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G = C23.231C24  order 128 = 27

84th central extension by C23 of C24

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

Aliases: C23.231C24, C24.556C23, C22.672+ 1+4, D4⋊6(C4⋊C4), (C4×D4)⋊22C4, C42⋊24(C2×C4), (C2×D4).36Q8, (C2×D4).342D4, C42⋊8C4⋊17C2, C2.4(D4⋊5D4), C2.3(D4⋊3Q8), C23.416(C2×D4), C23.114(C2×Q8), D4○3(C2.C42), (C23×C4).53C22, C23.7Q8⋊24C2, C23.8Q8⋊14C2, C22.39(C22×Q8), C22.122(C23×C4), (C2×C42).430C22, C23.130(C22×C4), C22.106(C22×D4), (C22×C4).1246C23, (C22×D4).610C22, C23.65C23⋊23C2, C2.26(C22.11C24), C2.C42.475C22, (C4×C4⋊C4)⋊37C2, C4⋊C4⋊44(C2×C4), C4.17(C2×C4⋊C4), (C2×C4×D4).34C2, C2.28(C4×C4○D4), C22.1(C2×C4⋊C4), C22⋊C4⋊41(C2×C4), (C22×C4)⋊32(C2×C4), C2.15(C22×C4⋊C4), (C2×C4).300(C2×Q8), (C2×D4).244(C2×C4), (C2×C4).1070(C2×D4), (C2×C4).720(C4○D4), (C2×C4⋊C4).187C22, (C2×C4).491(C22×C4), C22.116(C2×C4○D4), (C2×C2.C42)⋊19C2, (C2×D4)○3(C2.C42), (C2×C22⋊C4).439C22, SmallGroup(128,1081)

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

Derived series
C1 — C22 — C23.231C24
Chief series
C1 — C2 — C22 — C23 — C24 — C23×C4 — C2×C4×D4 — C23.231C24
Lower central
C1 — C22 — C23.231C24
Upper central
C1 — C23 — C23.231C24
Jennings
C1 — C23 — C23.231C24

Generators and relations for C23.231C24
 G = < a,b,c,d,e,f,g | a2=b2=c2=d2=f2=1, e2=d, g2=cb=bc, faf=ab=ba, ac=ca, ad=da, ae=ea, ag=ga, bd=db, fef=be=eb, bf=fb, bg=gb, cd=dc, geg-1=ce=ec, cf=fc, cg=gc, de=ed, df=fd, dg=gd, fg=gf >

Subgroups: 604 in 352 conjugacy classes, 184 normal (20 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C2×C4, C2×C4, D4, C23, C23, C23, C42, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C22×C4, C2×D4, C24, C2.C42, C2.C42, C2×C42, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C2×C4⋊C4, C4×D4, C4×D4, C23×C4, C22×D4, C2×C2.C42, C4×C4⋊C4, C23.7Q8, C42⋊8C4, C23.8Q8, C23.65C23, C2×C4×D4, C2×C4×D4, C23.231C24
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C23, C4⋊C4, C22×C4, C2×D4, C2×Q8, C4○D4, C24, C2×C4⋊C4, C23×C4, C22×D4, C22×Q8, C2×C4○D4, 2+ 1+4, C22×C4⋊C4, C4×C4○D4, C22.11C24, D4⋊5D4, D4⋊3Q8, C23.231C24

Smallest permutation representation of C23.231C24
►On 64 points
Generators in S64
(1 3)(2 4)(5 40)(6 37)(7 38)(8 39)(9 11)(10 12)(13 15)(14 16)(17 47)(18 48)(19 45)(20 46)(21 23)(22 24)(25 27)(26 28)(29 59)(30 60)(31 57)(32 58)(33 61)(34 62)(35 63)(36 64)(41 43)(42 44)(49 51)(50 52)(53 55)(54 56)

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

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

50 conjugacy classes

class 1 2A···2G2H···2O4A···4P4Q···4AH
order12···22···24···44···4
size11···12···22···24···4

50 irreducible representations

dim1111111112224
type+++++++++-+
imageC1C2C2C2C2C2C2C2C4D4Q8C4○D42+ 1+4
kernelC23.231C24C2×C2.C42C4×C4⋊C4C23.7Q8C42⋊8C4C23.8Q8C23.65C23C2×C4×D4C4×D4C2×D4C2×D4C2×C4C22
# reps12121423164482

Matrix representation of C23.231C24 ►in GL5(𝔽5)

10000
01000
00100
00040
00031
,
10000
01000
00100
00040
00004
,
10000
04000
00400
00010
00001
,
40000
01000
00100
00040
00004
,
20000
00100
01000
00030
00012
,
40000
01000
00100
00041
00001
,
40000
00100
04000
00020
00002

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

C23.231C24 in GAP, Magma, Sage, TeX 

C_2^3._{231}C_2^4
 
% in TeX
 
G:=Group("C2^3.231C2^4");
 
// GroupNames label
 
G:=SmallGroup(128,1081);
 
// by ID
 
G=gap.SmallGroup(128,1081);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,2,448,253,758,268,346]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=d^2=f^2=1,e^2=d,g^2=c*b=b*c,f*a*f=a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*g=g*a,b*d=d*b,f*e*f=b*e=e*b,b*f=f*b,b*g=g*b,c*d=d*c,g*e*g^-1=c*e=e*c,c*f=f*c,c*g=g*c,d*e=e*d,d*f=f*d,d*g=g*d,f*g=g*f>;
 
// generators/relations
 

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