Copied to
clipboard

G = C22.96C25order 128 = 27

77th central stem extension by C22 of C25

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

Aliases: C22.96C25, C42.88C23, C23.139C24, C4.412- 1+4, Q83Q818C2, D43Q823C2, (C2×C4).86C24, C4⋊C4.520C23, C4⋊Q8.344C22, (C2×D4).473C23, (C4×D4).235C22, (C4×Q8).222C22, (C2×Q8).451C23, C4⋊D4.242C22, C41D4.186C22, C22⋊C4.106C23, C422C2.1C22, (C2×C42).949C22, (C22×C4).367C23, C22⋊Q8.117C22, C2.26(C2×2- 1+4), C2.30(C2.C25), C22.33C245C2, C4.4D4.176C22, C42.C2.153C22, (C22×Q8).500C22, C22.D4.8C22, C23.38C2324C2, C42⋊C2.229C22, C23.36C2330C2, C22.53C2412C2, C23.37C2339C2, C23.33C2322C2, C22.50C2421C2, C22.46C2417C2, C22.26C24.50C2, (C2×C4×Q8)⋊60C2, C4.179(C2×C4○D4), C22.17(C2×C4○D4), C2.52(C22×C4○D4), (C2×C4).306(C4○D4), (C2×C4⋊C4).706C22, (C2×C4○D4).229C22, SmallGroup(128,2239)

Series: Derived Chief Lower central Upper central Jennings

C1C22 — C22.96C25
C1C2C22C23C22×C4C2×C42C2×C4×Q8 — C22.96C25
C1C22 — C22.96C25
C1C22 — C22.96C25
C1C22 — C22.96C25

Generators and relations for C22.96C25
 G = < a,b,c,d,e,f,g | a2=b2=c2=g2=1, d2=ba=ab, e2=b, f2=a, dcd-1=gcg=ac=ca, fdf-1=ad=da, ae=ea, af=fa, ag=ga, ece-1=bc=cb, bd=db, be=eb, bf=fb, bg=gb, cf=fc, de=ed, dg=gd, ef=fe, eg=ge, fg=gf >

Subgroups: 660 in 506 conjugacy classes, 390 normal (30 characteristic)
C1, C2 [×3], C2 [×6], C4 [×6], C4 [×23], C22, C22 [×2], C22 [×14], C2×C4 [×6], C2×C4 [×24], C2×C4 [×25], D4 [×18], Q8 [×18], C23, C23 [×4], C42 [×4], C42 [×20], C22⋊C4 [×36], C4⋊C4 [×56], C22×C4 [×3], C22×C4 [×16], C2×D4 [×10], C2×Q8 [×10], C2×Q8 [×4], C4○D4 [×8], C2×C42, C2×C42 [×2], C2×C4⋊C4, C2×C4⋊C4 [×6], C42⋊C2 [×14], C4×D4 [×20], C4×Q8 [×20], C4⋊D4 [×4], C22⋊Q8 [×20], C22.D4 [×16], C4.4D4 [×10], C42.C2 [×14], C422C2 [×16], C41D4, C4⋊Q8 [×3], C4⋊Q8 [×4], C22×Q8, C2×C4○D4 [×2], C2×C4×Q8, C23.33C23 [×2], C23.36C23 [×4], C22.26C24, C23.37C23, C23.38C23 [×2], C22.33C24 [×4], C22.46C24 [×4], D43Q8 [×2], C22.50C24 [×6], Q83Q8 [×2], C22.53C24 [×2], C22.96C25
Quotients: C1, C2 [×31], C22 [×155], C23 [×155], C4○D4 [×4], C24 [×31], C2×C4○D4 [×6], 2- 1+4 [×2], C25, C22×C4○D4, C2×2- 1+4, C2.C25, C22.96C25

Smallest permutation representation of C22.96C25
On 64 points
Generators in S64
(1 5)(2 6)(3 7)(4 8)(9 55)(10 56)(11 53)(12 54)(13 59)(14 60)(15 57)(16 58)(17 63)(18 64)(19 61)(20 62)(21 39)(22 40)(23 37)(24 38)(25 43)(26 44)(27 41)(28 42)(29 47)(30 48)(31 45)(32 46)(33 51)(34 52)(35 49)(36 50)
(1 7)(2 8)(3 5)(4 6)(9 53)(10 54)(11 55)(12 56)(13 57)(14 58)(15 59)(16 60)(17 61)(18 62)(19 63)(20 64)(21 37)(22 38)(23 39)(24 40)(25 41)(26 42)(27 43)(28 44)(29 45)(30 46)(31 47)(32 48)(33 49)(34 50)(35 51)(36 52)
(1 27)(2 42)(3 25)(4 44)(5 41)(6 28)(7 43)(8 26)(9 23)(10 38)(11 21)(12 40)(13 33)(14 52)(15 35)(16 50)(17 47)(18 30)(19 45)(20 32)(22 54)(24 56)(29 63)(31 61)(34 60)(36 58)(37 55)(39 53)(46 62)(48 64)(49 57)(51 59)
(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 15 7 59)(2 16 8 60)(3 13 5 57)(4 14 6 58)(9 61 53 17)(10 62 54 18)(11 63 55 19)(12 64 56 20)(21 45 37 29)(22 46 38 30)(23 47 39 31)(24 48 40 32)(25 49 41 33)(26 50 42 34)(27 51 43 35)(28 52 44 36)
(1 9 5 55)(2 56 6 10)(3 11 7 53)(4 54 8 12)(13 63 59 17)(14 18 60 64)(15 61 57 19)(16 20 58 62)(21 43 39 25)(22 26 40 44)(23 41 37 27)(24 28 38 42)(29 51 47 33)(30 34 48 52)(31 49 45 35)(32 36 46 50)
(1 23)(2 24)(3 21)(4 22)(5 37)(6 38)(7 39)(8 40)(9 41)(10 42)(11 43)(12 44)(13 45)(14 46)(15 47)(16 48)(17 49)(18 50)(19 51)(20 52)(25 53)(26 54)(27 55)(28 56)(29 57)(30 58)(31 59)(32 60)(33 61)(34 62)(35 63)(36 64)

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

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

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

44 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I4A···4P4Q···4AH
order12222222224···44···4
size11112244442···24···4

44 irreducible representations

dim1111111111111244
type+++++++++++++-
imageC1C2C2C2C2C2C2C2C2C2C2C2C2C4○D42- 1+4C2.C25
kernelC22.96C25C2×C4×Q8C23.33C23C23.36C23C22.26C24C23.37C23C23.38C23C22.33C24C22.46C24D43Q8C22.50C24Q83Q8C22.53C24C2×C4C4C2
# reps1124112442622822

Matrix representation of C22.96C25 in GL6(𝔽5)

100000
010000
004000
000400
000040
000004
,
400000
040000
001000
000100
000010
000001
,
400000
010000
000300
002000
001134
000132
,
300000
030000
000010
003342
004000
000002
,
010000
400000
001000
000100
000010
000001
,
100000
010000
003000
000300
000020
001102
,
400000
040000
000100
001000
003342
000001

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

C22.96C25 in GAP, Magma, Sage, TeX

C_2^2._{96}C_2^5
% in TeX

G:=Group("C2^2.96C2^5");
// GroupNames label

G:=SmallGroup(128,2239);
// by ID

G=gap.SmallGroup(128,2239);
# by ID

G:=PCGroup([7,-2,2,2,2,2,-2,2,477,680,1430,352,570,136,1684]);
// Polycyclic

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

׿
×
𝔽