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G = D10.D6order 240 = 24·3·5

6th non-split extension by D10 of D6 acting via D6/C6=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D102Dic3, D10.14D6, (C2×C6)⋊2F5, (C2×C30)⋊1C4, (C6×D5)⋊4C4, C5⋊(C6.D4), (C3×D5).5D4, C222(C3⋊F5), C32(C22⋊F5), C6.14(C2×F5), C152(C22⋊C4), C30.14(C2×C4), (C2×C10)⋊4Dic3, D5.3(C3⋊D4), C10.7(C2×Dic3), (C22×D5).3S3, (C6×D5).21C22, (C2×C3⋊F5)⋊3C2, C2.7(C2×C3⋊F5), (D5×C2×C6).4C2, SmallGroup(240,124)

Series: Derived Chief Lower central Upper central

C1C30 — D10.D6
C1C5C15C3×D5C6×D5C2×C3⋊F5 — D10.D6
C15C30 — D10.D6
C1C2C22

Generators and relations for D10.D6
 G = < a,b,c,d | a10=b2=c6=1, d2=a4b, bab=a-1, ac=ca, dad-1=a7, bc=cb, dbd-1=a6b, dcd-1=a5c-1 >

Subgroups: 320 in 68 conjugacy classes, 25 normal (19 characteristic)
C1, C2, C2 [×4], C3, C4 [×2], C22, C22 [×4], C5, C6, C6 [×4], C2×C4 [×2], C23, D5 [×2], D5, C10, C10, Dic3 [×2], C2×C6, C2×C6 [×4], C15, C22⋊C4, F5 [×2], D10 [×2], D10 [×2], C2×C10, C2×Dic3 [×2], C22×C6, C3×D5 [×2], C3×D5, C30, C30, C2×F5 [×2], C22×D5, C6.D4, C3⋊F5 [×2], C6×D5 [×2], C6×D5 [×2], C2×C30, C22⋊F5, C2×C3⋊F5 [×2], D5×C2×C6, D10.D6
Quotients: C1, C2 [×3], C4 [×2], C22, S3, C2×C4, D4 [×2], Dic3 [×2], D6, C22⋊C4, F5, C2×Dic3, C3⋊D4 [×2], C2×F5, C6.D4, C3⋊F5, C22⋊F5, C2×C3⋊F5, D10.D6

Character table of D10.D6

 class 12A2B2C2D2E34A4B4C4D56A6B6C6D6E6F6G10A10B10C15A15B30A30B30C30D30E30F
 size 112551023030303042221010101044444444444
ρ1111111111111111111111111111111    trivial
ρ211-111-11-11-1111-1-11-1-11-1-1111-1-1-111-1    linear of order 2
ρ311-111-111-11-111-1-11-1-11-1-1111-1-1-111-1    linear of order 2
ρ41111111-1-1-1-11111111111111111111    linear of order 2
ρ511-1-1-111-iii-i11-1-1-111-1-1-1111-1-1-111-1    linear of order 4
ρ6111-1-1-11ii-i-i1111-1-1-1-111111111111    linear of order 4
ρ7111-1-1-11-i-iii1111-1-1-1-111111111111    linear of order 4
ρ811-1-1-111i-i-ii11-1-1-111-1-1-1111-1-1-111-1    linear of order 4
ρ9222222-100002-1-1-1-1-1-1-1222-1-1-1-1-1-1-1-1    orthogonal lifted from S3
ρ102-20-220200002-200-200200-222000-2-20    orthogonal lifted from D4
ρ1122-222-2-100002-111-111-1-2-22-1-1111-1-11    orthogonal lifted from D6
ρ122-202-20200002-200200-200-222000-2-20    orthogonal lifted from D4
ρ13222-2-2-2-100002-1-1-11111222-1-1-1-1-1-1-1-1    symplectic lifted from Dic3, Schur index 2
ρ1422-2-2-22-100002-1111-1-11-2-22-1-1111-1-11    symplectic lifted from Dic3, Schur index 2
ρ152-20-220-1000021-3--31--3-3-100-2-1-1--3--3-311-3    complex lifted from C3⋊D4
ρ162-202-20-1000021-3--3-1-3--3100-2-1-1--3--3-311-3    complex lifted from C3⋊D4
ρ172-202-20-1000021--3-3-1--3-3100-2-1-1-3-3--311--3    complex lifted from C3⋊D4
ρ182-20-220-1000021--3-31-3--3-100-2-1-1-3-3--311--3    complex lifted from C3⋊D4
ρ1944-400040000-14-4-4000011-1-1-1111-1-11    orthogonal lifted from C2×F5
ρ2044400040000-14440000-1-1-1-1-1-1-1-1-1-1-1    orthogonal lifted from F5
ρ214-4000040000-1-40000005-51-1-15-5-5115    orthogonal lifted from C22⋊F5
ρ224-4000040000-1-4000000-551-1-1-55511-5    orthogonal lifted from C22⋊F5
ρ23444000-20000-1-2-2-20000-1-1-11--15/21+-15/21--15/21+-15/21--15/21--15/21+-15/21+-15/2    complex lifted from C3⋊F5
ρ244-40000-20000-122-3-2-300005-511--15/21+-15/2325352325453545-1+-15/2-1--15/235352    complex faithful
ρ254-40000-20000-122-3-2-30000-5511+-15/21--15/23254532535235352-1--15/2-1+-15/23545    complex faithful
ρ26444000-20000-1-2-2-20000-1-1-11+-15/21--15/21+-15/21--15/21+-15/21+-15/21--15/21--15/2    complex lifted from C3⋊F5
ρ274-40000-20000-12-2-32-300005-511+-15/21--15/235352354532545-1--15/2-1+-15/2325352    complex faithful
ρ2844-4000-20000-1-222000011-11+-15/21--15/2-1--15/2-1+-15/2-1--15/21+-15/21--15/2-1+-15/2    complex lifted from C2×C3⋊F5
ρ2944-4000-20000-1-222000011-11--15/21+-15/2-1+-15/2-1--15/2-1+-15/21--15/21+-15/2-1--15/2    complex lifted from C2×C3⋊F5
ρ304-40000-20000-12-2-32-30000-5511--15/21+-15/2354535352325352-1+-15/2-1--15/232545    complex faithful

Smallest permutation representation of D10.D6
On 60 points
Generators in S60
(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)
(1 10)(2 9)(3 8)(4 7)(5 6)(11 20)(12 19)(13 18)(14 17)(15 16)(21 28)(22 27)(23 26)(24 25)(29 30)(31 38)(32 37)(33 36)(34 35)(39 40)(41 48)(42 47)(43 46)(44 45)(49 50)(51 54)(52 53)(55 60)(56 59)(57 58)
(1 50 35 53 25 11)(2 41 36 54 26 12)(3 42 37 55 27 13)(4 43 38 56 28 14)(5 44 39 57 29 15)(6 45 40 58 30 16)(7 46 31 59 21 17)(8 47 32 60 22 18)(9 48 33 51 23 19)(10 49 34 52 24 20)
(1 11 6 16)(2 14 5 13)(3 17 4 20)(7 19 10 18)(8 12 9 15)(21 48 24 47)(22 41 23 44)(25 50 30 45)(26 43 29 42)(27 46 28 49)(31 51 34 60)(32 54 33 57)(35 53 40 58)(36 56 39 55)(37 59 38 52)

G:=sub<Sym(60)| (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), (1,10)(2,9)(3,8)(4,7)(5,6)(11,20)(12,19)(13,18)(14,17)(15,16)(21,28)(22,27)(23,26)(24,25)(29,30)(31,38)(32,37)(33,36)(34,35)(39,40)(41,48)(42,47)(43,46)(44,45)(49,50)(51,54)(52,53)(55,60)(56,59)(57,58), (1,50,35,53,25,11)(2,41,36,54,26,12)(3,42,37,55,27,13)(4,43,38,56,28,14)(5,44,39,57,29,15)(6,45,40,58,30,16)(7,46,31,59,21,17)(8,47,32,60,22,18)(9,48,33,51,23,19)(10,49,34,52,24,20), (1,11,6,16)(2,14,5,13)(3,17,4,20)(7,19,10,18)(8,12,9,15)(21,48,24,47)(22,41,23,44)(25,50,30,45)(26,43,29,42)(27,46,28,49)(31,51,34,60)(32,54,33,57)(35,53,40,58)(36,56,39,55)(37,59,38,52)>;

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), (1,10)(2,9)(3,8)(4,7)(5,6)(11,20)(12,19)(13,18)(14,17)(15,16)(21,28)(22,27)(23,26)(24,25)(29,30)(31,38)(32,37)(33,36)(34,35)(39,40)(41,48)(42,47)(43,46)(44,45)(49,50)(51,54)(52,53)(55,60)(56,59)(57,58), (1,50,35,53,25,11)(2,41,36,54,26,12)(3,42,37,55,27,13)(4,43,38,56,28,14)(5,44,39,57,29,15)(6,45,40,58,30,16)(7,46,31,59,21,17)(8,47,32,60,22,18)(9,48,33,51,23,19)(10,49,34,52,24,20), (1,11,6,16)(2,14,5,13)(3,17,4,20)(7,19,10,18)(8,12,9,15)(21,48,24,47)(22,41,23,44)(25,50,30,45)(26,43,29,42)(27,46,28,49)(31,51,34,60)(32,54,33,57)(35,53,40,58)(36,56,39,55)(37,59,38,52) );

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)], [(1,10),(2,9),(3,8),(4,7),(5,6),(11,20),(12,19),(13,18),(14,17),(15,16),(21,28),(22,27),(23,26),(24,25),(29,30),(31,38),(32,37),(33,36),(34,35),(39,40),(41,48),(42,47),(43,46),(44,45),(49,50),(51,54),(52,53),(55,60),(56,59),(57,58)], [(1,50,35,53,25,11),(2,41,36,54,26,12),(3,42,37,55,27,13),(4,43,38,56,28,14),(5,44,39,57,29,15),(6,45,40,58,30,16),(7,46,31,59,21,17),(8,47,32,60,22,18),(9,48,33,51,23,19),(10,49,34,52,24,20)], [(1,11,6,16),(2,14,5,13),(3,17,4,20),(7,19,10,18),(8,12,9,15),(21,48,24,47),(22,41,23,44),(25,50,30,45),(26,43,29,42),(27,46,28,49),(31,51,34,60),(32,54,33,57),(35,53,40,58),(36,56,39,55),(37,59,38,52)])

D10.D6 is a maximal subgroup of
D10.D12  D10.4D12  (C2×C60)⋊C4  C3⋊(C23⋊F5)  C22⋊F5.S3  F5×C3⋊D4  S3×C22⋊F5  (C2×C12)⋊6F5  D4×C3⋊F5
D10.D6 is a maximal quotient of
(C2×C60)⋊C4  C30.7M4(2)  (C2×C60).C4  D10.10D12  D20⋊Dic3  Dic10⋊Dic3  Dic102Dic3  D202Dic3  C3⋊(C23⋊F5)  C30.22M4(2)  C5⋊(C12.D4)

Matrix representation of D10.D6 in GL6(𝔽61)

6000000
0600000
0000601
0000600
0010600
0001600
,
100000
010000
0001600
0010600
0000600
0000601
,
29550000
18320000
00286055
00034655
00556340
00550628
,
14560000
39470000
00028556
00605534
00345506
00655280

G:=sub<GL(6,GF(61))| [60,0,0,0,0,0,0,60,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,60,60,60,60,0,0,1,0,0,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,60,60,60,60,0,0,0,0,0,1],[29,18,0,0,0,0,55,32,0,0,0,0,0,0,28,0,55,55,0,0,6,34,6,0,0,0,0,6,34,6,0,0,55,55,0,28],[14,39,0,0,0,0,56,47,0,0,0,0,0,0,0,6,34,6,0,0,28,0,55,55,0,0,55,55,0,28,0,0,6,34,6,0] >;

D10.D6 in GAP, Magma, Sage, TeX

D_{10}.D_6
% in TeX

G:=Group("D10.D6");
// GroupNames label

G:=SmallGroup(240,124);
// by ID

G=gap.SmallGroup(240,124);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-3,-5,24,121,964,5189,1745]);
// Polycyclic

G:=Group<a,b,c,d|a^10=b^2=c^6=1,d^2=a^4*b,b*a*b=a^-1,a*c=c*a,d*a*d^-1=a^7,b*c=c*b,d*b*d^-1=a^6*b,d*c*d^-1=a^5*c^-1>;
// generators/relations

Export

Character table of D10.D6 in TeX

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