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## G = C23.22M4(2)  order 128 = 27

### 4th non-split extension by C23 of M4(2) acting via M4(2)/C8=C2

p-group, metabelian, nilpotent (class 2), monomial

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C22 — C23.22M4(2)
 Chief series C1 — C2 — C4 — C2×C4 — C22×C4 — C23×C4 — C23×C8 — C23.22M4(2)
 Lower central C1 — C22 — C23.22M4(2)
 Upper central C1 — C22×C4 — C23.22M4(2)
 Jennings C1 — C2 — C2 — C22×C4 — C23.22M4(2)

Generators and relations for C23.22M4(2)
G = < a,b,c,d,e | a2=b2=c2=d8=e2=1, ab=ba, dad-1=eae=ac=ca, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede=bd5 >

Subgroups: 380 in 216 conjugacy classes, 80 normal (30 characteristic)
C1, C2 [×3], C2 [×4], C2 [×6], C4 [×2], C4 [×2], C4 [×6], C22 [×3], C22 [×8], C22 [×22], C8 [×6], C2×C4 [×2], C2×C4 [×8], C2×C4 [×22], D4 [×8], C23, C23 [×8], C23 [×10], C42 [×2], C22⋊C4 [×4], C4⋊C4 [×2], C2×C8 [×4], C2×C8 [×18], C22×C4 [×5], C22×C4 [×4], C22×C4 [×10], C2×D4 [×4], C2×D4 [×4], C24 [×2], C22⋊C8 [×6], C2×C42, C2×C22⋊C4 [×2], C2×C4⋊C4, C4×D4 [×4], C22×C8 [×4], C22×C8 [×6], C23×C4 [×2], C22×D4, C22.7C42 [×2], C2×C22⋊C8, C2×C22⋊C8 [×2], C2×C4×D4, C23×C8, C23.22M4(2)
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C8 [×4], C2×C4 [×6], D4 [×8], C23, C22⋊C4 [×4], C2×C8 [×6], M4(2) [×2], C22×C4, C2×D4 [×4], C4○D4 [×2], C22⋊C8 [×4], C2×C22⋊C4, C4×D4 [×2], C22≀C2, C4⋊D4 [×2], C22.D4, C22×C8, C2×M4(2), C8○D4 [×2], C23.23D4, C2×C22⋊C8, (C22×C8)⋊C2, C8×D4 [×2], C89D4 [×2], C23.22M4(2)

Smallest permutation representation of C23.22M4(2)
On 64 points
Generators in S64
```(1 59)(2 47)(3 61)(4 41)(5 63)(6 43)(7 57)(8 45)(9 24)(10 38)(11 18)(12 40)(13 20)(14 34)(15 22)(16 36)(17 30)(19 32)(21 26)(23 28)(25 33)(27 35)(29 37)(31 39)(42 55)(44 49)(46 51)(48 53)(50 58)(52 60)(54 62)(56 64)
(1 14)(2 15)(3 16)(4 9)(5 10)(6 11)(7 12)(8 13)(17 42)(18 43)(19 44)(20 45)(21 46)(22 47)(23 48)(24 41)(25 50)(26 51)(27 52)(28 53)(29 54)(30 55)(31 56)(32 49)(33 58)(34 59)(35 60)(36 61)(37 62)(38 63)(39 64)(40 57)
(1 51)(2 52)(3 53)(4 54)(5 55)(6 56)(7 49)(8 50)(9 29)(10 30)(11 31)(12 32)(13 25)(14 26)(15 27)(16 28)(17 38)(18 39)(19 40)(20 33)(21 34)(22 35)(23 36)(24 37)(41 62)(42 63)(43 64)(44 57)(45 58)(46 59)(47 60)(48 61)
(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 10)(3 12)(5 14)(7 16)(17 59)(18 39)(19 61)(20 33)(21 63)(22 35)(23 57)(24 37)(26 55)(28 49)(30 51)(32 53)(34 42)(36 44)(38 46)(40 48)(41 62)(43 64)(45 58)(47 60)```

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

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

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

56 conjugacy classes

 class 1 2A ··· 2G 2H 2I 2J 2K 2L 2M 4A ··· 4H 4I 4J 4K 4L 4M ··· 4R 8A ··· 8P 8Q ··· 8X order 1 2 ··· 2 2 2 2 2 2 2 4 ··· 4 4 4 4 4 4 ··· 4 8 ··· 8 8 ··· 8 size 1 1 ··· 1 2 2 2 2 4 4 1 ··· 1 2 2 2 2 4 ··· 4 2 ··· 2 4 ··· 4

56 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 2 2 2 2 type + + + + + + + image C1 C2 C2 C2 C2 C4 C4 C4 C8 D4 D4 C4○D4 M4(2) C8○D4 kernel C23.22M4(2) C22.7C42 C2×C22⋊C8 C2×C4×D4 C23×C8 C2×C22⋊C4 C2×C4⋊C4 C22×D4 C2×D4 C2×C8 C22×C4 C2×C4 C23 C22 # reps 1 2 3 1 1 4 2 2 16 4 4 4 4 8

Matrix representation of C23.22M4(2) in GL5(𝔽17)

 1 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 1 0
,
 16 0 0 0 0 0 16 0 0 0 0 0 16 0 0 0 0 0 1 0 0 0 0 0 1
,
 1 0 0 0 0 0 16 0 0 0 0 0 16 0 0 0 0 0 16 0 0 0 0 0 16
,
 8 0 0 0 0 0 8 0 0 0 0 0 9 0 0 0 0 0 0 9 0 0 0 8 0
,
 16 0 0 0 0 0 1 0 0 0 0 0 16 0 0 0 0 0 16 0 0 0 0 0 1

`G:=sub<GL(5,GF(17))| [1,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,1,0],[16,0,0,0,0,0,16,0,0,0,0,0,16,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,16,0,0,0,0,0,16,0,0,0,0,0,16,0,0,0,0,0,16],[8,0,0,0,0,0,8,0,0,0,0,0,9,0,0,0,0,0,0,8,0,0,0,9,0],[16,0,0,0,0,0,1,0,0,0,0,0,16,0,0,0,0,0,16,0,0,0,0,0,1] >;`

C23.22M4(2) in GAP, Magma, Sage, TeX

`C_2^3._{22}M_4(2)`
`% in TeX`

`G:=Group("C2^3.22M4(2)");`
`// GroupNames label`

`G:=SmallGroup(128,601);`
`// by ID`

`G=gap.SmallGroup(128,601);`
`# by ID`

`G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,288,422,124]);`
`// Polycyclic`

`G:=Group<a,b,c,d,e|a^2=b^2=c^2=d^8=e^2=1,a*b=b*a,d*a*d^-1=e*a*e=a*c=c*a,b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e=b*d^5>;`
`// generators/relations`

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