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G = C22.141C25  order 128 = 27

122nd central stem extension by C22 of C25

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

Aliases: C23.82C24, C42.124C23, C22.141C25, C4.1612+ 1+4, C4.1132- 1+4, C22.42- 1+4, (D4×Q8)⋊30C2, D4⋊6D4⋊41C2, Q8⋊3Q8⋊25C2, D4⋊3Q8⋊38C2, Q8⋊5D4⋊31C2, C4⋊C4.326C23, (C2×C4).131C24, C4⋊Q8.353C22, (C4×D4).251C22, (C2×D4).486C23, C22⋊C4.55C23, (C4×Q8).237C22, (C2×Q8).468C23, C4⋊D4.235C22, C42⋊2C2.7C22, (C2×C42).969C22, (C22×C4).401C23, C22⋊Q8.125C22, C2.70(C2×2+ 1+4), C2.47(C2×2- 1+4), C4.4D4.105C22, C42.C2.165C22, (C22×Q8).372C22, C23.38C23⋊35C2, C23.36C23⋊52C2, C42⋊C2.245C22, C22.47C24⋊35C2, C22.36C24⋊35C2, C22.50C24⋊35C2, C23.41C23⋊22C2, C22.57C24⋊12C2, C22.33C24⋊17C2, C22.35C24⋊19C2, C22.D4.17C22, (C2×C4⋊Q8)⋊61C2, (C2×C4⋊C4).722C22, (C2×C4○D4).244C22, SmallGroup(128,2284)

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

Derived series
C1 — C22 — C22.141C25
Chief series
C1 — C2 — C22 — C23 — C22×C4 — C2×C42 — C2×C4⋊Q8 — C22.141C25
Lower central
C1 — C22 — C22.141C25
Upper central
C1 — C22 — C22.141C25
Jennings
C1 — C22 — C22.141C25

Generators and relations for C22.141C25
 G = < a,b,c,d,e,f,g | a2=b2=c2=g2=1, d2=e2=b, f2=a, ab=ba, dcd-1=gcg=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: 700 in 502 conjugacy classes, 384 normal (50 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, C42, C42, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C2×Q8, C2×Q8, C4○D4, C2×C42, C2×C4⋊C4, C42⋊C2, C4×D4, C4×D4, C4×Q8, C4×Q8, C4⋊D4, C4⋊D4, C22⋊Q8, C22⋊Q8, C22.D4, C4.4D4, C4.4D4, C42.C2, C42.C2, C42⋊2C2, C4⋊Q8, C4⋊Q8, C22×Q8, C2×C4○D4, C23.36C23, C2×C4⋊Q8, C23.38C23, C22.33C24, C22.35C24, C22.36C24, C23.41C23, D4⋊6D4, Q8⋊5D4, D4×Q8, D4×Q8, C22.47C24, D4⋊3Q8, C22.50C24, Q8⋊3Q8, C22.57C24, C22.141C25
Quotients: C1, C2, C22, C23, C24, 2+ 1+4, 2- 1+4, C25, C2×2+ 1+4, C2×2- 1+4, C22.141C25

Smallest permutation representation of C22.141C25
►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 60)(10 45)(11 58)(12 47)(13 38)(14 63)(15 40)(16 61)(17 50)(19 52)(21 48)(22 57)(23 46)(24 59)(25 62)(26 39)(27 64)(28 37)(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 45 3 47)(2 48 4 46)(5 16 7 14)(6 15 8 13)(9 19 11 17)(10 18 12 20)(21 31 23 29)(22 30 24 32)(25 33 27 35)(26 36 28 34)(37 41 39 43)(38 44 40 42)(49 59 51 57)(50 58 52 60)(53 63 55 61)(54 62 56 64)

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

38 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I4A4B4C4D4E···4AB
order122222222244444···4
size111122444422224···4

38 irreducible representations

dim1111111111111111444
type+++++++++++++++++--
imageC1C2C2C2C2C2C2C2C2C2C2C2C2C2C2C22+ 1+42- 1+42- 1+4
kernelC22.141C25C23.36C23C2×C4⋊Q8C23.38C23C22.33C24C22.35C24C22.36C24C23.41C23D4⋊6D4Q8⋊5D4D4×Q8C22.47C24D4⋊3Q8C22.50C24Q8⋊3Q8C22.57C24C4C4C22
# reps1212422212322114222

Matrix representation of C22.141C25 ►in GL8(𝔽5)

40000000
04000000
00400000
00040000
00001000
00000100
00000010
00000001
,
10000000
01000000
00100000
00010000
00004000
00000400
00000040
00000004
,
40000000
01000000
00400000
00010000
00000030
00000002
00002000
00000300
,
01000000
10000000
00040000
00400000
00000010
00000001
00004000
00000400
,
40000000
04000000
00400000
00040000
00003000
00000300
00000020
00000002
,
00100000
00010000
40000000
04000000
00000100
00001000
00000001
00000010
,
01000000
10000000
00010000
00100000
00001000
00000100
00000010
00000001

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],[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],[4,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,3,0,0,0,0,3,0,0,0,0,0,0,0,0,2,0,0],[0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0],[4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,2],[0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0],[0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1] >;
 

C22.141C25 in GAP, Magma, Sage, TeX 

C_2^2._{141}C_2^5
 
% in TeX
 
G:=Group("C2^2.141C2^5");
 
// GroupNames label
 
G:=SmallGroup(128,2284);
 
// by ID
 
G=gap.SmallGroup(128,2284);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,2,-2,2,477,456,1430,723,352,2019,570,136,1684]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=g^2=1,d^2=e^2=b,f^2=a,a*b=b*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=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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