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G = C23.256C24  order 128 = 27

109th central extension by C23 of C24

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

Aliases: C23.256C24, C24.223C23, C22.872+ (1+4), C22.D4⋊10C4, (C2×C42).27C22, (C23×C4).61C22, C23.25(C22×C4), C22.147(C23×C4), (C22×C4).1255C23, C24.C22⋊24C2, C23.23D4.11C2, (C22×D4).111C22, C23.63C23⋊23C2, C2.36(C22.11C24), C2.9(C22.45C24), C24.3C22.30C2, C2.C42.483C22, C2.5(C22.53C24), C2.11(C22.47C24), (C4×C4⋊C4)⋊50C2, C4⋊C4⋊32(C2×C4), C2.43(C4×C4○D4), C22⋊C4⋊18(C2×C4), (C4×C22⋊C4)⋊12C2, (C22×C4)⋊14(C2×C4), (C2×D4).132(C2×C4), (C2×C4).53(C22×C4), (C2×C4).853(C4○D4), (C2×C4⋊C4).832C22, C22.141(C2×C4○D4), (C2×C22⋊C4).39C22, (C2×C22.D4).8C2, SmallGroup(128,1106)

Series: Derived ►Chief ►Lower central ►Upper central ►Jennings

Derived series
C1 — C22 — C23.256C24
Chief series
C1 — C2 — C22 — C23 — C22×C4 — C23×C4 — C4×C22⋊C4 — C23.256C24
Lower central
C1 — C22 — C23.256C24
Upper central
C1 — C23 — C23.256C24
Jennings
C1 — C23 — C23.256C24

Subgroups: 492 in 276 conjugacy classes, 140 normal (22 characteristic)
C1, C2 [×3], C2 [×4], C2 [×4], C4 [×22], C22 [×3], C22 [×4], C22 [×20], C2×C4 [×18], C2×C4 [×38], D4 [×4], C23, C23 [×4], C23 [×12], C42 [×9], C22⋊C4 [×12], C22⋊C4 [×13], C4⋊C4 [×8], C4⋊C4 [×5], C22×C4, C22×C4 [×16], C22×C4 [×7], C2×D4 [×4], C2×D4 [×2], C24 [×2], C2.C42 [×10], C2×C42, C2×C42 [×6], C2×C22⋊C4 [×2], C2×C22⋊C4 [×8], C2×C4⋊C4, C2×C4⋊C4 [×4], C22.D4 [×8], C23×C4 [×2], C22×D4, C4×C22⋊C4 [×4], C4×C4⋊C4 [×2], C23.23D4, C23.63C23 [×2], C24.C22 [×4], C24.3C22, C2×C22.D4, C23.256C24

Quotients:
C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], C23 [×15], C22×C4 [×14], C4○D4 [×8], C24, C23×C4, C2×C4○D4 [×4], 2+ (1+4) [×2], C4×C4○D4 [×2], C22.11C24, C22.45C24, C22.47C24 [×2], C22.53C24, C23.256C24

Generators and relations
 G = < a,b,c,d,e,f,g | a2=b2=c2=1, d2=e2=c, f2=g2=ba=ab, ac=ca, ede-1=ad=da, geg-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, dg=gd, ef=fe, fg=gf >

Smallest permutation representation
►On 64 points
Generators in S64
(1 11)(2 12)(3 9)(4 10)(5 38)(6 39)(7 40)(8 37)(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 53)(26 54)(27 55)(28 56)(29 57)(30 58)(31 59)(32 60)(33 63)(34 64)(35 61)(36 62)

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

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

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

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

Matrix representation ►G ⊆ GL6(𝔽5)

400000
040000
004000
000400
000010
000001
,
100000
010000
001000
000100
000040
000004
,
100000
010000
004000
000400
000010
000001
,
100000
040000
003000
001200
000043
000001
,
010000
100000
002300
000300
000040
000004
,
200000
020000
002000
000200
000030
000022
,
300000
020000
003000
001200
000020
000002

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

50 conjugacy classes

class 1 2A···2G2H2I2J2K4A···4X4Y···4AL
order12···222224···44···4
size11···144442···24···4

50 irreducible representations

dim11111111124
type+++++++++
imageC1C2C2C2C2C2C2C2C4C4○D42+ (1+4)
kernelC23.256C24C4×C22⋊C4C4×C4⋊C4C23.23D4C23.63C23C24.C22C24.3C22C2×C22.D4C22.D4C2×C4C22
# reps1421241116162

In GAP, Magma, Sage, TeX 

C_2^3._{256}C_2^4
 
% in TeX
 
G:=Group("C2^3.256C2^4");
 
// GroupNames label
 
G:=SmallGroup(128,1106);
 
// by ID
 
G=gap.SmallGroup(128,1106);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,2,448,253,456,758,268,346,192]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=1,d^2=e^2=c,f^2=g^2=b*a=a*b,a*c=c*a,e*d*e^-1=a*d=d*a,g*e*g^-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,d*g=g*d,e*f=f*e,f*g=g*f>;
 
// generators/relations
 

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