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G = C22.136C25  order 128 = 27

117th central stem extension by C22 of C25

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

Aliases: C23.77C24, C42.119C23, C22.136C25, C4.922+ 1+4, C4.1122- 1+4, (D4×Q8)⋊28C2, D4⋊6D4⋊39C2, D4⋊3Q8⋊36C2, Q8⋊5D4⋊29C2, C4⋊C4.323C23, (C2×C4).126C24, C4⋊Q8.225C22, (C4×D4).248C22, (C2×D4).328C23, C22⋊C4.51C23, (C4×Q8).234C22, (C2×Q8).307C23, C4⋊D4.119C22, C4⋊1D4.191C22, (C2×C42).966C22, (C22×C4).396C23, C22⋊Q8.124C22, C2.65(C2×2+ 1+4), C2.44(C2×2- 1+4), C2.52(C2.C25), C22.26C24⋊49C2, C42⋊2C2.23C22, C22.56C24⋊9C2, C4.4D4.103C22, C42.C2.163C22, (C22×Q8).370C22, C22.47C24⋊33C2, C22.34C24⋊22C2, C42⋊C2.242C22, C23.38C23⋊32C2, C22.46C24⋊34C2, C23.36C23⋊50C2, C23.41C23⋊21C2, C22.31C24⋊25C2, C22.36C24⋊33C2, C23.37C23⋊49C2, C22.49C24⋊23C2, C22.D4.14C22, (C2×C4⋊C4).720C22, (C2×C4○D4).241C22, SmallGroup(128,2279)

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

Derived series
C1 — C22 — C22.136C25
Chief series
C1 — C2 — C22 — C2×C4 — C22×C4 — C2×C42 — C22.26C24 — C22.136C25
Lower central
C1 — C22 — C22.136C25
Upper central
C1 — C22 — C22.136C25
Jennings
C1 — C22 — C22.136C25

Generators and relations for C22.136C25
 G = < a,b,c,d,e,f,g | a2=b2=d2=f2=1, c2=g2=a, e2=b, ab=ba, dcd=gcg-1=ac=ca, fdf=ad=da, ae=ea, af=fa, ag=ga, ece-1=fcf=bc=cb, ede-1=bd=db, be=eb, bf=fb, bg=gb, dg=gd, ef=fe, eg=ge, fg=gf >

Subgroups: 756 in 511 conjugacy classes, 382 normal (50 characteristic)
C1, C2, C2, C4, C4, 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⋊1D4, C4⋊Q8, C4⋊Q8, C22×Q8, C2×C4○D4, C23.36C23, C22.26C24, C23.37C23, C23.38C23, C22.31C24, C22.34C24, C22.36C24, C23.41C23, D4⋊6D4, Q8⋊5D4, D4×Q8, C22.46C24, C22.47C24, D4⋊3Q8, C22.49C24, C22.49C24, C22.56C24, C22.136C25
Quotients: C1, C2, C22, C23, C24, 2+ 1+4, 2- 1+4, C25, C2×2+ 1+4, C2×2- 1+4, C2.C25, C22.136C25

Smallest permutation representation of C22.136C25
►On 64 points
Generators in S64
(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 51)(2 52)(3 49)(4 50)(5 36)(6 33)(7 34)(8 35)(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)

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

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

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

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

38 conjugacy classes

class 1 2A2B2C2D···2J4A···4F4G···4AA
order12222···24···44···4
size11114···42···24···4

38 irreducible representations

dim11111111111111111444
type++++++++++++++++++-
imageC1C2C2C2C2C2C2C2C2C2C2C2C2C2C2C2C22+ 1+42- 1+4C2.C25
kernelC22.136C25C23.36C23C22.26C24C23.37C23C23.38C23C22.31C24C22.34C24C22.36C24C23.41C23D4⋊6D4Q8⋊5D4D4×Q8C22.46C24C22.47C24D4⋊3Q8C22.49C24C22.56C24C4C4C2
# reps11112224212122134222

Matrix representation of C22.136C25 ►in GL8(𝔽5)

40000000
04000000
00400000
00040000
00004000
00000400
00000040
00000004
,
10000000
01000000
00100000
00010000
00004000
00000400
00000040
00000004
,
00100000
00010000
40000000
04000000
00000010
00000001
00004000
00000400
,
02000000
30000000
00030000
00200000
00000300
00002000
00000002
00000030
,
40000000
04000000
00400000
00040000
00000300
00003000
00000002
00000020
,
01000000
10000000
00010000
00100000
00000100
00001000
00000004
00000040
,
20000000
02000000
00300000
00030000
00003000
00000300
00000020
00000002

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

C22.136C25 in GAP, Magma, Sage, TeX 

C_2^2._{136}C_2^5
 
% in TeX
 
G:=Group("C2^2.136C2^5");
 
// GroupNames label
 
G:=SmallGroup(128,2279);
 
// by ID
 
G=gap.SmallGroup(128,2279);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,2,-2,2,224,477,1430,723,352,2019,570,1684,102]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^2=b^2=d^2=f^2=1,c^2=g^2=a,e^2=b,a*b=b*a,d*c*d=g*c*g^-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=f*c*f=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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