32,109,95,46),(34,41,97,104),(35,112,98,49),(37,44,100,107),(40,47,103,110),(51,58,77,84),(52,71,78,66),(54,61,80,87),(55,74,81,69),(57,64,83,90),(60,67,86,72),(63,70,89,75)], [(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),(71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91),(92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112)], [(1,27,21,10),(2,26,15,9),(3,25,16,8),(4,24,17,14),(5,23,18,13),(6,22,19,12),(7,28,20,11),(29,54,92,80),(30,53,93,79),(31,52,94,78),(32,51,95,77),(33,50,96,76),(34,70,97,75),(35,69,98,74),(36,68,99,73),(37,67,100,72),(38,66,101,71),(39,65,102,91),(40,64,103,90),(41,63,104,89),(42,62,105,88),(43,61,106,87),(44,60,107,86),(45,59,108,85),(46,58,109,84),(47,57,110,83),(48,56,111,82),(49,55,112,81)])

Matrix representation of Q8.D21 in GL4(𝔽337) generated by

1000
0100
0031235
001125
,
1000
0100
0036326
0026301
,
6323400
10329700
003361
003360
,
123800
13121400
00142119
00261195
G:=sub<GL(4,GF(337))| [1,0,0,0,0,1,0,0,0,0,312,11,0,0,35,25],[1,0,0,0,0,1,0,0,0,0,36,26,0,0,326,301],[63,103,0,0,234,297,0,0,0,0,336,336,0,0,1,0],[123,131,0,0,8,214,0,0,0,0,142,261,0,0,119,195] >;

Q8.D21 in GAP, Magma, Sage, TeX

Q_8.D_{21}
% in TeX

G:=Group("Q8.D21");
// GroupNames label

G:=SmallGroup(336,118);
// by ID

G=gap.SmallGroup(336,118);
# by ID

G:=PCGroup([6,-2,-3,-7,-2,2,-2,1008,49,650,2019,3033,117,1264,1900,202,88]);
// Polycyclic

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

Export

Subgroup lattice of Q8.D21 in TeX
Character table of Q8.D21 in TeX

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