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G = Q16⋊5Q8  order 128 = 27

5th semidirect product of Q16 and Q8 acting via Q8/C4=C2

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

Aliases: Q16⋊5Q8, C42.71C23, C4.1052- (1+4), C8⋊Q8.3C2, C2.48(D4×Q8), C8.10(C2×Q8), C4⋊C4.393D4, Q8.14(C2×Q8), C8⋊4Q8.7C2, Q8.Q8.4C2, (C2×Q8).142D4, (C4×Q16).18C2, Q8⋊3Q8.8C2, C2.69(Q8○D8), C8.5Q8.8C2, C4.48(C22×Q8), C4⋊C8.149C22, C4⋊C4.279C23, (C2×C8).217C23, (C2×C4).582C24, (C4×C8).206C22, C4.Q16.12C2, Q16⋊C4.3C2, C4⋊Q8.211C22, C4.Q8.80C22, C2.D8.77C22, C8⋊C4.75C22, (C4×Q8).209C22, (C2×Q8).415C23, (C2×Q16).165C22, Q8⋊C4.93C22, C22.842(C22×D4), C42.C2.80C22, C2.107(D8⋊C22), (C2×C4).652(C2×D4), SmallGroup(128,2122)

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

Derived series
C1 — C2×C4 — Q16⋊5Q8
Chief series
C1 — C2 — C4 — C2×C4 — C42 — C4×Q8 — Q8⋊3Q8 — Q16⋊5Q8
Lower central
C1 — C2 — C2×C4 — Q16⋊5Q8
Upper central
C1 — C22 — C4×Q8 — Q16⋊5Q8
Jennings
C1 — C2 — C2 — C2×C4 — Q16⋊5Q8

Subgroups: 248 in 160 conjugacy classes, 94 normal (26 characteristic)
C1, C2 [×3], C4 [×2], C4 [×16], C22, C8 [×2], C8 [×3], C2×C4 [×3], C2×C4 [×4], C2×C4 [×8], Q8 [×4], Q8 [×7], C42, C42 [×2], C42 [×6], C4⋊C4, C4⋊C4 [×8], C4⋊C4 [×16], C2×C8 [×2], C2×C8 [×2], Q16 [×4], C2×Q8, C2×Q8 [×2], C2×Q8 [×2], C4×C8, C8⋊C4 [×2], Q8⋊C4 [×6], C4⋊C8, C4⋊C8 [×2], C4.Q8 [×4], C2.D8 [×3], C2.D8 [×2], C4×Q8, C4×Q8 [×6], C4×Q8 [×2], C42.C2 [×4], C42.C2 [×4], C4⋊Q8 [×2], C4⋊Q8 [×2], C2×Q16, C4×Q16, Q16⋊C4 [×2], C8⋊4Q8, C4.Q16 [×2], Q8.Q8 [×4], C8.5Q8, C8⋊Q8 [×2], Q8⋊3Q8 [×2], Q16⋊5Q8

Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], Q8 [×4], C23 [×15], C2×D4 [×6], C2×Q8 [×6], C24, C22×D4, C22×Q8, 2- (1+4), D4×Q8, D8⋊C22, Q8○D8, Q16⋊5Q8

Generators and relations
 G = < a,b,c,d | a8=c4=1, b2=a4, d2=c2, bab-1=a-1, ac=ca, dad-1=a5, cbc-1=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 51 5 55)(2 50 6 54)(3 49 7 53)(4 56 8 52)(9 126 13 122)(10 125 14 121)(11 124 15 128)(12 123 16 127)(17 107 21 111)(18 106 22 110)(19 105 23 109)(20 112 24 108)(25 104 29 100)(26 103 30 99)(27 102 31 98)(28 101 32 97)(33 63 37 59)(34 62 38 58)(35 61 39 57)(36 60 40 64)(41 70 45 66)(42 69 46 65)(43 68 47 72)(44 67 48 71)(73 114 77 118)(74 113 78 117)(75 120 79 116)(76 119 80 115)(81 91 85 95)(82 90 86 94)(83 89 87 93)(84 96 88 92)

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

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

Matrix representation ►G ⊆ GL6(𝔽17)

1600000
0160000
000020
000009
0015000
000800
,
100000
010000
000100
0016000
0000016
000010
,
420000
0130000
001000
0001600
000010
0000016
,
15100000
820000
000010
000001
001000
000100

G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,15,0,0,0,0,0,0,8,0,0,2,0,0,0,0,0,0,9,0,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,16,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,16,0],[4,0,0,0,0,0,2,13,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,16],[15,8,0,0,0,0,10,2,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,1,0,0] >;
 

Character table of Q16⋊5Q8

 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
ρ254-44-40-404000000000000000000000    symplectic lifted from 2- (1+4), Schur index 2
ρ2644-4-40000000000000000000002√22√200    symplectic lifted from Q8○D8, Schur index 2
ρ2744-4-40000000000000000000002√22√200    symplectic lifted from Q8○D8, Schur index 2
ρ284-4-444i04i0000000000000000000000    complex lifted from D8⋊C22
ρ294-4-444i04i0000000000000000000000    complex lifted from D8⋊C22

In GAP, Magma, Sage, TeX 

Q_{16}\rtimes_5Q_8
 
% in TeX
 
G:=Group("Q16:5Q8");
 
// GroupNames label
 
G:=SmallGroup(128,2122);
 
// by ID
 
G=gap.SmallGroup(128,2122);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,-2,112,253,120,758,723,352,346,304,4037,1027,124]);
 
// 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,c*b*c^-1=d*b*d^-1=a^4*b,d*c*d^-1=c^-1>;
 
// generators/relations
 

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