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G = Dic6.10D6  order 288 = 25·32

10th non-split extension by Dic6 of D6 acting via D6/C3=C22

metabelian, supersoluble, monomial

Aliases: Dic6.10D6, C3⋊C8.8D6, Q8.15S32, C3⋊Q16⋊3S3, C6.64(S3×D4), (C3×Q8).32D6, C3⋊4(Q16⋊S3), C3⋊Dic3.60D4, Dic3.D6⋊5C2, C32⋊5SD16⋊9C2, C12.31D6⋊6C2, C12.22(C22×S3), (C3×C12).22C23, C2.24(Dic3⋊D6), C12.26D6.2C2, C32⋊12(C8.C22), C12⋊S3.12C22, (Q8×C32).4C22, (C3×Dic6).18C22, C4.22(C2×S32), (C2×C3⋊S3).25D4, (C3×C3⋊Q16)⋊6C2, (C3×C6).137(C2×D4), (C3×C3⋊C8).15C22, (C4×C3⋊S3).20C22, SmallGroup(288,593)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C3×C12 — Dic6.10D6
Chief series
C1 — C3 — C32 — C3×C6 — C3×C12 — C3×Dic6 — Dic3.D6 — Dic6.10D6
Lower central
C32 — C3×C6 — C3×C12 — Dic6.10D6
Upper central
C1 — C2 — C4 — Q8

Generators and relations for Dic6.10D6
 G = < a,b,c,d | a12=1, b2=c6=a6, d2=a9, bab-1=a-1, cac-1=a7, ad=da, cbc-1=a9b, dbd-1=a3b, dcd-1=a9c5 >

Subgroups: 634 in 140 conjugacy classes, 38 normal (16 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, S3, C6, C6, C8, C2×C4, D4, Q8, Q8, C32, Dic3, C12, C12, D6, M4(2), SD16, Q16, C2×Q8, C4○D4, C3⋊S3, C3×C6, C3⋊C8, C24, Dic6, Dic6, C4×S3, D12, C3×Q8, C3×Q8, C8.C22, C3×Dic3, C3⋊Dic3, C3×C12, C3×C12, C2×C3⋊S3, C2×C3⋊S3, C8⋊S3, C24⋊C2, Q8⋊2S3, C3⋊Q16, C3×Q16, S3×Q8, Q8⋊3S3, C3×C3⋊C8, C6.D6, C32⋊2Q8, C3×Dic6, C4×C3⋊S3, C4×C3⋊S3, C12⋊S3, C12⋊S3, Q8×C32, Q16⋊S3, C12.31D6, C32⋊5SD16, C3×C3⋊Q16, Dic3.D6, C12.26D6, Dic6.10D6
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C22×S3, C8.C22, S32, S3×D4, C2×S32, Q16⋊S3, Dic3⋊D6, Dic6.10D6

Character table of Dic6.10D6

 class 12A2B2C3A3B3C4A4B4C4D4E6A6B6C8A8B12A12B12C12D12E12F12G12H12I24A24B24C24D
 size 1118362242412121822412124488888242412121212
ρ1111111111111111111111111111111    trivial
ρ211-111111-1-11-1111-1111-1-11-1-1-11-1-111    linear of order 2
ρ311-1-1111111-1-1111-1111111111-1-1-111    linear of order 2
ρ4111-11111-1-1-111111111-1-11-1-1-1-11111    linear of order 2
ρ5111111111-1-11111-1-11111111-1-1-1-1-1-1    linear of order 2
ρ611-111111-11-1-11111-111-1-11-1-11-111-1-1    linear of order 2
ρ711-1-111111-11-11111-11111111-1111-1-1    linear of order 2
ρ8111-11111-1111111-1-111-1-11-1-111-1-1-1-1    linear of order 2
ρ92200-12-122020-12-120-12-1-1-1-120-1-1-100    orthogonal lifted from S3
ρ102200-12-12-20-20-12-120-1211-11-201-1-100    orthogonal lifted from D6
ρ1122002-1-12-22002-1-10-22-111-1-21-100011    orthogonal lifted from D6
ρ1222002-1-12-2-2002-1-1022-111-1-211000-1-1    orthogonal lifted from D6
ρ1322-20222-2000222200-2-200-200000000    orthogonal lifted from D4
ρ142200-12-1220-20-12-1-20-12-1-1-1-12011100    orthogonal lifted from D6
ρ152200-12-12-2020-12-1-20-1211-11-20-11100    orthogonal lifted from D6
ρ1622002-1-1222002-1-1022-1-1-1-12-1-1000-1-1    orthogonal lifted from S3
ρ1722002-1-122-2002-1-10-22-1-1-1-12-1100011    orthogonal lifted from D6
ρ182220222-2000-222200-2-200-200000000    orthogonal lifted from D4
ρ1944004-2-2-400004-2-200-4200200000000    orthogonal lifted from S3×D4
ρ204400-2-21-40000-2-210022-33-100000000    orthogonal lifted from Dic3⋊D6
ρ214400-2-214-4000-2-2100-2-2-1-1122000000    orthogonal lifted from C2×S32
ρ224400-24-2-40000-24-2002-400200000000    orthogonal lifted from S3×D4
ρ234400-2-21-40000-2-2100223-3-100000000    orthogonal lifted from Dic3⋊D6
ρ244400-2-2144000-2-2100-2-2111-2-2000000    orthogonal lifted from S32
ρ254-40044400000-4-4-4000000000000000    symplectic lifted from C8.C22, Schur index 2
ρ264-4004-2-200000-4220000000000000-√-6√-6    complex lifted from Q16⋊S3
ρ274-4004-2-200000-4220000000000000√-6-√-6    complex lifted from Q16⋊S3
ρ284-400-24-2000002-4200000000000√-6-√-600    complex lifted from Q16⋊S3
ρ294-400-24-2000002-4200000000000-√-6√-600    complex lifted from Q16⋊S3
ρ308-800-4-420000044-2000000000000000    orthogonal faithful, Schur index 2

Smallest permutation representation of Dic6.10D6
►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 25 7 31)(2 36 8 30)(3 35 9 29)(4 34 10 28)(5 33 11 27)(6 32 12 26)(13 41 19 47)(14 40 20 46)(15 39 21 45)(16 38 22 44)(17 37 23 43)(18 48 24 42)

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

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

Matrix representation of Dic6.10D6 ►in GL8(𝔽73)

072000000
11000000
000720000
00110000
000017100
000017200
0000542722
00007256721
,
00100000
0072720000
10000000
7272000000
0000611200
0000671200
00004572012
000082960
,
01000000
7272000000
00110000
007200000
000064330
0000371203
00003370970
000036123661
,
0072720000
00100000
072000000
11000000
0000671036
00001918180
000068535535
00006754167

G:=sub<GL(8,GF(73))| [0,1,0,0,0,0,0,0,72,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,72,1,0,0,0,0,0,0,0,0,1,1,54,72,0,0,0,0,71,72,2,56,0,0,0,0,0,0,72,72,0,0,0,0,0,0,2,1],[0,0,1,72,0,0,0,0,0,0,0,72,0,0,0,0,1,72,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,0,0,61,67,45,8,0,0,0,0,12,12,72,29,0,0,0,0,0,0,0,6,0,0,0,0,0,0,12,0],[0,72,0,0,0,0,0,0,1,72,0,0,0,0,0,0,0,0,1,72,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,64,37,33,36,0,0,0,0,3,12,70,12,0,0,0,0,3,0,9,36,0,0,0,0,0,3,70,61],[0,0,0,1,0,0,0,0,0,0,72,1,0,0,0,0,72,1,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,0,0,0,6,19,68,67,0,0,0,0,71,18,53,54,0,0,0,0,0,18,55,1,0,0,0,0,36,0,35,67] >;
 

Dic6.10D6 in GAP, Magma, Sage, TeX 

{\rm Dic}_6._{10}D_6
 
% in TeX
 
G:=Group("Dic6.10D6");
 
// GroupNames label
 
G:=SmallGroup(288,593);
 
// by ID
 
G=gap.SmallGroup(288,593);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,112,120,254,303,100,675,346,185,80,1356,9414]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^12=1,b^2=c^6=a^6,d^2=a^9,b*a*b^-1=a^-1,c*a*c^-1=a^7,a*d=d*a,c*b*c^-1=a^9*b,d*b*d^-1=a^3*b,d*c*d^-1=a^9*c^5>;
 
// generators/relations
 

Export

Character table of Dic6.10D6 in TeX

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