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## G = C10.532+ 1+4order 320 = 26·5

### 53rd non-split extension by C10 of 2+ 1+4 acting via 2+ 1+4/C2×D4=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — C10.532+ 1+4
 Chief series C1 — C5 — C10 — C2×C10 — C22×D5 — C23×D5 — D5×C22⋊C4 — C10.532+ 1+4
 Lower central C5 — C2×C10 — C10.532+ 1+4
 Upper central C1 — C22 — C22⋊Q8

Generators and relations for C10.532+ 1+4
G = < a,b,c,d,e | a10=b4=c2=1, d2=a5b2, e2=a5, ab=ba, ac=ca, dad-1=a-1, ae=ea, cbc=a5b-1, dbd-1=ebe-1=a5b, cd=dc, ce=ec, ede-1=b2d >

Subgroups: 982 in 248 conjugacy classes, 97 normal (91 characteristic)
C1, C2 [×3], C2 [×6], C4 [×11], C22, C22 [×2], C22 [×16], C5, C2×C4 [×6], C2×C4 [×12], D4 [×5], Q8, C23, C23 [×8], D5 [×4], C10 [×3], C10 [×2], C42 [×3], C22⋊C4 [×2], C22⋊C4 [×12], C4⋊C4 [×3], C4⋊C4 [×5], C22×C4, C22×C4 [×4], C2×D4 [×3], C2×Q8, C24, Dic5 [×5], C20 [×6], D10 [×2], D10 [×12], C2×C10, C2×C10 [×2], C2×C10 [×2], C2×C22⋊C4 [×2], C42⋊C2 [×2], C4×D4 [×2], C22≀C2, C22⋊Q8, C22⋊Q8, C22.D4 [×3], C4.4D4, C422C2 [×2], C4×D5 [×4], D20 [×3], C2×Dic5 [×5], C2×Dic5 [×2], C5⋊D4 [×2], C2×C20 [×6], C2×C20, C5×Q8, C22×D5 [×3], C22×D5 [×5], C22×C10, C22.45C24, C4×Dic5 [×3], C10.D4 [×3], C4⋊Dic5 [×2], D10⋊C4 [×11], C23.D5, C5×C22⋊C4 [×2], C5×C4⋊C4 [×3], C2×C4×D5 [×3], C2×D20 [×2], C22×Dic5, C2×C5⋊D4, C22×C20, Q8×C10, C23×D5, C23.11D10, D5×C22⋊C4, C22⋊D20, C22.D20, C4⋊C47D5, D208C4, D10.13D4 [×2], C4⋊C4⋊D5 [×2], C2×D10⋊C4, C4×C5⋊D4, D103Q8, C20.23D4, C5×C22⋊Q8, C10.532+ 1+4
Quotients: C1, C2 [×15], C22 [×35], C23 [×15], D5, C4○D4 [×4], C24, D10 [×7], C2×C4○D4 [×2], 2+ 1+4, C22×D5 [×7], C22.45C24, Q82D5 [×2], C23×D5, D46D10, C2×Q82D5, D5×C4○D4, C10.532+ 1+4

Smallest permutation representation of C10.532+ 1+4
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 48 18 53)(2 49 19 54)(3 50 20 55)(4 41 11 56)(5 42 12 57)(6 43 13 58)(7 44 14 59)(8 45 15 60)(9 46 16 51)(10 47 17 52)(21 66 36 71)(22 67 37 72)(23 68 38 73)(24 69 39 74)(25 70 40 75)(26 61 31 76)(27 62 32 77)(28 63 33 78)(29 64 34 79)(30 65 35 80)
(1 6)(2 7)(3 8)(4 9)(5 10)(11 16)(12 17)(13 18)(14 19)(15 20)(21 26)(22 27)(23 28)(24 29)(25 30)(31 36)(32 37)(33 38)(34 39)(35 40)(41 56)(42 57)(43 58)(44 59)(45 60)(46 51)(47 52)(48 53)(49 54)(50 55)(61 76)(62 77)(63 78)(64 79)(65 80)(66 71)(67 72)(68 73)(69 74)(70 75)
(1 33 13 23)(2 32 14 22)(3 31 15 21)(4 40 16 30)(5 39 17 29)(6 38 18 28)(7 37 19 27)(8 36 20 26)(9 35 11 25)(10 34 12 24)(41 80 51 70)(42 79 52 69)(43 78 53 68)(44 77 54 67)(45 76 55 66)(46 75 56 65)(47 74 57 64)(48 73 58 63)(49 72 59 62)(50 71 60 61)
(1 28 6 23)(2 29 7 24)(3 30 8 25)(4 21 9 26)(5 22 10 27)(11 36 16 31)(12 37 17 32)(13 38 18 33)(14 39 19 34)(15 40 20 35)(41 61 46 66)(42 62 47 67)(43 63 48 68)(44 64 49 69)(45 65 50 70)(51 71 56 76)(52 72 57 77)(53 73 58 78)(54 74 59 79)(55 75 60 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), (1,48,18,53)(2,49,19,54)(3,50,20,55)(4,41,11,56)(5,42,12,57)(6,43,13,58)(7,44,14,59)(8,45,15,60)(9,46,16,51)(10,47,17,52)(21,66,36,71)(22,67,37,72)(23,68,38,73)(24,69,39,74)(25,70,40,75)(26,61,31,76)(27,62,32,77)(28,63,33,78)(29,64,34,79)(30,65,35,80), (1,6)(2,7)(3,8)(4,9)(5,10)(11,16)(12,17)(13,18)(14,19)(15,20)(21,26)(22,27)(23,28)(24,29)(25,30)(31,36)(32,37)(33,38)(34,39)(35,40)(41,56)(42,57)(43,58)(44,59)(45,60)(46,51)(47,52)(48,53)(49,54)(50,55)(61,76)(62,77)(63,78)(64,79)(65,80)(66,71)(67,72)(68,73)(69,74)(70,75), (1,33,13,23)(2,32,14,22)(3,31,15,21)(4,40,16,30)(5,39,17,29)(6,38,18,28)(7,37,19,27)(8,36,20,26)(9,35,11,25)(10,34,12,24)(41,80,51,70)(42,79,52,69)(43,78,53,68)(44,77,54,67)(45,76,55,66)(46,75,56,65)(47,74,57,64)(48,73,58,63)(49,72,59,62)(50,71,60,61), (1,28,6,23)(2,29,7,24)(3,30,8,25)(4,21,9,26)(5,22,10,27)(11,36,16,31)(12,37,17,32)(13,38,18,33)(14,39,19,34)(15,40,20,35)(41,61,46,66)(42,62,47,67)(43,63,48,68)(44,64,49,69)(45,65,50,70)(51,71,56,76)(52,72,57,77)(53,73,58,78)(54,74,59,79)(55,75,60,80)>;`

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

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

53 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 4A 4B 4C ··· 4G 4H ··· 4M 4N 4O 5A 5B 10A ··· 10F 10G 10H 10I 10J 20A ··· 20H 20I ··· 20P order 1 2 2 2 2 2 2 2 2 2 4 4 4 ··· 4 4 ··· 4 4 4 5 5 10 ··· 10 10 10 10 10 20 ··· 20 20 ··· 20 size 1 1 1 1 2 2 10 10 20 20 2 2 4 ··· 4 10 ··· 10 20 20 2 2 2 ··· 2 4 4 4 4 4 ··· 4 8 ··· 8

53 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 D5 C4○D4 C4○D4 D10 D10 D10 D10 2+ 1+4 Q8⋊2D5 D4⋊6D10 D5×C4○D4 kernel C10.532+ 1+4 C23.11D10 D5×C22⋊C4 C22⋊D20 C22.D20 C4⋊C4⋊7D5 D20⋊8C4 D10.13D4 C4⋊C4⋊D5 C2×D10⋊C4 C4×C5⋊D4 D10⋊3Q8 C20.23D4 C5×C22⋊Q8 C22⋊Q8 D10 C2×C10 C22⋊C4 C4⋊C4 C22×C4 C2×Q8 C10 C22 C2 C2 # reps 1 1 1 1 1 1 1 2 2 1 1 1 1 1 2 4 4 4 6 2 2 1 4 4 4

Matrix representation of C10.532+ 1+4 in GL6(𝔽41)

 7 7 0 0 0 0 34 40 0 0 0 0 0 0 40 0 0 0 0 0 0 40 0 0 0 0 0 0 40 0 0 0 0 0 0 40
,
 40 0 0 0 0 0 0 40 0 0 0 0 0 0 0 40 0 0 0 0 40 0 0 0 0 0 0 0 9 0 0 0 0 0 2 32
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 40 0 0 0 0 0 0 1 0 0 0 0 0 0 40 0 0 0 0 0 0 40
,
 40 0 0 0 0 0 7 1 0 0 0 0 0 0 32 0 0 0 0 0 0 9 0 0 0 0 0 0 32 40 0 0 0 0 39 9
,
 40 0 0 0 0 0 0 40 0 0 0 0 0 0 32 0 0 0 0 0 0 9 0 0 0 0 0 0 32 40 0 0 0 0 0 9

`G:=sub<GL(6,GF(41))| [7,34,0,0,0,0,7,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,0,40,0,0,0,0,40,0,0,0,0,0,0,0,9,2,0,0,0,0,0,32],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[40,7,0,0,0,0,0,1,0,0,0,0,0,0,32,0,0,0,0,0,0,9,0,0,0,0,0,0,32,39,0,0,0,0,40,9],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,32,0,0,0,0,0,0,9,0,0,0,0,0,0,32,0,0,0,0,0,40,9] >;`

C10.532+ 1+4 in GAP, Magma, Sage, TeX

`C_{10}._{53}2_+^{1+4}`
`% in TeX`

`G:=Group("C10.53ES+(2,2)");`
`// GroupNames label`

`G:=SmallGroup(320,1309);`
`// by ID`

`G=gap.SmallGroup(320,1309);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,219,184,1571,297,136,12550]);`
`// Polycyclic`

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

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