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## G = Dic5.7S4order 480 = 25·3·5

### 2nd non-split extension by Dic5 of S4 acting through Inn(Dic5)

Aliases: Dic5.7S4, CSU2(𝔽3)⋊3D5, SL2(𝔽3).3D10, C2.6(D5×S4), C10.3(C2×S4), Q8⋊D153C2, Q8.3(S3×D5), (C5×Q8).3D6, C51(C4.6S4), Q82D52S3, Dic5.A42C2, (C5×CSU2(𝔽3))⋊1C2, (C5×SL2(𝔽3)).3C22, SmallGroup(480,969)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — Q8 — C5×SL2(𝔽3) — Dic5.7S4
 Chief series C1 — C2 — Q8 — C5×Q8 — C5×SL2(𝔽3) — Dic5.A4 — Dic5.7S4
 Lower central C5×SL2(𝔽3) — Dic5.7S4
 Upper central C1 — C2

Generators and relations for Dic5.7S4
G = < a,b,c,d,e,f | a10=e3=1, b2=c2=d2=f2=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-1=cd, ede-1=c, fdf-1=a5d, fef-1=e-1 >

Subgroups: 634 in 78 conjugacy classes, 15 normal (all characteristic)
C1, C2, C2 [×2], C3, C4 [×3], C22 [×2], C5, S3 [×2], C6, C8 [×2], C2×C4 [×2], D4 [×3], Q8, Q8, D5 [×2], C10, Dic3, C12, D6, C15, C2×C8, D8, SD16 [×2], Q16, C4○D4 [×2], Dic5, C20 [×2], D10 [×2], SL2(𝔽3), C4×S3, D15 [×2], C30, C4○D8, C52C8, C40, C4×D5 [×2], D20 [×3], C5×Q8, C5×Q8, CSU2(𝔽3), GL2(𝔽3), C4.A4, C5×Dic3, C3×Dic5, D30, C8×D5, D40, Q8⋊D5 [×2], C5×Q16, Q82D5, Q82D5, C4.6S4, D30.C2, C5×SL2(𝔽3), Q8.D10, C5×CSU2(𝔽3), Q8⋊D15, Dic5.A4, Dic5.7S4
Quotients: C1, C2 [×3], C22, S3, D5, D6, D10, S4, C2×S4, S3×D5, C4.6S4, D5×S4, Dic5.7S4

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

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

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

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

32 conjugacy classes

 class 1 2A 2B 2C 3 4A 4B 4C 4D 5A 5B 6 8A 8B 8C 8D 10A 10B 12A 12B 15A 15B 20A 20B 20C 20D 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 12 12 15 15 20 20 20 20 30 30 40 40 40 40 size 1 1 30 60 8 5 5 6 12 2 2 8 6 6 30 30 2 2 40 40 16 16 12 12 24 24 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 Dic5.7S4 D5×S4 Dic5.7S4 kernel Dic5.7S4 C5×CSU2(𝔽3) Q8⋊D15 Dic5.A4 Q8⋊2D5 CSU2(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 Dic5.7S4 in GL4(𝔽241) generated by

 240 1 0 0 50 190 0 0 0 0 240 0 0 0 0 240
,
 51 1 0 0 51 190 0 0 0 0 64 0 0 0 0 64
,
 1 0 0 0 0 1 0 0 0 0 68 174 0 0 105 173
,
 1 0 0 0 0 1 0 0 0 0 173 105 0 0 174 68
,
 1 0 0 0 0 1 0 0 0 0 67 135 0 0 68 173
,
 1 0 0 0 0 1 0 0 0 0 177 0 0 0 64 64
G:=sub<GL(4,GF(241))| [240,50,0,0,1,190,0,0,0,0,240,0,0,0,0,240],[51,51,0,0,1,190,0,0,0,0,64,0,0,0,0,64],[1,0,0,0,0,1,0,0,0,0,68,105,0,0,174,173],[1,0,0,0,0,1,0,0,0,0,173,174,0,0,105,68],[1,0,0,0,0,1,0,0,0,0,67,68,0,0,135,173],[1,0,0,0,0,1,0,0,0,0,177,64,0,0,0,64] >;

Dic5.7S4 in GAP, Magma, Sage, TeX

{\rm Dic}_5._7S_4
% in TeX

G:=Group("Dic5.7S4");
// GroupNames label

G:=SmallGroup(480,969);
// by ID

G=gap.SmallGroup(480,969);
# by ID

G:=PCGroup([7,-2,-2,-3,-5,-2,2,-2,1680,1688,93,1347,2111,3168,172,1272,1909,285,124]);
// Polycyclic

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

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