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G = D8:2Dic3order 192 = 26·3

2nd semidirect product of D8 and Dic3 acting via Dic3/C6=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D8:2Dic3, C24.40D4, Q16:2Dic3, C12.37SD16, (C3xD8):4C4, (C2xC6).4D8, (C3xQ16):4C4, C4oD8.2S3, (C2xC8).51D6, C3:3(D8:2C4), C24.27(C2xC4), C8:Dic3:23C2, C12.C8:8C2, C8.3(C2xDic3), (C2xC12).118D4, C8.30(C3:D4), C4.12(D4.S3), C22.3(D4:S3), C6.30(D4:C4), C12.17(C22:C4), (C2xC24).154C22, C4.5(C6.D4), C2.10(D4:Dic3), (C3xC4oD8).5C2, (C2xC4).26(C3:D4), SmallGroup(192,125)

Series: Derived Chief Lower central Upper central

C1C24 — D8:2Dic3
C1C3C6C12C2xC12C2xC24C8:Dic3 — D8:2Dic3
C3C6C12C24 — D8:2Dic3
C1C2C2xC4C2xC8C4oD8

Generators and relations for D8:2Dic3
 G = < a,b,c,d | a8=b2=c6=1, d2=c3, bab=a-1, ac=ca, dad-1=a3, cbc-1=a4b, dbd-1=a5b, dcd-1=c-1 >

Subgroups: 168 in 58 conjugacy classes, 27 normal (all characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C6, C6, C8, C2xC4, C2xC4, D4, Q8, Dic3, C12, C12, C2xC6, C2xC6, C16, C4:C4, C2xC8, D8, SD16, Q16, C4oD4, C24, C2xDic3, C2xC12, C2xC12, C3xD4, C3xQ8, C4.Q8, M5(2), C4oD8, C3:C16, C4:Dic3, C2xC24, C3xD8, C3xSD16, C3xQ16, C3xC4oD4, D8:2C4, C12.C8, C8:Dic3, C3xC4oD8, D8:2Dic3
Quotients: C1, C2, C4, C22, S3, C2xC4, D4, Dic3, D6, C22:C4, D8, SD16, C2xDic3, C3:D4, D4:C4, D4:S3, D4.S3, C6.D4, D8:2C4, D4:Dic3, D8:2Dic3

Character table of D8:2Dic3

 class 12A2B2C34A4B4C4D4E6A6B6C6D8A8B8C12A12B12C12D12E16A16B16C16D24A24B24C24D
 size 112822282424248822422488121212124444
ρ1111111111111111111111111111111    trivial
ρ211111111-1-1111111111111-1-1-1-11111    linear of order 2
ρ3111-1111-11111-1-1111111-1-1-1-1-1-11111    linear of order 2
ρ4111-1111-1-1-111-1-1111111-1-111111111    linear of order 2
ρ511-1-11-111-ii1-1-1-1-1-11-1-1111i-ii-i1-11-1    linear of order 4
ρ611-1-11-111i-i1-1-1-1-1-11-1-1111-ii-ii1-11-1    linear of order 4
ρ711-111-11-1-ii1-111-1-11-1-11-1-1-ii-ii1-11-1    linear of order 4
ρ811-111-11-1i-i1-111-1-11-1-11-1-1i-ii-i1-11-1    linear of order 4
ρ9222-2-122-200-1-111222-1-1-1110000-1-1-1-1    orthogonal lifted from D6
ρ1022-202-220002-20022-2-2-22000000-22-22    orthogonal lifted from D4
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
ρ1322202-2-20002200000-2-2-200-2-2220000    orthogonal lifted from D8
ρ1422202-2-20002200000-2-2-20022-2-20000    orthogonal lifted from D8
ρ1522-2-2-1-22200-1111-2-2211-1-1-10000-11-11    symplectic lifted from Dic3, Schur index 2
ρ1622-22-1-22-200-11-1-1-2-2211-1110000-11-11    symplectic lifted from Dic3, Schur index 2
ρ1722-20-1-22000-11-3--322-211-1--3-300001-11-1    complex lifted from C3:D4
ρ1822-20-1-22000-11--3-322-211-1-3--300001-11-1    complex lifted from C3:D4
ρ1922-2022-20002-20000022-200--2-2-2--20000    complex lifted from SD16
ρ2022-2022-20002-20000022-200-2--2--2-20000    complex lifted from SD16
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
ρ234440-2-4-4000-2-2000002220000000000    orthogonal lifted from D4:S3, Schur index 2
ρ2444-40-24-4000-2200000-2-220000000000    symplectic lifted from D4.S3, Schur index 2
ρ254-400400000-40002-2-2-200000000000-2-202-2    complex lifted from D8:2C4
ρ264-400400000-4000-2-22-2000000000002-20-2-2    complex lifted from D8:2C4
ρ274-400-2000002000-2-22-20-23230000000--6--2-6-2    complex faithful
ρ284-400-20000020002-2-2-20-23230000000-6-2--6--2    complex faithful
ρ294-400-20000020002-2-2-2023-230000000--6-2-6--2    complex faithful
ρ304-400-2000002000-2-22-2023-230000000-6--2--6-2    complex faithful

Smallest permutation representation of D8:2Dic3
On 48 points
Generators in S48
(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)
(1 12)(2 11)(3 10)(4 9)(5 16)(6 15)(7 14)(8 13)(17 31)(18 30)(19 29)(20 28)(21 27)(22 26)(23 25)(24 32)(33 46)(34 45)(35 44)(36 43)(37 42)(38 41)(39 48)(40 47)
(1 26 34)(2 27 35)(3 28 36)(4 29 37)(5 30 38)(6 31 39)(7 32 40)(8 25 33)(9 23 42 13 19 46)(10 24 43 14 20 47)(11 17 44 15 21 48)(12 18 45 16 22 41)
(2 4)(3 7)(6 8)(9 10 13 14)(11 16 15 12)(17 45 21 41)(18 48 22 44)(19 43 23 47)(20 46 24 42)(25 39)(26 34)(27 37)(28 40)(29 35)(30 38)(31 33)(32 36)

G:=sub<Sym(48)| (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), (1,12)(2,11)(3,10)(4,9)(5,16)(6,15)(7,14)(8,13)(17,31)(18,30)(19,29)(20,28)(21,27)(22,26)(23,25)(24,32)(33,46)(34,45)(35,44)(36,43)(37,42)(38,41)(39,48)(40,47), (1,26,34)(2,27,35)(3,28,36)(4,29,37)(5,30,38)(6,31,39)(7,32,40)(8,25,33)(9,23,42,13,19,46)(10,24,43,14,20,47)(11,17,44,15,21,48)(12,18,45,16,22,41), (2,4)(3,7)(6,8)(9,10,13,14)(11,16,15,12)(17,45,21,41)(18,48,22,44)(19,43,23,47)(20,46,24,42)(25,39)(26,34)(27,37)(28,40)(29,35)(30,38)(31,33)(32,36)>;

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), (1,12)(2,11)(3,10)(4,9)(5,16)(6,15)(7,14)(8,13)(17,31)(18,30)(19,29)(20,28)(21,27)(22,26)(23,25)(24,32)(33,46)(34,45)(35,44)(36,43)(37,42)(38,41)(39,48)(40,47), (1,26,34)(2,27,35)(3,28,36)(4,29,37)(5,30,38)(6,31,39)(7,32,40)(8,25,33)(9,23,42,13,19,46)(10,24,43,14,20,47)(11,17,44,15,21,48)(12,18,45,16,22,41), (2,4)(3,7)(6,8)(9,10,13,14)(11,16,15,12)(17,45,21,41)(18,48,22,44)(19,43,23,47)(20,46,24,42)(25,39)(26,34)(27,37)(28,40)(29,35)(30,38)(31,33)(32,36) );

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)], [(1,12),(2,11),(3,10),(4,9),(5,16),(6,15),(7,14),(8,13),(17,31),(18,30),(19,29),(20,28),(21,27),(22,26),(23,25),(24,32),(33,46),(34,45),(35,44),(36,43),(37,42),(38,41),(39,48),(40,47)], [(1,26,34),(2,27,35),(3,28,36),(4,29,37),(5,30,38),(6,31,39),(7,32,40),(8,25,33),(9,23,42,13,19,46),(10,24,43,14,20,47),(11,17,44,15,21,48),(12,18,45,16,22,41)], [(2,4),(3,7),(6,8),(9,10,13,14),(11,16,15,12),(17,45,21,41),(18,48,22,44),(19,43,23,47),(20,46,24,42),(25,39),(26,34),(27,37),(28,40),(29,35),(30,38),(31,33),(32,36)]])

Matrix representation of D8:2Dic3 in GL4(F97) generated by

618900
85300
73513689
3394844
,
57436952
83491769
89783256
53703256
,
96100
96000
2212096
177811
,
0100
1000
5314448
79206153
G:=sub<GL(4,GF(97))| [61,8,73,33,89,53,51,94,0,0,36,8,0,0,89,44],[57,83,89,53,43,49,78,70,69,17,32,32,52,69,56,56],[96,96,22,17,1,0,12,78,0,0,0,1,0,0,96,1],[0,1,53,79,1,0,14,20,0,0,44,61,0,0,8,53] >;

D8:2Dic3 in GAP, Magma, Sage, TeX

D_8\rtimes_2{\rm Dic}_3
% in TeX

G:=Group("D8:2Dic3");
// GroupNames label

G:=SmallGroup(192,125);
// by ID

G=gap.SmallGroup(192,125);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,28,141,387,675,794,80,1684,851,102,6278]);
// Polycyclic

G:=Group<a,b,c,d|a^8=b^2=c^6=1,d^2=c^3,b*a*b=a^-1,a*c=c*a,d*a*d^-1=a^3,c*b*c^-1=a^4*b,d*b*d^-1=a^5*b,d*c*d^-1=c^-1>;
// generators/relations

Export

Character table of D8:2Dic3 in TeX

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