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G = D8.9D6order 192 = 26·3

4th non-split extension by D8 of D6 acting via D6/C6=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D8.9D6, C24.33D4, C12.52D8, Q16.10D6, C24.31C23, Dic12.14C22, D8.S3:6C2, C4oD8.3S3, C3:Q32:6C2, (C2xC8).99D6, (C2xC6).11D8, C6.70(C2xD8), C8.8(C3:D4), C3:5(Q32:C2), C3:C16.4C22, C12.C8:7C2, C4.25(D4:S3), C12.193(C2xD4), (C2xC12).187D4, C8.37(C22xS3), (C3xD8).9C22, (C2xDic12):22C2, C22.6(D4:S3), (C2xC24).105C22, (C3xQ16).10C22, C2.25(C2xD4:S3), (C3xC4oD8).4C2, C4.19(C2xC3:D4), (C2xC4).82(C3:D4), SmallGroup(192,754)

Series: Derived Chief Lower central Upper central

C1C24 — D8.9D6
C1C3C6C12C24Dic12C2xDic12 — D8.9D6
C3C6C12C24 — D8.9D6
C1C2C2xC4C2xC8C4oD8

Generators and relations for D8.9D6
 G = < a,b,c,d | a8=b2=c6=1, d2=a4, bab=dad-1=a-1, ac=ca, cbc-1=a4b, dbd-1=ab, dcd-1=c-1 >

Subgroups: 232 in 82 conjugacy classes, 35 normal (27 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C6, C6, C8, C2xC4, C2xC4, D4, Q8, Dic3, C12, C12, C2xC6, C2xC6, C16, C2xC8, D8, SD16, Q16, Q16, C2xQ8, C4oD4, C24, Dic6, C2xDic3, C2xC12, C2xC12, C3xD4, C3xQ8, M5(2), SD32, Q32, C2xQ16, C4oD8, C3:C16, Dic12, Dic12, C2xC24, C3xD8, C3xSD16, C3xQ16, C2xDic6, C3xC4oD4, Q32:C2, C12.C8, D8.S3, C3:Q32, C2xDic12, C3xC4oD8, D8.9D6
Quotients: C1, C2, C22, S3, D4, C23, D6, D8, C2xD4, C3:D4, C22xS3, C2xD8, D4:S3, C2xC3:D4, Q32:C2, C2xD4:S3, D8.9D6

Character table of D8.9D6

 class 12A2B2C34A4B4C4D4E6A6B6C6D8A8B8C12A12B12C12D12E16A16B16C16D24A24B24C24D
 size 112822282424248822422488121212124444
ρ1111111111111111111111111111111    trivial
ρ211-1-11-1111-11-1-1-111-1-1-1111-111-1-11-11    linear of order 2
ρ311111111-1-1111111111111-1-1-1-11111    linear of order 2
ρ411-1-11-111-111-1-1-111-1-1-11111-1-11-11-11    linear of order 2
ρ5111-1111-11111-1-1111111-1-1-1-1-1-11111    linear of order 2
ρ611-111-11-11-11-11111-1-1-11-1-11-1-11-11-11    linear of order 2
ρ7111-1111-1-1-111-1-1111111-1-111111111    linear of order 2
ρ811-111-11-1-111-11111-1-1-11-1-1-111-1-11-11    linear of order 2
ρ9222-2-122-200-1-111222-1-1-1110000-1-1-1-1    orthogonal lifted from D6
ρ1022-2-2-1-22200-111122-211-1-1-100001-11-1    orthogonal lifted from D6
ρ1122202220002200-2-2-2222000000-2-2-2-2    orthogonal lifted from D4
ρ122222-122200-1-1-1-1222-1-1-1-1-10000-1-1-1-1    orthogonal lifted from S3
ρ1322-22-1-22-200-11-1-122-211-11100001-11-1    orthogonal lifted from D6
ρ1422-202-220002-200-2-22-2-220000002-22-2    orthogonal lifted from D4
ρ1522202-2-20002200000-2-2-20022-2-20000    orthogonal lifted from D8
ρ1622-2022-20002-20000022-2002-22-20000    orthogonal lifted from D8
ρ1722202-2-20002200000-2-2-200-2-2220000    orthogonal lifted from D8
ρ1822-2022-20002-20000022-200-22-220000    orthogonal lifted from D8
ρ1922-20-1-22000-11--3-3-2-2211-1--3-30000-11-11    complex lifted from C3:D4
ρ2022-20-1-22000-11-3--3-2-2211-1-3--30000-11-11    complex lifted from C3:D4
ρ212220-122000-1-1--3-3-2-2-2-1-1-1-3--300001111    complex lifted from C3:D4
ρ222220-122000-1-1-3--3-2-2-2-1-1-1--3-300001111    complex lifted from C3:D4
ρ2344-40-24-4000-2200000-2-220000000000    orthogonal lifted from D4:S3, Schur index 2
ρ244440-2-4-4000-2-2000002220000000000    orthogonal lifted from D4:S3, Schur index 2
ρ254-400400000-400022-2200000000000-22022    symplectic lifted from Q32:C2, Schur index 2
ρ264-400400000-4000-222200000000000220-22    symplectic lifted from Q32:C2, Schur index 2
ρ274-400-200000200022-220-2323000000062-6-2    symplectic faithful, Schur index 2
ρ284-400-200000200022-22023-230000000-626-2    symplectic faithful, Schur index 2
ρ294-400-2000002000-2222023-2300000006-2-62    symplectic faithful, Schur index 2
ρ304-400-2000002000-22220-23230000000-6-262    symplectic faithful, Schur index 2

Smallest permutation representation of D8.9D6
On 96 points
Generators in S96
(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)
(1 8)(2 7)(3 6)(4 5)(9 15)(10 14)(11 13)(18 24)(19 23)(20 22)(25 30)(26 29)(27 28)(31 32)(33 36)(34 35)(37 40)(38 39)(41 46)(42 45)(43 44)(47 48)(49 52)(50 51)(53 56)(54 55)(57 62)(58 61)(59 60)(63 64)(66 72)(67 71)(68 70)(73 75)(76 80)(77 79)(81 85)(82 84)(86 88)(89 93)(90 92)(94 96)
(1 46 39 62 32 49)(2 47 40 63 25 50)(3 48 33 64 26 51)(4 41 34 57 27 52)(5 42 35 58 28 53)(6 43 36 59 29 54)(7 44 37 60 30 55)(8 45 38 61 31 56)(9 73 96 82 22 72)(10 74 89 83 23 65)(11 75 90 84 24 66)(12 76 91 85 17 67)(13 77 92 86 18 68)(14 78 93 87 19 69)(15 79 94 88 20 70)(16 80 95 81 21 71)
(1 94 5 90)(2 93 6 89)(3 92 7 96)(4 91 8 95)(9 33 13 37)(10 40 14 36)(11 39 15 35)(12 38 16 34)(17 31 21 27)(18 30 22 26)(19 29 23 25)(20 28 24 32)(41 76 45 80)(42 75 46 79)(43 74 47 78)(44 73 48 77)(49 88 53 84)(50 87 54 83)(51 86 55 82)(52 85 56 81)(57 67 61 71)(58 66 62 70)(59 65 63 69)(60 72 64 68)

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

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

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

Matrix representation of D8.9D6 in GL4(F97) generated by

70090
46149446
536500
7007
,
7007
510351
536500
70090
,
52294381
90391122
106960
682900
,
73618689
74164894
21758237
74426823
G:=sub<GL(4,GF(97))| [7,46,53,7,0,14,65,0,0,94,0,0,90,46,0,7],[7,51,53,7,0,0,65,0,0,3,0,0,7,51,0,90],[52,90,10,68,29,39,69,29,43,11,6,0,81,22,0,0],[73,74,21,74,61,16,75,42,86,48,82,68,89,94,37,23] >;

D8.9D6 in GAP, Magma, Sage, TeX

D_8._9D_6
% in TeX

G:=Group("D8.9D6");
// GroupNames label

G:=SmallGroup(192,754);
// by ID

G=gap.SmallGroup(192,754);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,224,254,387,675,185,192,1684,438,102,6278]);
// Polycyclic

G:=Group<a,b,c,d|a^8=b^2=c^6=1,d^2=a^4,b*a*b=d*a*d^-1=a^-1,a*c=c*a,c*b*c^-1=a^4*b,d*b*d^-1=a*b,d*c*d^-1=c^-1>;
// generators/relations

Export

Character table of D8.9D6 in TeX

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