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## G = C10.1202+ 1+4order 320 = 26·5

### 29th non-split extension by C10 of 2+ 1+4 acting via 2+ 1+4/C4○D4=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — C10.1202+ 1+4
 Chief series C1 — C5 — C10 — C2×C10 — C22×D5 — C23×D5 — C2×D4×D5 — C10.1202+ 1+4
 Lower central C5 — C2×C10 — C10.1202+ 1+4
 Upper central C1 — C22 — C22.D4

Generators and relations for C10.1202+ 1+4
G = < a,b,c,d,e | a10=b4=c2=e2=1, d2=a5b2, ab=ba, ac=ca, ad=da, eae=a-1, cbc=a5b-1, dbd-1=ebe=a5b, cd=dc, ce=ec, ede=a5b2d >

Subgroups: 1694 in 346 conjugacy classes, 103 normal (39 characteristic)
C1, C2, C2, C2, C4, C22, C22, C22, C5, C2×C4, C2×C4, C2×C4, D4, C23, C23, D5, C10, C10, C10, C22⋊C4, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×D4, C24, Dic5, C20, D10, D10, C2×C10, C2×C10, C2×C10, C2×C22⋊C4, C22≀C2, C4⋊D4, C22.D4, C22.D4, C22×D4, C4×D5, D20, C2×Dic5, C2×Dic5, C5⋊D4, C2×C20, C2×C20, C2×C20, C5×D4, C22×D5, C22×D5, C22×D5, C22×C10, C233D4, C10.D4, D10⋊C4, C23.D5, C5×C22⋊C4, C5×C22⋊C4, C5×C4⋊C4, C2×C4×D5, C2×C4×D5, C2×D20, C2×D20, C2×D20, D4×D5, C2×C5⋊D4, C2×C5⋊D4, C22×C20, D4×C10, C23×D5, D5×C22⋊C4, C22⋊D20, C22⋊D20, D10⋊D4, D10.13D4, C4⋊D20, C23.23D10, C23⋊D10, C5×C22.D4, C22×D20, C2×D4×D5, C10.1202+ 1+4
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, C24, D10, C22×D4, 2+ 1+4, C22×D5, C233D4, D4×D5, C23×D5, C2×D4×D5, D48D10, C10.1202+ 1+4

Smallest permutation representation of C10.1202+ 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 68 18 73)(2 69 19 74)(3 70 20 75)(4 61 11 76)(5 62 12 77)(6 63 13 78)(7 64 14 79)(8 65 15 80)(9 66 16 71)(10 67 17 72)(21 56 36 41)(22 57 37 42)(23 58 38 43)(24 59 39 44)(25 60 40 45)(26 51 31 46)(27 52 32 47)(28 53 33 48)(29 54 34 49)(30 55 35 50)
(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 34 14 24)(3 35 15 25)(4 36 16 26)(5 37 17 27)(6 38 18 28)(7 39 19 29)(8 40 20 30)(9 31 11 21)(10 32 12 22)(41 76 51 66)(42 77 52 67)(43 78 53 68)(44 79 54 69)(45 80 55 70)(46 71 56 61)(47 72 57 62)(48 73 58 63)(49 74 59 64)(50 75 60 65)
(1 28)(2 27)(3 26)(4 25)(5 24)(6 23)(7 22)(8 21)(9 30)(10 29)(11 40)(12 39)(13 38)(14 37)(15 36)(16 35)(17 34)(18 33)(19 32)(20 31)(41 75)(42 74)(43 73)(44 72)(45 71)(46 80)(47 79)(48 78)(49 77)(50 76)(51 65)(52 64)(53 63)(54 62)(55 61)(56 70)(57 69)(58 68)(59 67)(60 66)```

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

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

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

50 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 2J 2K 2L 2M 4A ··· 4E 4F 4G 4H 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 2 2 2 2 4 ··· 4 4 4 4 5 5 10 ··· 10 10 10 10 10 10 10 20 ··· 20 20 ··· 20 size 1 1 1 1 2 2 4 10 10 10 10 20 20 20 4 ··· 4 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 2 2 2 2 2 2 4 4 4 type + + + + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 D4 D5 D10 D10 D10 D10 2+ 1+4 D4×D5 D4⋊8D10 kernel C10.1202+ 1+4 D5×C22⋊C4 C22⋊D20 D10⋊D4 D10.13D4 C4⋊D20 C23.23D10 C23⋊D10 C5×C22.D4 C22×D20 C2×D4×D5 C22×D5 C22.D4 C22⋊C4 C4⋊C4 C22×C4 C2×D4 C10 C22 C2 # reps 1 1 3 2 2 2 1 1 1 1 1 4 2 6 4 2 2 2 4 8

Matrix representation of C10.1202+ 1+4 in GL6(𝔽41)

 40 0 0 0 0 0 0 40 0 0 0 0 0 0 40 7 0 0 0 0 34 7 0 0 0 0 14 0 7 7 0 0 0 27 34 40
,
 0 1 0 0 0 0 40 0 0 0 0 0 0 0 20 29 27 39 0 0 25 22 27 27 0 0 13 9 19 12 0 0 21 13 16 21
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 33 17 40 0 0 0 35 33 0 40
,
 40 0 0 0 0 0 0 1 0 0 0 0 0 0 11 32 0 0 0 0 9 30 0 0 0 0 23 0 30 32 0 0 0 18 9 11
,
 40 0 0 0 0 0 0 1 0 0 0 0 0 0 14 14 0 0 0 0 30 27 0 0 0 0 5 26 30 32 0 0 20 13 27 11

`G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,34,14,0,0,0,7,7,0,27,0,0,0,0,7,34,0,0,0,0,7,40],[0,40,0,0,0,0,1,0,0,0,0,0,0,0,20,25,13,21,0,0,29,22,9,13,0,0,27,27,19,16,0,0,39,27,12,21],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,33,35,0,0,0,1,17,33,0,0,0,0,40,0,0,0,0,0,0,40],[40,0,0,0,0,0,0,1,0,0,0,0,0,0,11,9,23,0,0,0,32,30,0,18,0,0,0,0,30,9,0,0,0,0,32,11],[40,0,0,0,0,0,0,1,0,0,0,0,0,0,14,30,5,20,0,0,14,27,26,13,0,0,0,0,30,27,0,0,0,0,32,11] >;`

C10.1202+ 1+4 in GAP, Magma, Sage, TeX

`C_{10}._{120}2_+^{1+4}`
`% in TeX`

`G:=Group("C10.120ES+(2,2)");`
`// GroupNames label`

`G:=SmallGroup(320,1325);`
`// by ID`

`G=gap.SmallGroup(320,1325);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,219,675,297,80,12550]);`
`// 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,a*d=d*a,e*a*e=a^-1,c*b*c=a^5*b^-1,d*b*d^-1=e*b*e=a^5*b,c*d=d*c,c*e=e*c,e*d*e=a^5*b^2*d>;`
`// generators/relations`

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