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G = C87Q16order 128 = 27

1st semidirect product of C8 and Q16 acting via Q16/Q8=C2

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

Aliases: C87Q16, Q8.1D8, C42.225C23, C4.34(C2×D8), (C8×Q8).5C2, C81C8.7C2, C4⋊C4.200D4, (C2×C8).344D4, C82Q8.4C2, C4.40(C2×Q16), C4.87(C4○D8), C2.9(C87D4), C4⋊C8.22C22, (C4×C8).54C22, (C2×Q8).150D4, C42Q16.5C2, C4⋊Q8.48C22, C2.7(C42Q16), C4.10D8.3C2, C2.9(D4.5D4), (C4×Q8).265C22, C4.113(C8.C22), C22.186(C4⋊D4), (C2×C4).10(C4○D4), (C2×C4).1260(C2×D4), SmallGroup(128,406)

Series: Derived Chief Lower central Upper central Jennings

C1C42 — C87Q16
C1C2C22C2×C4C42C4×Q8C8×Q8 — C87Q16
C1C22C42 — C87Q16
C1C22C42 — C87Q16
C1C22C22C42 — C87Q16

Generators and relations for C87Q16
 G = < a,b,c | a8=b8=1, c2=b4, bab-1=a-1, ac=ca, cbc-1=b-1 >

Subgroups: 160 in 78 conjugacy classes, 36 normal (24 characteristic)
C1, C2 [×3], C4 [×4], C4 [×6], C22, C8 [×2], C8 [×4], C2×C4 [×3], C2×C4 [×4], Q8 [×2], Q8 [×5], C42, C42, C4⋊C4, C4⋊C4 [×5], C2×C8 [×2], C2×C8 [×3], Q16 [×4], C2×Q8, C2×Q8 [×2], C4×C8, C4×C8, Q8⋊C4 [×2], C4⋊C8, C4⋊C8 [×2], C4⋊C8, C2.D8 [×2], C4×Q8, C4⋊Q8 [×2], C2×Q16 [×2], C4.10D8 [×2], C81C8, C8×Q8, C42Q16 [×2], C82Q8, C87Q16
Quotients: C1, C2 [×7], C22 [×7], D4 [×4], C23, D8 [×2], Q16 [×2], C2×D4 [×2], C4○D4, C4⋊D4, C2×D8, C2×Q16, C4○D8, C8.C22, C42Q16, C87D4, D4.5D4, C87Q16

Character table of C87Q16

 class 12A2B2C4A4B4C4D4E4F4G4H4I4J4K8A8B8C8D8E8F8G8H8I8J8K8L8M8N
 size 1111222244444161622224444448888
ρ111111111111111111111111111111    trivial
ρ211111111-1-1-11-1-11-1-1-1-1111-1-11-11-11    linear of order 2
ρ31111111111111-11-1-1-1-1-1-1-1-1-1-11-11-1    linear of order 2
ρ411111111-1-1-11-1111111-1-1-111-1-1-1-1-1    linear of order 2
ρ511111111111111-1-1-1-1-1-1-1-1-1-1-1-11-11    linear of order 2
ρ611111111-1-1-11-1-1-11111-1-1-111-11111    linear of order 2
ρ71111111111111-1-11111111111-1-1-1-1    linear of order 2
ρ811111111-1-1-11-11-1-1-1-1-1111-1-111-11-1    linear of order 2
ρ922222-22-2000-2000-2-2-2-20002200000    orthogonal lifted from D4
ρ102222-22-222-2-2-220000000000000000    orthogonal lifted from D4
ρ112222-22-22-222-2-20000000000000000    orthogonal lifted from D4
ρ1222222-22-2000-20002222000-2-200000    orthogonal lifted from D4
ρ1322-2-2020-20-220000-222-22-2-22-220000    orthogonal lifted from D8
ρ1422-2-2020-202-200002-2-222-2-2-2220000    orthogonal lifted from D8
ρ1522-2-2020-202-20000-222-2-2222-2-20000    orthogonal lifted from D8
ρ1622-2-2020-20-2200002-2-22-222-22-20000    orthogonal lifted from D8
ρ172-22-2-20200000000-22-220000002-2-22    symplectic lifted from Q16, Schur index 2
ρ182-22-2-202000000002-22-2000000-2-222    symplectic lifted from Q16, Schur index 2
ρ192-22-2-202000000002-22-200000022-2-2    symplectic lifted from Q16, Schur index 2
ρ202-22-2-20200000000-22-22000000-222-2    symplectic lifted from Q16, Schur index 2
ρ2122-2-20-2022i000-2i002-2-22-2--2-22-2--20000    complex lifted from C4○D8
ρ2222-2-20-202-2i0002i00-222-2-2--2-2-22--20000    complex lifted from C4○D8
ρ2322-2-20-2022i000-2i00-222-2--2-2--2-22-20000    complex lifted from C4○D8
ρ2422-2-20-202-2i0002i002-2-22--2-2--22-2-20000    complex lifted from C4○D8
ρ252222-2-2-2-2000200000002i2i-2i00-2i0000    complex lifted from C4○D4
ρ262222-2-2-2-200020000000-2i-2i2i002i0000    complex lifted from C4○D4
ρ274-44-440-40000000000000000000000    symplectic lifted from C8.C22, Schur index 2
ρ284-4-44000000000002222-22-220000000000    symplectic lifted from D4.5D4, Schur index 2
ρ294-4-4400000000000-22-2222220000000000    symplectic lifted from D4.5D4, Schur index 2

Smallest permutation representation of C87Q16
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)
(1 115 19 16 57 46 73 87)(2 114 20 15 58 45 74 86)(3 113 21 14 59 44 75 85)(4 120 22 13 60 43 76 84)(5 119 23 12 61 42 77 83)(6 118 24 11 62 41 78 82)(7 117 17 10 63 48 79 81)(8 116 18 9 64 47 80 88)(25 123 51 111 71 89 101 36)(26 122 52 110 72 96 102 35)(27 121 53 109 65 95 103 34)(28 128 54 108 66 94 104 33)(29 127 55 107 67 93 97 40)(30 126 56 106 68 92 98 39)(31 125 49 105 69 91 99 38)(32 124 50 112 70 90 100 37)
(1 97 57 55)(2 98 58 56)(3 99 59 49)(4 100 60 50)(5 101 61 51)(6 102 62 52)(7 103 63 53)(8 104 64 54)(9 108 88 33)(10 109 81 34)(11 110 82 35)(12 111 83 36)(13 112 84 37)(14 105 85 38)(15 106 86 39)(16 107 87 40)(17 65 79 27)(18 66 80 28)(19 67 73 29)(20 68 74 30)(21 69 75 31)(22 70 76 32)(23 71 77 25)(24 72 78 26)(41 122 118 96)(42 123 119 89)(43 124 120 90)(44 125 113 91)(45 126 114 92)(46 127 115 93)(47 128 116 94)(48 121 117 95)

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

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

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

Matrix representation of C87Q16 in GL4(𝔽17) generated by

2000
0900
00160
00016
,
0100
16000
001111
0030
,
1000
01600
00113
00166
G:=sub<GL(4,GF(17))| [2,0,0,0,0,9,0,0,0,0,16,0,0,0,0,16],[0,16,0,0,1,0,0,0,0,0,11,3,0,0,11,0],[1,0,0,0,0,16,0,0,0,0,11,16,0,0,3,6] >;

C87Q16 in GAP, Magma, Sage, TeX

C_8\rtimes_7Q_{16}
% in TeX

G:=Group("C8:7Q16");
// GroupNames label

G:=SmallGroup(128,406);
// by ID

G=gap.SmallGroup(128,406);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,2,448,141,288,422,352,1123,136,2804,718,172]);
// Polycyclic

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

Export

Character table of C87Q16 in TeX

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