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G = C22.120C25  order 128 = 27

101st central stem extension by C22 of C25

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

Aliases: C23.61C24, C22.120C25, C42.584C23, C4.1332- 1+4, C4⋊C4.309C23, (C2×C4).110C24, C4⋊Q8.350C22, (C4×D4).245C22, (C2×D4).313C23, C22⋊C4.39C23, (C2×Q8).298C23, (C4×Q8).232C22, C4⋊D4.232C22, (C2×C42).964C22, C22⋊Q8.234C22, C2.38(C2×2- 1+4), C2.42(C2.C25), C4○(C22.58C24), C42⋊2C2.22C22, C22.58C24⋊6C2, (C22×C4).1218C23, C4.4D4.180C22, C42.C2.161C22, C4○2(C22.57C24), C4○2(C22.56C24), C22.56C24⋊21C2, C23.37C23⋊48C2, C42⋊C2.240C22, C23.36C23⋊45C2, C22.57C24⋊25C2, C22.D4.37C22, (C2×C4)○(C22.58C24), SmallGroup(128,2263)

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

Derived series
C1 — C22 — C22.120C25
Chief series
C1 — C2 — C22 — C2×C4 — C22×C4 — C2×C42 — C23.37C23 — C22.120C25
Lower central
C1 — C22 — C22.120C25
Upper central
C1 — C2×C4 — C22.120C25
Jennings
C1 — C22 — C22.120C25

Generators and relations for C22.120C25
 G = < a,b,c,d,e,f,g | a2=b2=c2=f2=1, d2=e2=b, g2=a, ab=ba, dcd-1=ac=ca, fdf=ad=da, ae=ea, af=fa, ag=ga, ece-1=bc=cb, bd=db, be=eb, bf=fb, bg=gb, fcf=abc, cg=gc, ede-1=abd, dg=gd, ef=fe, eg=ge, fg=gf >

Subgroups: 620 in 465 conjugacy classes, 380 normal (6 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C42, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C2×Q8, C2×C42, C42⋊C2, C4×D4, C4×Q8, C4⋊D4, C22⋊Q8, C22.D4, C4.4D4, C42.C2, C42⋊2C2, C4⋊Q8, C23.36C23, C23.37C23, C22.56C24, C22.57C24, C22.58C24, C22.120C25
Quotients: C1, C2, C22, C23, C24, 2- 1+4, C25, C2×2- 1+4, C2.C25, C22.120C25

Smallest permutation representation of C22.120C25
►On 64 points
Generators in S64
(1 27)(2 28)(3 25)(4 26)(5 36)(6 33)(7 34)(8 35)(9 13)(10 14)(11 15)(12 16)(17 44)(18 41)(19 42)(20 43)(21 29)(22 30)(23 31)(24 32)(37 46)(38 47)(39 48)(40 45)(49 58)(50 59)(51 60)(52 57)(53 62)(54 63)(55 64)(56 61)

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

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

(2 28)(4 26)(5 36)(7 34)(10 14)(12 16)(17 42)(18 20)(19 44)(22 30)(24 32)(37 39)(38 45)(40 47)(41 43)(46 48)(49 51)(50 57)(52 59)(53 55)(54 61)(56 63)(58 60)(62 64)

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

38 conjugacy classes

class 1 2A2B2C2D···2H4A4B4C4D4E···4AC
order12222···244444···4
size11114···411114···4

38 irreducible representations

dim11111144
type++++++-
imageC1C2C2C2C2C22- 1+4C2.C25
kernelC22.120C25C23.36C23C23.37C23C22.56C24C22.57C24C22.58C24C4C2
# reps1105510124

Matrix representation of C22.120C25 ►in GL8(𝔽5)

10000000
01000000
00100000
00010000
00004000
00000400
00000040
00000004
,
40000000
04000000
00400000
00040000
00004000
00000400
00000040
00000004
,
00100000
00010000
10000000
01000000
00000400
00004000
00000001
00000010
,
02000000
20000000
00020000
00200000
00000010
00000001
00004000
00000400
,
01000000
40000000
00040000
00100000
00000100
00004000
00000001
00000040
,
10000000
01000000
00400000
00040000
00001000
00000100
00000040
00000004
,
40000000
04000000
00400000
00040000
00003000
00000300
00000030
00000003

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

C22.120C25 in GAP, Magma, Sage, TeX 

C_2^2._{120}C_2^5
 
% in TeX
 
G:=Group("C2^2.120C2^5");
 
// GroupNames label
 
G:=SmallGroup(128,2263);
 
// by ID
 
G=gap.SmallGroup(128,2263);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,2,-2,2,477,456,1430,1059,352,2915,570,102]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=f^2=1,d^2=e^2=b,g^2=a,a*b=b*a,d*c*d^-1=a*c=c*a,f*d*f=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,f*c*f=a*b*c,c*g=g*c,e*d*e^-1=a*b*d,d*g=g*d,e*f=f*e,e*g=g*e,f*g=g*f>;
 
// generators/relations
 

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