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## G = C24.198C23order 128 = 27

### 38th non-split extension by C24 of C23 acting via C23/C2=C22

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C22 — C24.198C23
 Chief series C1 — C2 — C22 — C23 — C24 — C23×C4 — C2×C2.C42 — C24.198C23
 Lower central C1 — C22 — C24.198C23
 Upper central C1 — C23 — C24.198C23
 Jennings C1 — C23 — C24.198C23

Generators and relations for C24.198C23
G = < a,b,c,d,e,f,g | a2=b2=c2=g2=1, d2=e2=c, f2=b, ab=ba, ac=ca, ede-1=gdg=ad=da, fef-1=ae=ea, af=fa, ag=ga, bc=cb, fdf-1=bd=db, be=eb, bf=fb, bg=gb, cd=dc, ce=ec, cf=fc, cg=gc, eg=ge, fg=gf >

Subgroups: 668 in 348 conjugacy classes, 148 normal (30 characteristic)
C1, C2 [×7], C2 [×8], C4 [×18], C22 [×7], C22 [×4], C22 [×32], C2×C4 [×12], C2×C4 [×50], D4 [×20], C23, C23 [×10], C23 [×16], C42 [×4], C22⋊C4 [×8], C22⋊C4 [×12], C4⋊C4 [×4], C4⋊C4 [×6], C22×C4 [×2], C22×C4 [×18], C22×C4 [×16], C2×D4 [×12], C2×D4 [×10], C24, C24 [×2], C2.C42 [×10], C2×C42 [×2], C2×C22⋊C4 [×10], C2×C4⋊C4 [×3], C2×C4⋊C4 [×2], C4×D4 [×8], C4⋊D4 [×8], C23×C4 [×3], C23×C4 [×2], C22×D4, C22×D4 [×2], C2×C2.C42, C23.7Q8, C23.8Q8 [×2], C23.23D4 [×4], C23.63C23 [×2], C24.C22 [×2], C2×C4×D4 [×2], C2×C4⋊D4, C24.198C23
Quotients: C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], D4 [×4], C23 [×15], C22×C4 [×14], C2×D4 [×6], C4○D4 [×2], C24, C4×D4 [×4], C23×C4, C22×D4, C2×C4○D4, 2+ 1+4 [×3], 2- 1+4, C2×C4×D4, C22.11C24, C23.33C23, C233D4, C22.31C24, C22.32C24, C22.33C24, C24.198C23

Smallest permutation representation of C24.198C23
On 64 points
Generators in S64
```(1 12)(2 9)(3 10)(4 11)(5 38)(6 39)(7 40)(8 37)(13 36)(14 33)(15 34)(16 35)(17 60)(18 57)(19 58)(20 59)(21 56)(22 53)(23 54)(24 55)(25 48)(26 45)(27 46)(28 47)(29 52)(30 49)(31 50)(32 51)(41 64)(42 61)(43 62)(44 63)
(1 53)(2 54)(3 55)(4 56)(5 61)(6 62)(7 63)(8 64)(9 23)(10 24)(11 21)(12 22)(13 31)(14 32)(15 29)(16 30)(17 26)(18 27)(19 28)(20 25)(33 51)(34 52)(35 49)(36 50)(37 41)(38 42)(39 43)(40 44)(45 60)(46 57)(47 58)(48 59)
(1 3)(2 4)(5 7)(6 8)(9 11)(10 12)(13 15)(14 16)(17 19)(18 20)(21 23)(22 24)(25 27)(26 28)(29 31)(30 32)(33 35)(34 36)(37 39)(38 40)(41 43)(42 44)(45 47)(46 48)(49 51)(50 52)(53 55)(54 56)(57 59)(58 60)(61 63)(62 64)
(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 34 3 36)(2 16 4 14)(5 17 7 19)(6 57 8 59)(9 35 11 33)(10 13 12 15)(18 37 20 39)(21 51 23 49)(22 29 24 31)(25 43 27 41)(26 63 28 61)(30 56 32 54)(38 60 40 58)(42 45 44 47)(46 64 48 62)(50 53 52 55)
(1 38 53 42)(2 43 54 39)(3 40 55 44)(4 41 56 37)(5 22 61 12)(6 9 62 23)(7 24 63 10)(8 11 64 21)(13 58 31 47)(14 48 32 59)(15 60 29 45)(16 46 30 57)(17 52 26 34)(18 35 27 49)(19 50 28 36)(20 33 25 51)
(1 16)(2 36)(3 14)(4 34)(5 27)(6 47)(7 25)(8 45)(9 13)(10 33)(11 15)(12 35)(17 41)(18 61)(19 43)(20 63)(21 29)(22 49)(23 31)(24 51)(26 37)(28 39)(30 53)(32 55)(38 46)(40 48)(42 57)(44 59)(50 54)(52 56)(58 62)(60 64)```

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

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

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

44 conjugacy classes

 class 1 2A ··· 2G 2H 2I 2J 2K 2L 2M 2N 2O 4A ··· 4H 4I ··· 4AB order 1 2 ··· 2 2 2 2 2 2 2 2 2 4 ··· 4 4 ··· 4 size 1 1 ··· 1 2 2 2 2 4 4 4 4 2 ··· 2 4 ··· 4

44 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 4 4 type + + + + + + + + + + + - image C1 C2 C2 C2 C2 C2 C2 C2 C2 C4 D4 C4○D4 2+ 1+4 2- 1+4 kernel C24.198C23 C2×C2.C42 C23.7Q8 C23.8Q8 C23.23D4 C23.63C23 C24.C22 C2×C4×D4 C2×C4⋊D4 C4⋊D4 C22×C4 C23 C22 C22 # reps 1 1 1 2 4 2 2 2 1 16 4 4 3 1

Matrix representation of C24.198C23 in GL8(𝔽5)

 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 4
,
 4 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 4
,
 4 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 4
,
 3 0 0 0 0 0 0 0 1 2 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 3 0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 0 3
,
 3 0 0 0 0 0 0 0 0 3 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 2 0 0 0 0 0 0 0 0 0 0 3 0 0 0 0 0 0 3 0
,
 4 1 0 0 0 0 0 0 3 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 4 0 0 0 0 0 0 0 0 4 0 0
,
 4 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0

`G:=sub<GL(8,GF(5))| [1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4],[4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4],[4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4],[3,1,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,3],[3,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,3,0],[4,3,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0],[4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0] >;`

C24.198C23 in GAP, Magma, Sage, TeX

`C_2^4._{198}C_2^3`
`% in TeX`

`G:=Group("C2^4.198C2^3");`
`// GroupNames label`

`G:=SmallGroup(128,1057);`
`// by ID`

`G=gap.SmallGroup(128,1057);`
`# by ID`

`G:=PCGroup([7,-2,2,2,2,-2,2,2,448,253,456,758,219,184,675]);`
`// Polycyclic`

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

׿
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