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G = C22.146C25  order 128 = 27

127th central stem extension by C22 of C25

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

Aliases: C23.87C24, C22.146C25, C42.129C23, C4.492- 1+4, Q8⋊3Q8⋊29C2, D4⋊3Q8⋊42C2, C4⋊C4.507C23, (C2×C4).136C24, C4⋊Q8.229C22, (C4×D4).255C22, (C2×D4).335C23, C22⋊C4.60C23, (C4×Q8).242C22, (C2×Q8).312C23, C4⋊1D4.120C22, C4⋊D4.122C22, (C2×C42).974C22, C42⋊2C2.9C22, (C22×C4).405C23, C22⋊Q8.129C22, C2.52(C2×2- 1+4), C2.57(C2.C25), C4.4D4.108C22, C42.C2.167C22, C22.35C24⋊22C2, C22.57C24⋊15C2, C22.50C24⋊37C2, C42⋊C2.250C22, C23.36C23⋊55C2, C22.33C24⋊20C2, C22.36C24⋊37C2, C22.53C24⋊24C2, C23.37C23⋊54C2, C22.46C24⋊38C2, C22.34C24.5C2, C22.D4.38C22, (C2×C4⋊C4).726C22, SmallGroup(128,2289)

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

Derived series
C1 — C22 — C22.146C25
Chief series
C1 — C2 — C22 — C2×C4 — C22×C4 — C2×C42 — C23.37C23 — C22.146C25
Lower central
C1 — C22 — C22.146C25
Upper central
C1 — C22 — C22.146C25
Jennings
C1 — C22 — C22.146C25

Generators and relations for C22.146C25
 G = < a,b,c,d,e,f,g | a2=b2=c2=1, d2=b, e2=ba=ab, f2=g2=a, dcd-1=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: 620 in 465 conjugacy classes, 380 normal (24 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, C42, C42, C22⋊C4, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×Q8, C2×C42, C2×C4⋊C4, C42⋊C2, C42⋊C2, C4×D4, C4×Q8, C4⋊D4, C22⋊Q8, C22.D4, C4.4D4, C42.C2, C42.C2, C42⋊2C2, C4⋊1D4, C4⋊Q8, C4⋊Q8, C23.36C23, C23.37C23, C22.33C24, C22.34C24, C22.35C24, C22.36C24, C22.46C24, D4⋊3Q8, C22.50C24, Q8⋊3Q8, C22.53C24, C22.57C24, C22.146C25
Quotients: C1, C2, C22, C23, C24, 2- 1+4, C25, C2×2- 1+4, C2.C25, C22.146C25

Smallest permutation representation of C22.146C25
►On 64 points
Generators in S64
(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 18)(2 31)(3 20)(4 29)(5 53)(6 42)(7 55)(8 44)(9 48)(10 57)(11 46)(12 59)(13 62)(14 39)(15 64)(16 37)(17 50)(19 52)(21 60)(22 45)(23 58)(24 47)(25 38)(26 63)(27 40)(28 61)(30 51)(32 49)(33 54)(34 43)(35 56)(36 41)

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

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

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

38 conjugacy classes

class 1 2A2B2C2D···2H4A···4F4G···4AC
order12222···24···44···4
size11114···42···24···4

38 irreducible representations

dim111111111111144
type+++++++++++++-
imageC1C2C2C2C2C2C2C2C2C2C2C2C22- 1+4C2.C25
kernelC22.146C25C23.36C23C23.37C23C22.33C24C22.34C24C22.35C24C22.36C24C22.46C24D4⋊3Q8C22.50C24Q8⋊3Q8C22.53C24C22.57C24C4C2
# reps121411642222424

Matrix representation of C22.146C25 ►in GL8(𝔽5)

40000000
04000000
00400000
00040000
00001000
00000100
00000010
00000001
,
40000000
04000000
00400000
00040000
00004000
00000400
00000040
00000004
,
13000000
04000000
41040000
41400000
00000002
00000030
00000200
00003000
,
30100000
00320000
00200000
02200000
00000200
00002000
00000003
00000030
,
20400000
00230000
30300000
32300000
00000010
00000001
00004000
00000400
,
40300000
00410000
10100000
14100000
00000100
00001000
00000001
00000010
,
13000000
14000000
01010000
41400000
00004000
00000400
00000040
00000004

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

C22.146C25 in GAP, Magma, Sage, TeX 

C_2^2._{146}C_2^5
 
% in TeX
 
G:=Group("C2^2.146C2^5");
 
// GroupNames label
 
G:=SmallGroup(128,2289);
 
// by ID
 
G=gap.SmallGroup(128,2289);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,2,-2,2,477,456,1430,723,520,2019,570,136,1684,102]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=1,d^2=b,e^2=b*a=a*b,f^2=g^2=a,d*c*d^-1=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
 

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