Copied to
clipboard

G = C23.567C24  order 128 = 27

284th central stem extension by C23 of C24

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

Aliases: C23.567C24, C24.380C23, C22.3412+ 1+4, C22.2562- 1+4, C23.70(C2×Q8), (C22×C4).62Q8, (C23×C4).440C22, (C22×C4).172C23, (C2×C42).631C22, C2.12(C23⋊2Q8), C23.7Q8.63C2, C23.Q8.26C2, C23.4Q8.18C2, C22.141(C22×Q8), C23.34D4.25C2, C23.81C23⋊74C2, C23.63C23⋊123C2, C23.65C23⋊112C2, C2.C42.281C22, C2.56(C22.33C24), C2.37(C22.34C24), C2.28(C23.41C23), C2.44(C23.37C23), (C2×C4).170(C2×Q8), (C4×C22⋊C4).75C2, (C2×C4).187(C4○D4), (C2×C4⋊C4).388C22, C22.434(C2×C4○D4), (C2×C22⋊C4).523C22, SmallGroup(128,1399)

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

Derived series
C1 — C23 — C23.567C24
Chief series
C1 — C2 — C22 — C23 — C22×C4 — C2.C42 — C23.65C23 — C23.567C24
Lower central
C1 — C23 — C23.567C24
Upper central
C1 — C23 — C23.567C24
Jennings
C1 — C23 — C23.567C24

Generators and relations for C23.567C24
 G = < a,b,c,d,e,f,g | a2=b2=c2=d2=1, e2=d, f2=db=bd, g2=c, eae-1=ab=ba, faf-1=ac=ca, ad=da, ag=ga, bc=cb, be=eb, gfg-1=bf=fb, bg=gb, cd=dc, ce=ec, cf=fc, cg=gc, fef-1=de=ed, df=fd, dg=gd, eg=ge >

Subgroups: 388 in 204 conjugacy classes, 96 normal (22 characteristic)
C1, C2, C2, C2, C4, C22, C22, C22, C2×C4, C2×C4, C23, C23, C23, C42, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C22×C4, C24, C2.C42, C2.C42, C2×C42, C2×C22⋊C4, C2×C22⋊C4, C2×C4⋊C4, C23×C4, C4×C22⋊C4, C23.7Q8, C23.34D4, C23.63C23, C23.65C23, C23.Q8, C23.81C23, C23.4Q8, C23.567C24
Quotients: C1, C2, C22, Q8, C23, C2×Q8, C4○D4, C24, C22×Q8, C2×C4○D4, 2+ 1+4, 2- 1+4, C23.37C23, C22.33C24, C22.34C24, C23⋊2Q8, C23.41C23, C23.567C24

Smallest permutation representation of C23.567C24
►On 64 points
Generators in S64
(2 10)(4 12)(5 36)(6 63)(7 34)(8 61)(14 42)(16 44)(17 31)(18 60)(19 29)(20 58)(22 50)(24 52)(26 54)(28 56)(30 48)(32 46)(33 39)(35 37)(38 62)(40 64)(45 59)(47 57)

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

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

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

32 conjugacy classes

class 1 2A···2G2H2I4A4B4C4D4E···4N4O···4V
order12···22244444···44···4
size11···14422224···48···8

32 irreducible representations

dim1111111112244
type+++++++++-+-
imageC1C2C2C2C2C2C2C2C2Q8C4○D42+ 1+42- 1+4
kernelC23.567C24C4×C22⋊C4C23.7Q8C23.34D4C23.63C23C23.65C23C23.Q8C23.81C23C23.4Q8C22×C4C2×C4C22C22
# reps1111222424831

Matrix representation of C23.567C24 ►in GL8(𝔽5)

10000000
04000000
00400000
00040000
00001000
00000100
00000040
00004304
,
10000000
01000000
00100000
00010000
00004000
00000400
00000040
00000004
,
40000000
04000000
00100000
00010000
00001000
00000100
00000010
00000001
,
40000000
04000000
00400000
00040000
00001000
00000100
00000010
00000001
,
30000000
02000000
00020000
00200000
00000010
00001212
00001000
00001113
,
01000000
40000000
00040000
00100000
00004200
00004100
00002414
00001224
,
20000000
02000000
00400000
00040000
00004000
00004100
00000040
00002031

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

C23.567C24 in GAP, Magma, Sage, TeX 

C_2^3._{567}C_2^4
 
% in TeX
 
G:=Group("C2^3.567C2^4");
 
// GroupNames label
 
G:=SmallGroup(128,1399);
 
// by ID
 
G=gap.SmallGroup(128,1399);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,2,253,456,758,723,436,185,136]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=d^2=1,e^2=d,f^2=d*b=b*d,g^2=c,e*a*e^-1=a*b=b*a,f*a*f^-1=a*c=c*a,a*d=d*a,a*g=g*a,b*c=c*b,b*e=e*b,g*f*g^-1=b*f=f*b,b*g=g*b,c*d=d*c,c*e=e*c,c*f=f*c,c*g=g*c,f*e*f^-1=d*e=e*d,d*f=f*d,d*g=g*d,e*g=g*e>;
 
// generators/relations
 

׿
×
⋊
ℤ
𝔽
○
≀
ℚ
◁