Copied to
clipboard

?

G = Q16⋊4Q8  order 128 = 27

4th semidirect product of Q16 and Q8 acting via Q8/C4=C2

p-group, metabelian, nilpotent (class 3), monomial

Aliases: Q16⋊4Q8, C42.68C23, C4.1022- (1+4), C8⋊Q8.2C2, C8.7(C2×Q8), (Q82).5C2, C2.45(D4×Q8), C4⋊C4.390D4, Q8.12(C2×Q8), C8⋊3Q8.3C2, C8⋊4Q8.6C2, Q8.Q8.3C2, (C2×Q8).251D4, Q8⋊Q8.2C2, (C4×Q16).17C2, Q8⋊3Q8.7C2, C4.45(C22×Q8), C4⋊C4.276C23, C4⋊C8.146C22, (C2×C8).376C23, (C4×C8).203C22, (C2×C4).579C24, Q16⋊C4.2C2, C4.Q16.11C2, C4⋊Q8.208C22, C8⋊C4.72C22, C4.82(C8.C22), (C4×Q8).206C22, (C2×Q8).413C23, C2.D8.140C22, C4.Q8.119C22, C2.109(D4○SD16), (C2×Q16).164C22, Q8⋊C4.92C22, C22.839(C22×D4), C42.C2.77C22, (C2×C4).649(C2×D4), C2.91(C2×C8.C22), SmallGroup(128,2119)

Series: Derived ►Chief ►Lower central ►Upper central ►Jennings

Derived series
C1 — C2×C4 — Q16⋊4Q8
Chief series
C1 — C2 — C4 — C2×C4 — C42 — C4×Q8 — Q8⋊3Q8 — Q16⋊4Q8
Lower central
C1 — C2 — C2×C4 — Q16⋊4Q8
Upper central
C1 — C22 — C4×Q8 — Q16⋊4Q8
Jennings
C1 — C2 — C2 — C2×C4 — Q16⋊4Q8

Subgroups: 264 in 163 conjugacy classes, 96 normal (38 characteristic)
C1, C2 [×3], C4 [×2], C4 [×2], C4 [×15], C22, C8 [×2], C8 [×3], C2×C4 [×3], C2×C4 [×4], C2×C4 [×8], Q8 [×4], Q8 [×9], C42, C42 [×2], C42 [×6], C4⋊C4 [×3], C4⋊C4 [×6], C4⋊C4 [×14], C2×C8 [×2], C2×C8 [×2], Q16 [×4], C2×Q8 [×3], C2×Q8 [×4], C4×C8, C8⋊C4 [×2], Q8⋊C4 [×2], Q8⋊C4 [×4], C4⋊C8, C4⋊C8 [×2], C4.Q8 [×6], C2.D8, C2.D8 [×2], C4×Q8 [×3], C4×Q8 [×4], C4×Q8 [×2], C42.C2 [×2], C42.C2 [×2], C4⋊Q8 [×2], C4⋊Q8 [×2], C4⋊Q8 [×4], C2×Q16, C4×Q16, Q16⋊C4 [×2], C8⋊4Q8, Q8⋊Q8, Q8⋊Q8 [×2], C4.Q16, Q8.Q8 [×2], C8⋊3Q8, C8⋊Q8 [×2], Q8⋊3Q8, Q82, Q16⋊4Q8

Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], Q8 [×4], C23 [×15], C2×D4 [×6], C2×Q8 [×6], C24, C8.C22 [×2], C22×D4, C22×Q8, 2- (1+4), D4×Q8, C2×C8.C22, D4○SD16, Q16⋊4Q8

Generators and relations
 G = < a,b,c,d | a8=c4=1, b2=a4, d2=c2, bab-1=a-1, ac=ca, dad-1=a5, bc=cb, dbd-1=a4b, dcd-1=c-1 >

Smallest permutation representation
►Regular action on 128 points
Generators in S128
(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)(113 114 115 116 117 118 119 120)(121 122 123 124 125 126 127 128)

(1 55 5 51)(2 54 6 50)(3 53 7 49)(4 52 8 56)(9 123 13 127)(10 122 14 126)(11 121 15 125)(12 128 16 124)(17 107 21 111)(18 106 22 110)(19 105 23 109)(20 112 24 108)(25 113 29 117)(26 120 30 116)(27 119 31 115)(28 118 32 114)(33 75 37 79)(34 74 38 78)(35 73 39 77)(36 80 40 76)(41 69 45 65)(42 68 46 72)(43 67 47 71)(44 66 48 70)(57 99 61 103)(58 98 62 102)(59 97 63 101)(60 104 64 100)(81 95 85 91)(82 94 86 90)(83 93 87 89)(84 92 88 96)

(1 57 42 107)(2 58 43 108)(3 59 44 109)(4 60 45 110)(5 61 46 111)(6 62 47 112)(7 63 48 105)(8 64 41 106)(9 82 115 38)(10 83 116 39)(11 84 117 40)(12 85 118 33)(13 86 119 34)(14 87 120 35)(15 88 113 36)(16 81 114 37)(17 51 103 72)(18 52 104 65)(19 53 97 66)(20 54 98 67)(21 55 99 68)(22 56 100 69)(23 49 101 70)(24 50 102 71)(25 76 121 92)(26 77 122 93)(27 78 123 94)(28 79 124 95)(29 80 125 96)(30 73 126 89)(31 74 127 90)(32 75 128 91)

(1 74 42 90)(2 79 43 95)(3 76 44 92)(4 73 45 89)(5 78 46 94)(6 75 47 91)(7 80 48 96)(8 77 41 93)(9 103 115 17)(10 100 116 22)(11 97 117 19)(12 102 118 24)(13 99 119 21)(14 104 120 18)(15 101 113 23)(16 98 114 20)(25 109 121 59)(26 106 122 64)(27 111 123 61)(28 108 124 58)(29 105 125 63)(30 110 126 60)(31 107 127 57)(32 112 128 62)(33 71 85 50)(34 68 86 55)(35 65 87 52)(36 70 88 49)(37 67 81 54)(38 72 82 51)(39 69 83 56)(40 66 84 53)
 
G:=sub<Sym(128)| (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)(113,114,115,116,117,118,119,120)(121,122,123,124,125,126,127,128), (1,55,5,51)(2,54,6,50)(3,53,7,49)(4,52,8,56)(9,123,13,127)(10,122,14,126)(11,121,15,125)(12,128,16,124)(17,107,21,111)(18,106,22,110)(19,105,23,109)(20,112,24,108)(25,113,29,117)(26,120,30,116)(27,119,31,115)(28,118,32,114)(33,75,37,79)(34,74,38,78)(35,73,39,77)(36,80,40,76)(41,69,45,65)(42,68,46,72)(43,67,47,71)(44,66,48,70)(57,99,61,103)(58,98,62,102)(59,97,63,101)(60,104,64,100)(81,95,85,91)(82,94,86,90)(83,93,87,89)(84,92,88,96), (1,57,42,107)(2,58,43,108)(3,59,44,109)(4,60,45,110)(5,61,46,111)(6,62,47,112)(7,63,48,105)(8,64,41,106)(9,82,115,38)(10,83,116,39)(11,84,117,40)(12,85,118,33)(13,86,119,34)(14,87,120,35)(15,88,113,36)(16,81,114,37)(17,51,103,72)(18,52,104,65)(19,53,97,66)(20,54,98,67)(21,55,99,68)(22,56,100,69)(23,49,101,70)(24,50,102,71)(25,76,121,92)(26,77,122,93)(27,78,123,94)(28,79,124,95)(29,80,125,96)(30,73,126,89)(31,74,127,90)(32,75,128,91), (1,74,42,90)(2,79,43,95)(3,76,44,92)(4,73,45,89)(5,78,46,94)(6,75,47,91)(7,80,48,96)(8,77,41,93)(9,103,115,17)(10,100,116,22)(11,97,117,19)(12,102,118,24)(13,99,119,21)(14,104,120,18)(15,101,113,23)(16,98,114,20)(25,109,121,59)(26,106,122,64)(27,111,123,61)(28,108,124,58)(29,105,125,63)(30,110,126,60)(31,107,127,57)(32,112,128,62)(33,71,85,50)(34,68,86,55)(35,65,87,52)(36,70,88,49)(37,67,81,54)(38,72,82,51)(39,69,83,56)(40,66,84,53)>;
 
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)(81,82,83,84,85,86,87,88)(89,90,91,92,93,94,95,96)(97,98,99,100,101,102,103,104)(105,106,107,108,109,110,111,112)(113,114,115,116,117,118,119,120)(121,122,123,124,125,126,127,128), (1,55,5,51)(2,54,6,50)(3,53,7,49)(4,52,8,56)(9,123,13,127)(10,122,14,126)(11,121,15,125)(12,128,16,124)(17,107,21,111)(18,106,22,110)(19,105,23,109)(20,112,24,108)(25,113,29,117)(26,120,30,116)(27,119,31,115)(28,118,32,114)(33,75,37,79)(34,74,38,78)(35,73,39,77)(36,80,40,76)(41,69,45,65)(42,68,46,72)(43,67,47,71)(44,66,48,70)(57,99,61,103)(58,98,62,102)(59,97,63,101)(60,104,64,100)(81,95,85,91)(82,94,86,90)(83,93,87,89)(84,92,88,96), (1,57,42,107)(2,58,43,108)(3,59,44,109)(4,60,45,110)(5,61,46,111)(6,62,47,112)(7,63,48,105)(8,64,41,106)(9,82,115,38)(10,83,116,39)(11,84,117,40)(12,85,118,33)(13,86,119,34)(14,87,120,35)(15,88,113,36)(16,81,114,37)(17,51,103,72)(18,52,104,65)(19,53,97,66)(20,54,98,67)(21,55,99,68)(22,56,100,69)(23,49,101,70)(24,50,102,71)(25,76,121,92)(26,77,122,93)(27,78,123,94)(28,79,124,95)(29,80,125,96)(30,73,126,89)(31,74,127,90)(32,75,128,91), (1,74,42,90)(2,79,43,95)(3,76,44,92)(4,73,45,89)(5,78,46,94)(6,75,47,91)(7,80,48,96)(8,77,41,93)(9,103,115,17)(10,100,116,22)(11,97,117,19)(12,102,118,24)(13,99,119,21)(14,104,120,18)(15,101,113,23)(16,98,114,20)(25,109,121,59)(26,106,122,64)(27,111,123,61)(28,108,124,58)(29,105,125,63)(30,110,126,60)(31,107,127,57)(32,112,128,62)(33,71,85,50)(34,68,86,55)(35,65,87,52)(36,70,88,49)(37,67,81,54)(38,72,82,51)(39,69,83,56)(40,66,84,53) );
 
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),(81,82,83,84,85,86,87,88),(89,90,91,92,93,94,95,96),(97,98,99,100,101,102,103,104),(105,106,107,108,109,110,111,112),(113,114,115,116,117,118,119,120),(121,122,123,124,125,126,127,128)], [(1,55,5,51),(2,54,6,50),(3,53,7,49),(4,52,8,56),(9,123,13,127),(10,122,14,126),(11,121,15,125),(12,128,16,124),(17,107,21,111),(18,106,22,110),(19,105,23,109),(20,112,24,108),(25,113,29,117),(26,120,30,116),(27,119,31,115),(28,118,32,114),(33,75,37,79),(34,74,38,78),(35,73,39,77),(36,80,40,76),(41,69,45,65),(42,68,46,72),(43,67,47,71),(44,66,48,70),(57,99,61,103),(58,98,62,102),(59,97,63,101),(60,104,64,100),(81,95,85,91),(82,94,86,90),(83,93,87,89),(84,92,88,96)], [(1,57,42,107),(2,58,43,108),(3,59,44,109),(4,60,45,110),(5,61,46,111),(6,62,47,112),(7,63,48,105),(8,64,41,106),(9,82,115,38),(10,83,116,39),(11,84,117,40),(12,85,118,33),(13,86,119,34),(14,87,120,35),(15,88,113,36),(16,81,114,37),(17,51,103,72),(18,52,104,65),(19,53,97,66),(20,54,98,67),(21,55,99,68),(22,56,100,69),(23,49,101,70),(24,50,102,71),(25,76,121,92),(26,77,122,93),(27,78,123,94),(28,79,124,95),(29,80,125,96),(30,73,126,89),(31,74,127,90),(32,75,128,91)], [(1,74,42,90),(2,79,43,95),(3,76,44,92),(4,73,45,89),(5,78,46,94),(6,75,47,91),(7,80,48,96),(8,77,41,93),(9,103,115,17),(10,100,116,22),(11,97,117,19),(12,102,118,24),(13,99,119,21),(14,104,120,18),(15,101,113,23),(16,98,114,20),(25,109,121,59),(26,106,122,64),(27,111,123,61),(28,108,124,58),(29,105,125,63),(30,110,126,60),(31,107,127,57),(32,112,128,62),(33,71,85,50),(34,68,86,55),(35,65,87,52),(36,70,88,49),(37,67,81,54),(38,72,82,51),(39,69,83,56),(40,66,84,53)])
 

Matrix representation ►G ⊆ GL6(𝔽17)

100000
010000
00125010
00512107
0012500
0012101210
,
1600000
0160000
0051134
004151010
004101616
005141215
,
010000
1600000
000010
00116115
0016000
0016101
,
400000
0130000
0034812
00161236
001691013
0036139

G:=sub<GL(6,GF(17))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,5,12,12,0,0,5,12,5,10,0,0,0,10,0,12,0,0,10,7,0,10],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,5,4,4,5,0,0,11,15,10,14,0,0,3,10,16,12,0,0,4,10,16,15],[0,16,0,0,0,0,1,0,0,0,0,0,0,0,0,1,16,16,0,0,0,16,0,1,0,0,1,1,0,0,0,0,0,15,0,1],[4,0,0,0,0,0,0,13,0,0,0,0,0,0,3,16,16,3,0,0,4,12,9,6,0,0,8,3,10,13,0,0,12,6,13,9] >;
 

Character table of Q16⋊4Q8

 class 12A2B2C4A4B4C4D4E4F4G4H4I4J4K4L4M4N4O4P4Q4R4S8A8B8C8D8E8F
 size 11112222444444444888888444488
ρ111111111111111111111111111111    trivial
ρ21111-11-111-1-111-1-11-1111-1-1-1-1-111-11    linear of order 2
ρ31111-11-11-11-11-11-11-11-1-111-1-1-1111-1    linear of order 2
ρ411111111-1-111-1-11111-1-1-1-111111-1-1    linear of order 2
ρ511111111-1-11-1-1-1-1-1-1-11111-11111-1-1    linear of order 2
ρ61111-11-11-11-1-1-111-11-111-1-11-1-1111-1    linear of order 2
ρ71111-11-111-1-1-11-11-11-1-1-1111-1-111-11    linear of order 2
ρ811111111111-111-1-1-1-1-1-1-1-1-1111111    linear of order 2
ρ911111111-1-111-1-1-11-1-1-111-11-1-1-1-111    linear of order 2
ρ101111-11-11-11-11-11111-1-11-11-111-1-1-11    linear of order 2
ρ111111-11-111-1-111-1111-11-11-1-111-1-11-1    linear of order 2
ρ1211111111111111-11-1-11-1-111-1-1-1-1-1-1    linear of order 2
ρ1311111111111-1111-111-111-1-1-1-1-1-1-1-1    linear of order 2
ρ141111-11-111-1-1-11-1-1-1-11-11-11111-1-11-1    linear of order 2
ρ151111-11-11-11-1-1-11-1-1-111-11-1111-1-1-11    linear of order 2
ρ1611111111-1-11-1-1-11-1111-1-11-1-1-1-1-111    linear of order 2
ρ1722222-22-222-20-2-2000000000000000    orthogonal lifted from D4
ρ182222-2-2-2-22-220-22000000000000000    orthogonal lifted from D4
ρ1922222-22-2-2-2-2022000000000000000    orthogonal lifted from D4
ρ202222-2-2-2-2-22202-2000000000000000    orthogonal lifted from D4
ρ212-22-2020-2000-200-2220000002-20000    symplectic lifted from Q8, Schur index 2
ρ222-22-2020-20002002-2-20000002-20000    symplectic lifted from Q8, Schur index 2
ρ232-22-2020-2000-20022-2000000-220000    symplectic lifted from Q8, Schur index 2
ρ242-22-2020-2000200-2-22000000-220000    symplectic lifted from Q8, Schur index 2
ρ254-4-4440-40000000000000000000000    symplectic lifted from C8.C22, Schur index 2
ρ264-44-40-404000000000000000000000    symplectic lifted from 2- (1+4), Schur index 2
ρ274-4-44-4040000000000000000000000    symplectic lifted from C8.C22, Schur index 2
ρ2844-4-40000000000000000000002√-22√-200    complex lifted from D4○SD16
ρ2944-4-40000000000000000000002√-22√-200    complex lifted from D4○SD16

In GAP, Magma, Sage, TeX 

Q_{16}\rtimes_4Q_8
 
% in TeX
 
G:=Group("Q16:4Q8");
 
// GroupNames label
 
G:=SmallGroup(128,2119);
 
// by ID
 
G=gap.SmallGroup(128,2119);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,-2,560,253,120,758,723,352,346,80,4037,1027,124]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^8=c^4=1,b^2=a^4,d^2=c^2,b*a*b^-1=a^-1,a*c=c*a,d*a*d^-1=a^5,b*c=c*b,d*b*d^-1=a^4*b,d*c*d^-1=c^-1>;
 
// generators/relations
 

׿
×
⋊
ℤ
𝔽
○
≀