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G = C23.334C24order 128 = 27

51st central stem extension by C23 of C24

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C23 — C23.334C24
 Chief series C1 — C2 — C22 — C23 — C22×C4 — C2×C42 — C4×C22⋊C4 — C23.334C24
 Lower central C1 — C23 — C23.334C24
 Upper central C1 — C23 — C23.334C24
 Jennings C1 — C23 — C23.334C24

Generators and relations for C23.334C24
G = < a,b,c,d,e,f,g | a2=b2=c2=d2=1, e2=g2=b, f2=c, ab=ba, ac=ca, faf-1=ad=da, ae=ea, ag=ga, bc=cb, bd=db, geg-1=be=eb, bf=fb, bg=gb, cd=dc, fef-1=ce=ec, cf=fc, cg=gc, de=ed, gfg-1=df=fd, dg=gd >

Subgroups: 676 in 400 conjugacy classes, 140 normal (16 characteristic)
C1, C2 [×3], C2 [×4], C2 [×4], C4 [×4], C4 [×22], C22, C22 [×10], C22 [×12], C2×C4 [×24], C2×C4 [×54], Q8 [×36], C23, C23 [×6], C23 [×4], C42 [×4], C22⋊C4 [×8], C22⋊C4 [×4], C4⋊C4 [×22], C22×C4 [×2], C22×C4 [×16], C22×C4 [×16], C2×Q8 [×8], C2×Q8 [×58], C24, C2.C42 [×10], C2×C42 [×2], C2×C22⋊C4 [×4], C2×C4⋊C4, C2×C4⋊C4 [×10], C22⋊Q8 [×4], C4⋊Q8 [×8], C23×C4, C23×C4 [×2], C22×Q8, C22×Q8 [×4], C22×Q8 [×12], C4×C22⋊C4, C23.8Q8 [×4], C23.67C23 [×2], C23.78C23 [×4], C2×C22⋊Q8, C2×C4⋊Q8 [×2], Q8×C23, C23.334C24
Quotients: C1, C2 [×15], C22 [×35], D4 [×12], Q8 [×8], C23 [×15], C2×D4 [×18], C2×Q8 [×12], C24, C22≀C2 [×4], C4⋊Q8 [×4], C22×D4 [×3], C22×Q8 [×2], 2- 1+4 [×2], C2×C22≀C2, C2×C4⋊Q8, C23.38C23, D4×Q8 [×4], C23.334C24

Smallest permutation representation of C23.334C24
On 64 points
Generators in S64
```(1 9)(2 10)(3 11)(4 12)(5 24)(6 21)(7 22)(8 23)(13 41)(14 42)(15 43)(16 44)(17 61)(18 62)(19 63)(20 64)(25 55)(26 56)(27 53)(28 54)(29 59)(30 60)(31 57)(32 58)(33 46)(34 47)(35 48)(36 45)(37 50)(38 51)(39 52)(40 49)
(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 9)(2 10)(3 11)(4 12)(5 39)(6 40)(7 37)(8 38)(13 41)(14 42)(15 43)(16 44)(17 45)(18 46)(19 47)(20 48)(21 49)(22 50)(23 51)(24 52)(25 55)(26 56)(27 53)(28 54)(29 59)(30 60)(31 57)(32 58)(33 62)(34 63)(35 64)(36 61)
(1 53)(2 54)(3 55)(4 56)(5 52)(6 49)(7 50)(8 51)(9 27)(10 28)(11 25)(12 26)(13 31)(14 32)(15 29)(16 30)(17 36)(18 33)(19 34)(20 35)(21 40)(22 37)(23 38)(24 39)(41 57)(42 58)(43 59)(44 60)(45 61)(46 62)(47 63)(48 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 17 9 45)(2 46 10 18)(3 19 11 47)(4 48 12 20)(5 30 39 60)(6 57 40 31)(7 32 37 58)(8 59 38 29)(13 49 41 21)(14 22 42 50)(15 51 43 23)(16 24 44 52)(25 63 55 34)(26 35 56 64)(27 61 53 36)(28 33 54 62)
(1 13 3 15)(2 16 4 14)(5 20 7 18)(6 19 8 17)(9 41 11 43)(10 44 12 42)(21 63 23 61)(22 62 24 64)(25 59 27 57)(26 58 28 60)(29 53 31 55)(30 56 32 54)(33 52 35 50)(34 51 36 49)(37 46 39 48)(38 45 40 47)```

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

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

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

38 conjugacy classes

 class 1 2A ··· 2G 2H 2I 2J 2K 4A 4B 4C 4D 4E ··· 4V 4W 4X 4Y 4Z order 1 2 ··· 2 2 2 2 2 4 4 4 4 4 ··· 4 4 4 4 4 size 1 1 ··· 1 2 2 2 2 2 2 2 2 4 ··· 4 8 8 8 8

38 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 4 type + + + + + + + + - + + - image C1 C2 C2 C2 C2 C2 C2 C2 Q8 D4 D4 2- 1+4 kernel C23.334C24 C4×C22⋊C4 C23.8Q8 C23.67C23 C23.78C23 C2×C22⋊Q8 C2×C4⋊Q8 Q8×C23 C22⋊C4 C22×C4 C2×Q8 C22 # reps 1 1 4 2 4 1 2 1 8 4 8 2

Matrix representation of C23.334C24 in GL6(𝔽5)

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

`G:=sub<GL(6,GF(5))| [4,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,4,0,0,0,0,0,1],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[3,0,0,0,0,0,0,2,0,0,0,0,0,0,2,0,0,0,0,0,1,3,0,0,0,0,0,0,1,1,0,0,0,0,0,4],[0,4,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,1,0,0,0,0,3,4],[0,4,0,0,0,0,1,0,0,0,0,0,0,0,1,1,0,0,0,0,3,4,0,0,0,0,0,0,1,1,0,0,0,0,0,4] >;`

C23.334C24 in GAP, Magma, Sage, TeX

`C_2^3._{334}C_2^4`
`% in TeX`

`G:=Group("C2^3.334C2^4");`
`// GroupNames label`

`G:=SmallGroup(128,1166);`
`// by ID`

`G=gap.SmallGroup(128,1166);`
`# by ID`

`G:=PCGroup([7,-2,2,2,2,-2,2,2,112,253,120,758,723,184,675,80]);`
`// Polycyclic`

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

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