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## G = Dic5.6S4order 480 = 25·3·5

### 1st non-split extension by Dic5 of S4 acting through Inn(Dic5)

Aliases: Dic5.6S4, GL2(𝔽3)⋊3D5, SL2(𝔽3).2D10, C2.5(D5×S4), C10.2(C2×S4), (C5×Q8).2D6, Q8.2(S3×D5), C52(C4.6S4), Q82D51S3, Q8.D154C2, Dic5.A41C2, (C5×GL2(𝔽3))⋊3C2, (C5×SL2(𝔽3)).2C22, SmallGroup(480,968)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — Q8 — C5×SL2(𝔽3) — Dic5.6S4
 Chief series C1 — C2 — Q8 — C5×Q8 — C5×SL2(𝔽3) — Dic5.A4 — Dic5.6S4
 Lower central C5×SL2(𝔽3) — Dic5.6S4
 Upper central C1 — C2

Generators and relations for Dic5.6S4
G = < a,b,c,d,e,f | a10=e3=f2=1, b2=c2=d2=a5, bab-1=a-1, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, dcd-1=a5c, ece-1=a5cd, fcf=cd, ede-1=c, fdf=a5d, fef=e-1 >

Subgroups: 554 in 78 conjugacy classes, 15 normal (all characteristic)
C1, C2, C2, C3, C4, C22, C5, S3, C6, C8, C2×C4, D4, Q8, Q8, D5, C10, C10, Dic3, C12, D6, C15, C2×C8, D8, SD16, Q16, C4○D4, Dic5, Dic5, C20, D10, C2×C10, SL2(𝔽3), C4×S3, C5×S3, C30, C4○D8, C52C8, C40, Dic10, C4×D5, D20, C2×Dic5, C5⋊D4, C5×D4, C5×Q8, CSU2(𝔽3), GL2(𝔽3), C4.A4, C3×Dic5, Dic15, S3×C10, C8×D5, C40⋊C2, D4⋊D5, C5⋊Q16, C5×SD16, D42D5, Q82D5, C4.6S4, S3×Dic5, C5×SL2(𝔽3), SD163D5, C5×GL2(𝔽3), Q8.D15, Dic5.A4, Dic5.6S4
Quotients: C1, C2, C22, S3, D5, D6, D10, S4, C2×S4, S3×D5, C4.6S4, D5×S4, Dic5.6S4

Smallest permutation representation of Dic5.6S4
On 80 points
Generators in S80
(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)
(1 74 6 79)(2 73 7 78)(3 72 8 77)(4 71 9 76)(5 80 10 75)(11 70 16 65)(12 69 17 64)(13 68 18 63)(14 67 19 62)(15 66 20 61)(21 60 26 55)(22 59 27 54)(23 58 28 53)(24 57 29 52)(25 56 30 51)(31 50 36 45)(32 49 37 44)(33 48 38 43)(34 47 39 42)(35 46 40 41)
(1 54 6 59)(2 55 7 60)(3 56 8 51)(4 57 9 52)(5 58 10 53)(11 40 16 35)(12 31 17 36)(13 32 18 37)(14 33 19 38)(15 34 20 39)(21 78 26 73)(22 79 27 74)(23 80 28 75)(24 71 29 76)(25 72 30 77)(41 65 46 70)(42 66 47 61)(43 67 48 62)(44 68 49 63)(45 69 50 64)
(1 46 6 41)(2 47 7 42)(3 48 8 43)(4 49 9 44)(5 50 10 45)(11 27 16 22)(12 28 17 23)(13 29 18 24)(14 30 19 25)(15 21 20 26)(31 80 36 75)(32 71 37 76)(33 72 38 77)(34 73 39 78)(35 74 40 79)(51 62 56 67)(52 63 57 68)(53 64 58 69)(54 65 59 70)(55 66 60 61)
(11 27 35)(12 28 36)(13 29 37)(14 30 38)(15 21 39)(16 22 40)(17 23 31)(18 24 32)(19 25 33)(20 26 34)(41 65 59)(42 66 60)(43 67 51)(44 68 52)(45 69 53)(46 70 54)(47 61 55)(48 62 56)(49 63 57)(50 64 58)
(1 6)(2 7)(3 8)(4 9)(5 10)(11 27)(12 28)(13 29)(14 30)(15 21)(16 22)(17 23)(18 24)(19 25)(20 26)(51 67)(52 68)(53 69)(54 70)(55 61)(56 62)(57 63)(58 64)(59 65)(60 66)(71 76)(72 77)(73 78)(74 79)(75 80)

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

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

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

32 conjugacy classes

 class 1 2A 2B 2C 3 4A 4B 4C 4D 5A 5B 6 8A 8B 8C 8D 10A 10B 10C 10D 12A 12B 15A 15B 20A 20B 30A 30B 40A 40B 40C 40D order 1 2 2 2 3 4 4 4 4 5 5 6 8 8 8 8 10 10 10 10 12 12 15 15 20 20 30 30 40 40 40 40 size 1 1 12 30 8 5 5 6 60 2 2 8 6 6 30 30 2 2 24 24 40 40 16 16 12 12 16 16 12 12 12 12

32 irreducible representations

 dim 1 1 1 1 2 2 2 2 2 3 3 4 4 4 6 8 type + + + + + + + + + + + + - image C1 C2 C2 C2 S3 D5 D6 D10 C4.6S4 S4 C2×S4 S3×D5 C4.6S4 Dic5.6S4 D5×S4 Dic5.6S4 kernel Dic5.6S4 C5×GL2(𝔽3) Q8.D15 Dic5.A4 Q8⋊2D5 GL2(𝔽3) C5×Q8 SL2(𝔽3) C5 Dic5 C10 Q8 C5 C1 C2 C1 # reps 1 1 1 1 1 2 1 2 4 2 2 2 2 4 4 2

Matrix representation of Dic5.6S4 in GL4(𝔽241) generated by

 240 1 0 0 50 190 0 0 0 0 240 0 0 0 0 240
,
 20 89 0 0 93 221 0 0 0 0 64 0 0 0 0 64
,
 1 0 0 0 0 1 0 0 0 0 216 12 0 0 229 25
,
 1 0 0 0 0 1 0 0 0 0 228 229 0 0 215 13
,
 1 0 0 0 0 1 0 0 0 0 12 13 0 0 25 228
,
 240 0 0 0 0 240 0 0 0 0 240 1 0 0 0 1
G:=sub<GL(4,GF(241))| [240,50,0,0,1,190,0,0,0,0,240,0,0,0,0,240],[20,93,0,0,89,221,0,0,0,0,64,0,0,0,0,64],[1,0,0,0,0,1,0,0,0,0,216,229,0,0,12,25],[1,0,0,0,0,1,0,0,0,0,228,215,0,0,229,13],[1,0,0,0,0,1,0,0,0,0,12,25,0,0,13,228],[240,0,0,0,0,240,0,0,0,0,240,0,0,0,1,1] >;

Dic5.6S4 in GAP, Magma, Sage, TeX

{\rm Dic}_5._6S_4
% in TeX

G:=Group("Dic5.6S4");
// GroupNames label

G:=SmallGroup(480,968);
// by ID

G=gap.SmallGroup(480,968);
# by ID

G:=PCGroup([7,-2,-2,-3,-5,-2,2,-2,1680,93,1347,2111,3168,172,1272,1909,285,124]);
// Polycyclic

G:=Group<a,b,c,d,e,f|a^10=e^3=f^2=1,b^2=c^2=d^2=a^5,b*a*b^-1=a^-1,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,b*c=c*b,b*d=d*b,b*e=e*b,b*f=f*b,d*c*d^-1=a^5*c,e*c*e^-1=a^5*c*d,f*c*f=c*d,e*d*e^-1=c,f*d*f=a^5*d,f*e*f=e^-1>;
// generators/relations

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