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## G = C10.612+ 1+4order 320 = 26·5

### 61st 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 — C10.612+ 1+4
 Chief series C1 — C5 — C10 — C2×C10 — C22×D5 — C23×D5 — D5×C22⋊C4 — C10.612+ 1+4
 Lower central C5 — C2×C10 — C10.612+ 1+4
 Upper central C1 — C22 — C22.D4

Generators and relations for C10.612+ 1+4
G = < a,b,c,d,e | a10=b4=c2=e2=1, d2=a5b2, ab=ba, ac=ca, dad-1=a-1, ae=ea, cbc=a5b-1, dbd-1=ebe=a5b, dcd-1=ece=a5c, ede=a5b2d >

Subgroups: 1070 in 250 conjugacy classes, 93 normal (91 characteristic)
C1, C2, C2, C4, C22, C22, C5, C2×C4, C2×C4, D4, Q8, C23, C23, D5, C10, C10, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C24, Dic5, C20, D10, D10, C2×C10, C2×C10, C2×C22⋊C4, C4×D4, C22≀C2, C4⋊D4, C22⋊Q8, C22.D4, C22.D4, C4.4D4, C422C2, Dic10, C4×D5, D20, C2×Dic5, C5⋊D4, C2×C20, C2×C20, C5×D4, C22×D5, C22×D5, C22×C10, C22.32C24, C4×Dic5, C10.D4, C4⋊Dic5, D10⋊C4, C23.D5, C5×C22⋊C4, C5×C4⋊C4, C2×Dic10, C2×C4×D5, C2×D20, C2×C5⋊D4, C22×C20, D4×C10, C23×D5, C23.D10, D5×C22⋊C4, C22⋊D20, D10⋊D4, Dic5.5D4, D208C4, D10.13D4, D102Q8, C4⋊C4⋊D5, C4×C5⋊D4, C207D4, C23⋊D10, C202D4, C5×C22.D4, C10.612+ 1+4
Quotients: C1, C2, C22, C23, D5, C4○D4, C24, D10, C2×C4○D4, 2+ 1+4, C22×D5, C22.32C24, C23×D5, D46D10, D5×C4○D4, D48D10, C10.612+ 1+4

Smallest permutation representation of C10.612+ 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 43 18 58)(2 44 19 59)(3 45 20 60)(4 46 11 51)(5 47 12 52)(6 48 13 53)(7 49 14 54)(8 50 15 55)(9 41 16 56)(10 42 17 57)(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)(41 56)(42 57)(43 58)(44 59)(45 60)(46 51)(47 52)(48 53)(49 54)(50 55)(61 71)(62 72)(63 73)(64 74)(65 75)(66 76)(67 77)(68 78)(69 79)(70 80)
(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 75 51 65)(42 74 52 64)(43 73 53 63)(44 72 54 62)(45 71 55 61)(46 80 56 70)(47 79 57 69)(48 78 58 68)(49 77 59 67)(50 76 60 66)
(1 23)(2 24)(3 25)(4 26)(5 27)(6 28)(7 29)(8 30)(9 21)(10 22)(11 31)(12 32)(13 33)(14 34)(15 35)(16 36)(17 37)(18 38)(19 39)(20 40)(41 61)(42 62)(43 63)(44 64)(45 65)(46 66)(47 67)(48 68)(49 69)(50 70)(51 71)(52 72)(53 73)(54 74)(55 75)(56 76)(57 77)(58 78)(59 79)(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), (1,43,18,58)(2,44,19,59)(3,45,20,60)(4,46,11,51)(5,47,12,52)(6,48,13,53)(7,49,14,54)(8,50,15,55)(9,41,16,56)(10,42,17,57)(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)(41,56)(42,57)(43,58)(44,59)(45,60)(46,51)(47,52)(48,53)(49,54)(50,55)(61,71)(62,72)(63,73)(64,74)(65,75)(66,76)(67,77)(68,78)(69,79)(70,80), (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)(41,75,51,65)(42,74,52,64)(43,73,53,63)(44,72,54,62)(45,71,55,61)(46,80,56,70)(47,79,57,69)(48,78,58,68)(49,77,59,67)(50,76,60,66), (1,23)(2,24)(3,25)(4,26)(5,27)(6,28)(7,29)(8,30)(9,21)(10,22)(11,31)(12,32)(13,33)(14,34)(15,35)(16,36)(17,37)(18,38)(19,39)(20,40)(41,61)(42,62)(43,63)(44,64)(45,65)(46,66)(47,67)(48,68)(49,69)(50,70)(51,71)(52,72)(53,73)(54,74)(55,75)(56,76)(57,77)(58,78)(59,79)(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), (1,43,18,58)(2,44,19,59)(3,45,20,60)(4,46,11,51)(5,47,12,52)(6,48,13,53)(7,49,14,54)(8,50,15,55)(9,41,16,56)(10,42,17,57)(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)(41,56)(42,57)(43,58)(44,59)(45,60)(46,51)(47,52)(48,53)(49,54)(50,55)(61,71)(62,72)(63,73)(64,74)(65,75)(66,76)(67,77)(68,78)(69,79)(70,80), (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)(41,75,51,65)(42,74,52,64)(43,73,53,63)(44,72,54,62)(45,71,55,61)(46,80,56,70)(47,79,57,69)(48,78,58,68)(49,77,59,67)(50,76,60,66), (1,23)(2,24)(3,25)(4,26)(5,27)(6,28)(7,29)(8,30)(9,21)(10,22)(11,31)(12,32)(13,33)(14,34)(15,35)(16,36)(17,37)(18,38)(19,39)(20,40)(41,61)(42,62)(43,63)(44,64)(45,65)(46,66)(47,67)(48,68)(49,69)(50,70)(51,71)(52,72)(53,73)(54,74)(55,75)(56,76)(57,77)(58,78)(59,79)(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)], [(1,43,18,58),(2,44,19,59),(3,45,20,60),(4,46,11,51),(5,47,12,52),(6,48,13,53),(7,49,14,54),(8,50,15,55),(9,41,16,56),(10,42,17,57),(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),(41,56),(42,57),(43,58),(44,59),(45,60),(46,51),(47,52),(48,53),(49,54),(50,55),(61,71),(62,72),(63,73),(64,74),(65,75),(66,76),(67,77),(68,78),(69,79),(70,80)], [(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),(41,75,51,65),(42,74,52,64),(43,73,53,63),(44,72,54,62),(45,71,55,61),(46,80,56,70),(47,79,57,69),(48,78,58,68),(49,77,59,67),(50,76,60,66)], [(1,23),(2,24),(3,25),(4,26),(5,27),(6,28),(7,29),(8,30),(9,21),(10,22),(11,31),(12,32),(13,33),(14,34),(15,35),(16,36),(17,37),(18,38),(19,39),(20,40),(41,61),(42,62),(43,63),(44,64),(45,65),(46,66),(47,67),(48,68),(49,69),(50,70),(51,71),(52,72),(53,73),(54,74),(55,75),(56,76),(57,77),(58,78),(59,79),(60,80)]])`

50 conjugacy classes

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

50 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 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 C2 D5 C4○D4 D10 D10 D10 D10 2+ 1+4 D4⋊6D10 D5×C4○D4 D4⋊8D10 kernel C10.612+ 1+4 C23.D10 D5×C22⋊C4 C22⋊D20 D10⋊D4 Dic5.5D4 D20⋊8C4 D10.13D4 D10⋊2Q8 C4⋊C4⋊D5 C4×C5⋊D4 C20⋊7D4 C23⋊D10 C20⋊2D4 C5×C22.D4 C22.D4 D10 C22⋊C4 C4⋊C4 C22×C4 C2×D4 C10 C2 C2 C2 # reps 1 1 1 1 1 2 1 1 1 1 1 1 1 1 1 2 4 6 4 2 2 2 4 4 4

Matrix representation of C10.612+ 1+4 in GL6(𝔽41)

 40 0 0 0 0 0 0 40 0 0 0 0 0 0 35 35 0 0 0 0 6 40 0 0 0 0 0 0 35 35 0 0 0 0 6 40
,
 32 0 0 0 0 0 16 9 0 0 0 0 0 0 0 0 40 0 0 0 0 0 0 40 0 0 1 0 0 0 0 0 0 1 0 0
,
 40 0 0 0 0 0 20 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 40 0 0 0 0 0 0 40
,
 40 4 0 0 0 0 0 1 0 0 0 0 0 0 23 6 0 0 0 0 21 18 0 0 0 0 0 0 23 6 0 0 0 0 21 18
,
 1 37 0 0 0 0 0 40 0 0 0 0 0 0 23 6 0 0 0 0 35 18 0 0 0 0 0 0 23 6 0 0 0 0 35 18

`G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,35,6,0,0,0,0,35,40,0,0,0,0,0,0,35,6,0,0,0,0,35,40],[32,16,0,0,0,0,0,9,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,40,0,0,0,0,0,0,40,0,0],[40,20,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[40,0,0,0,0,0,4,1,0,0,0,0,0,0,23,21,0,0,0,0,6,18,0,0,0,0,0,0,23,21,0,0,0,0,6,18],[1,0,0,0,0,0,37,40,0,0,0,0,0,0,23,35,0,0,0,0,6,18,0,0,0,0,0,0,23,35,0,0,0,0,6,18] >;`

C10.612+ 1+4 in GAP, Magma, Sage, TeX

`C_{10}._{61}2_+^{1+4}`
`% in TeX`

`G:=Group("C10.61ES+(2,2)");`
`// GroupNames label`

`G:=SmallGroup(320,1329);`
`// by ID`

`G=gap.SmallGroup(320,1329);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,219,184,675,570,12550]);`
`// Polycyclic`

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

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