Copied to
clipboard

G = C22.137C25  order 128 = 27

118th central stem extension by C22 of C25

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

Aliases: C23.78C24, C22.137C25, C42.120C23, C4.442- 1+4, C4.932+ 1+4, D4⋊6D4⋊40C2, D4⋊3Q8⋊37C2, Q8⋊6D4⋊29C2, C4⋊C4.324C23, (C2×C4).127C24, C4⋊Q8.352C22, (C4×D4).249C22, (C2×D4).329C23, C22⋊C4.52C23, (C4×Q8).235C22, (C2×Q8).467C23, C4⋊D4.120C22, C4⋊1D4.118C22, (C22×C4).397C23, (C2×C42).967C22, C22⋊Q8.235C22, C2.66(C2×2+ 1+4), C2.45(C2×2- 1+4), C42.C2.84C22, C22.26C24⋊50C2, C4.4D4.181C22, C22.34C24⋊23C2, C22.31C24⋊26C2, C42⋊C2.243C22, C22.56C24⋊10C2, C23.37C23⋊50C2, C22.D4.15C22, (C2×C4⋊C4).721C22, (C2×C4○D4).242C22, SmallGroup(128,2280)

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

Derived series
C1 — C22 — C22.137C25
Chief series
C1 — C2 — C22 — C2×C4 — C22×C4 — C2×C42 — C22.26C24 — C22.137C25
Lower central
C1 — C22 — C22.137C25
Upper central
C1 — C22 — C22.137C25
Jennings
C1 — C22 — C22.137C25

Generators and relations for C22.137C25
 G = < a,b,c,d,e,f,g | a2=b2=c2=d2=1, e2=b, f2=g2=a, ab=ba, dcd=gcg-1=ac=ca, fdf-1=ad=da, ae=ea, af=fa, ag=ga, ece-1=fcf-1=bc=cb, ede-1=bd=db, be=eb, bf=fb, bg=gb, dg=gd, ef=fe, eg=ge, fg=gf >

Subgroups: 892 in 557 conjugacy classes, 384 normal (12 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, C42, C42, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×Q8, C4○D4, C2×C42, C2×C4⋊C4, C42⋊C2, C4×D4, C4×Q8, C4⋊D4, C22⋊Q8, C22.D4, C4.4D4, C42.C2, C4⋊1D4, C4⋊Q8, C2×C4○D4, C22.26C24, C23.37C23, C22.31C24, C22.34C24, D4⋊6D4, Q8⋊6D4, D4⋊3Q8, C22.56C24, C22.137C25
Quotients: C1, C2, C22, C23, C24, 2+ 1+4, 2- 1+4, C25, C2×2+ 1+4, C2×2- 1+4, C22.137C25

Smallest permutation representation of C22.137C25
►On 64 points
Generators in S64
(1 27)(2 28)(3 25)(4 26)(5 29)(6 30)(7 31)(8 32)(9 13)(10 14)(11 15)(12 16)(17 50)(18 51)(19 52)(20 49)(21 36)(22 33)(23 34)(24 35)(37 60)(38 57)(39 58)(40 59)(41 64)(42 61)(43 62)(44 63)(45 56)(46 53)(47 54)(48 55)

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

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

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

38 conjugacy classes

class 1 2A2B2C2D···2L4A···4F4G···4Y
order12222···24···44···4
size11114···42···24···4

38 irreducible representations

dim11111111144
type++++++++++-
imageC1C2C2C2C2C2C2C2C22+ 1+42- 1+4
kernelC22.137C25C22.26C24C23.37C23C22.31C24C22.34C24D4⋊6D4Q8⋊6D4D4⋊3Q8C22.56C24C4C4
# reps12184444442

Matrix representation of C22.137C25 ►in GL8(𝔽5)

40000000
04000000
00400000
00040000
00004000
00000400
00000040
00000004
,
10000000
01000000
00100000
00010000
00004000
00000400
00000040
00000004
,
12000000
04000000
44010000
11100000
00001000
00000400
00000040
00000001
,
40300000
00110000
00100000
01400000
00000010
00000001
00001000
00000100
,
40000000
04000000
00400000
00040000
00000400
00001000
00000001
00000040
,
30000000
03000000
20200000
30020000
00000100
00004000
00000004
00000010
,
43000000
11000000
01010000
44400000
00000100
00004000
00000001
00000040

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

C22.137C25 in GAP, Magma, Sage, TeX 

C_2^2._{137}C_2^5
 
% in TeX
 
G:=Group("C2^2.137C2^5");
 
// GroupNames label
 
G:=SmallGroup(128,2280);
 
// by ID
 
G=gap.SmallGroup(128,2280);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,2,-2,2,477,1430,723,352,2019,570,136,1684,102]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=d^2=1,e^2=b,f^2=g^2=a,a*b=b*a,d*c*d=g*c*g^-1=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=f*c*f^-1=b*c=c*b,e*d*e^-1=b*d=d*b,b*e=e*b,b*f=f*b,b*g=g*b,d*g=g*d,e*f=f*e,e*g=g*e,f*g=g*f>;
 
// generators/relations
 

׿
×
⋊
ℤ
𝔽
○
≀
ℚ
◁