Copied to
clipboard

## G = C22.M5(2)  order 128 = 27

### 2nd non-split extension by C22 of M5(2) acting via M5(2)/C2×C8=C2

p-group, metabelian, nilpotent (class 3), monomial

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C22 — C22.M5(2)
 Chief series C1 — C2 — C4 — C2×C4 — C2×C8 — C22×C8 — C2×C4⋊C8 — C22.M5(2)
 Lower central C1 — C2 — C22 — C22.M5(2)
 Upper central C1 — C2×C4 — C22×C8 — C22.M5(2)
 Jennings C1 — C2 — C2 — C2 — C2 — C4 — C2×C4 — C22×C8 — C22.M5(2)

Generators and relations for C22.M5(2)
G = < a,b,c,d | a2=b2=c16=1, d2=b, cac-1=ab=ba, ad=da, bc=cb, bd=db, dcd-1=abc9 >

Smallest permutation representation of C22.M5(2)
On 64 points
Generators in S64
```(1 9)(2 24)(3 11)(4 26)(5 13)(6 28)(7 15)(8 30)(10 32)(12 18)(14 20)(16 22)(17 25)(19 27)(21 29)(23 31)(33 56)(34 42)(35 58)(36 44)(37 60)(38 46)(39 62)(40 48)(41 64)(43 50)(45 52)(47 54)(49 57)(51 59)(53 61)(55 63)
(1 31)(2 32)(3 17)(4 18)(5 19)(6 20)(7 21)(8 22)(9 23)(10 24)(11 25)(12 26)(13 27)(14 28)(15 29)(16 30)(33 64)(34 49)(35 50)(36 51)(37 52)(38 53)(39 54)(40 55)(41 56)(42 57)(43 58)(44 59)(45 60)(46 61)(47 62)(48 63)
(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)
(1 59 31 44)(2 45 32 60)(3 46 17 61)(4 62 18 47)(5 63 19 48)(6 33 20 64)(7 34 21 49)(8 50 22 35)(9 51 23 36)(10 37 24 52)(11 38 25 53)(12 54 26 39)(13 55 27 40)(14 41 28 56)(15 42 29 57)(16 58 30 43)```

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

`G:=Group( (1,9)(2,24)(3,11)(4,26)(5,13)(6,28)(7,15)(8,30)(10,32)(12,18)(14,20)(16,22)(17,25)(19,27)(21,29)(23,31)(33,56)(34,42)(35,58)(36,44)(37,60)(38,46)(39,62)(40,48)(41,64)(43,50)(45,52)(47,54)(49,57)(51,59)(53,61)(55,63), (1,31)(2,32)(3,17)(4,18)(5,19)(6,20)(7,21)(8,22)(9,23)(10,24)(11,25)(12,26)(13,27)(14,28)(15,29)(16,30)(33,64)(34,49)(35,50)(36,51)(37,52)(38,53)(39,54)(40,55)(41,56)(42,57)(43,58)(44,59)(45,60)(46,61)(47,62)(48,63), (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), (1,59,31,44)(2,45,32,60)(3,46,17,61)(4,62,18,47)(5,63,19,48)(6,33,20,64)(7,34,21,49)(8,50,22,35)(9,51,23,36)(10,37,24,52)(11,38,25,53)(12,54,26,39)(13,55,27,40)(14,41,28,56)(15,42,29,57)(16,58,30,43) );`

`G=PermutationGroup([(1,9),(2,24),(3,11),(4,26),(5,13),(6,28),(7,15),(8,30),(10,32),(12,18),(14,20),(16,22),(17,25),(19,27),(21,29),(23,31),(33,56),(34,42),(35,58),(36,44),(37,60),(38,46),(39,62),(40,48),(41,64),(43,50),(45,52),(47,54),(49,57),(51,59),(53,61),(55,63)], [(1,31),(2,32),(3,17),(4,18),(5,19),(6,20),(7,21),(8,22),(9,23),(10,24),(11,25),(12,26),(13,27),(14,28),(15,29),(16,30),(33,64),(34,49),(35,50),(36,51),(37,52),(38,53),(39,54),(40,55),(41,56),(42,57),(43,58),(44,59),(45,60),(46,61),(47,62),(48,63)], [(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)], [(1,59,31,44),(2,45,32,60),(3,46,17,61),(4,62,18,47),(5,63,19,48),(6,33,20,64),(7,34,21,49),(8,50,22,35),(9,51,23,36),(10,37,24,52),(11,38,25,53),(12,54,26,39),(13,55,27,40),(14,41,28,56),(15,42,29,57),(16,58,30,43)])`

44 conjugacy classes

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 8A ··· 8H 8I 8J 8K 8L 16A ··· 16P order 1 2 2 2 2 2 4 4 4 4 4 4 4 4 4 4 8 ··· 8 8 8 8 8 16 ··· 16 size 1 1 1 1 2 2 1 1 1 1 2 2 4 4 4 4 2 ··· 2 4 4 4 4 4 ··· 4

44 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 4 4 4 type + + + + + - image C1 C2 C2 C4 C4 C8 C16 D4 M4(2) M5(2) C23⋊C4 C4.10D4 C23.C8 kernel C22.M5(2) C22⋊C16 C2×C4⋊C8 C2×C42 C22×C8 C22×C4 C2×C4 C2×C8 C2×C4 C22 C4 C4 C2 # reps 1 2 1 2 2 8 16 2 2 4 1 1 2

Matrix representation of C22.M5(2) in GL6(𝔽17)

 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 3 12 16 0 0 0 3 15 0 16
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 16 0 0 0 0 0 0 16 0 0 0 0 0 0 16 0 0 0 0 0 0 16
,
 7 7 0 0 0 0 3 10 0 0 0 0 0 0 11 10 4 0 0 0 7 6 9 9 0 0 16 4 9 3 0 0 16 5 14 8
,
 16 0 0 0 0 0 2 1 0 0 0 0 0 0 13 0 0 0 0 0 0 4 0 0 0 0 0 14 13 0 0 0 10 3 8 4

`G:=sub<GL(6,GF(17))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,3,3,0,0,0,1,12,15,0,0,0,0,16,0,0,0,0,0,0,16],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[7,3,0,0,0,0,7,10,0,0,0,0,0,0,11,7,16,16,0,0,10,6,4,5,0,0,4,9,9,14,0,0,0,9,3,8],[16,2,0,0,0,0,0,1,0,0,0,0,0,0,13,0,0,10,0,0,0,4,14,3,0,0,0,0,13,8,0,0,0,0,0,4] >;`

C22.M5(2) in GAP, Magma, Sage, TeX

`C_2^2.M_5(2)`
`% in TeX`

`G:=Group("C2^2.M5(2)");`
`// GroupNames label`

`G:=SmallGroup(128,54);`
`// by ID`

`G=gap.SmallGroup(128,54);`
`# by ID`

`G:=PCGroup([7,-2,2,-2,2,-2,2,-2,56,85,120,422,346,136,124]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^2=b^2=c^16=1,d^2=b,c*a*c^-1=a*b=b*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=a*b*c^9>;`
`// generators/relations`

Export

׿
×
𝔽