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G = C22.143C25  order 128 = 27

124th central stem extension by C22 of C25

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

Aliases: C23.84C24, C42.126C23, C22.143C25, C4.462- 1+4, C4.1622+ 1+4, (D4×Q8)⋊31C2, D4⋊6D4⋊42C2, Q8⋊5D4⋊32C2, D4⋊3Q8⋊40C2, Q8⋊6D4⋊30C2, Q8⋊3Q8⋊26C2, C4⋊C4.328C23, (C2×C4).133C24, C4⋊Q8.354C22, (C4×D4).253C22, (C2×D4).487C23, C22⋊C4.57C23, (C4×Q8).239C22, (C2×Q8).469C23, C4⋊D4.121C22, C4⋊1D4.193C22, (C22×C4).403C23, (C2×C42).971C22, C22⋊Q8.127C22, C2.49(C2×2- 1+4), C2.71(C2×2+ 1+4), C42.C2.85C22, C2.55(C2.C25), C42⋊2C2.25C22, C22.26C24⋊53C2, C4.4D4.106C22, (C22×Q8).373C22, C22.33C24⋊19C2, C23.36C23⋊54C2, C23.37C23⋊52C2, C22.36C24⋊36C2, C22.56C24⋊12C2, C22.53C24⋊23C2, C23.38C23⋊36C2, C42⋊C2.247C22, C22.34C24⋊25C2, C22.D4.19C22, (C2×C4⋊C4).724C22, (C2×C4○D4).245C22, SmallGroup(128,2286)

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

Derived series
C1 — C22 — C22.143C25
Chief series
C1 — C2 — C22 — C2×C4 — C22×C4 — C2×C42 — C23.37C23 — C22.143C25
Lower central
C1 — C22 — C22.143C25
Upper central
C1 — C22 — C22.143C25
Jennings
C1 — C22 — C22.143C25

Generators and relations for C22.143C25
 G = < a,b,c,d,e,f,g | a2=b2=c2=f2=1, d2=e2=b, g2=a, ab=ba, dcd-1=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⋊1D4, C4⋊Q8, C4⋊Q8, C22×Q8, C2×C4○D4, C23.36C23, C22.26C24, C23.37C23, C23.38C23, C22.33C24, C22.34C24, C22.36C24, D4⋊6D4, Q8⋊5D4, D4×Q8, Q8⋊6D4, D4⋊3Q8, Q8⋊3Q8, C22.53C24, C22.53C24, C22.56C24, C22.143C25
Quotients: C1, C2, C22, C23, C24, 2+ 1+4, 2- 1+4, C25, C2×2+ 1+4, C2×2- 1+4, C2.C25, C22.143C25

Smallest permutation representation of C22.143C25
►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 19)(2 32)(3 17)(4 30)(5 56)(6 41)(7 54)(8 43)(9 45)(10 58)(11 47)(12 60)(13 39)(14 64)(15 37)(16 62)(18 50)(20 52)(21 57)(22 46)(23 59)(24 48)(25 63)(26 40)(27 61)(28 38)(29 49)(31 51)(33 53)(34 42)(35 55)(36 44)

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

(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), (1,19)(2,32)(3,17)(4,30)(5,56)(6,41)(7,54)(8,43)(9,45)(10,58)(11,47)(12,60)(13,39)(14,64)(15,37)(16,62)(18,50)(20,52)(21,57)(22,46)(23,59)(24,48)(25,63)(26,40)(27,61)(28,38)(29,49)(31,51)(33,53)(34,42)(35,55)(36,44), (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), (1,43)(2,56)(3,41)(4,54)(5,30)(6,19)(7,32)(8,17)(9,13)(10,26)(11,15)(12,28)(14,22)(16,24)(18,36)(20,34)(21,25)(23,27)(29,35)(31,33)(37,45)(38,58)(39,47)(40,60)(42,50)(44,52)(46,62)(48,64)(49,53)(51,55)(57,61)(59,63), (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), (1,19)(2,32)(3,17)(4,30)(5,56)(6,41)(7,54)(8,43)(9,45)(10,58)(11,47)(12,60)(13,39)(14,64)(15,37)(16,62)(18,50)(20,52)(21,57)(22,46)(23,59)(24,48)(25,63)(26,40)(27,61)(28,38)(29,49)(31,51)(33,53)(34,42)(35,55)(36,44), (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), (1,43)(2,56)(3,41)(4,54)(5,30)(6,19)(7,32)(8,17)(9,13)(10,26)(11,15)(12,28)(14,22)(16,24)(18,36)(20,34)(21,25)(23,27)(29,35)(31,33)(37,45)(38,58)(39,47)(40,60)(42,50)(44,52)(46,62)(48,64)(49,53)(51,55)(57,61)(59,63), (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)], [(1,19),(2,32),(3,17),(4,30),(5,56),(6,41),(7,54),(8,43),(9,45),(10,58),(11,47),(12,60),(13,39),(14,64),(15,37),(16,62),(18,50),(20,52),(21,57),(22,46),(23,59),(24,48),(25,63),(26,40),(27,61),(28,38),(29,49),(31,51),(33,53),(34,42),(35,55),(36,44)], [(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)], [(1,43),(2,56),(3,41),(4,54),(5,30),(6,19),(7,32),(8,17),(9,13),(10,26),(11,15),(12,28),(14,22),(16,24),(18,36),(20,34),(21,25),(23,27),(29,35),(31,33),(37,45),(38,58),(39,47),(40,60),(42,50),(44,52),(46,62),(48,64),(49,53),(51,55),(57,61),(59,63)], [(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···2J4A···4F4G···4AA
order12222···24···44···4
size11114···42···24···4

38 irreducible representations

dim1111111111111111444
type+++++++++++++++++-
imageC1C2C2C2C2C2C2C2C2C2C2C2C2C2C2C22+ 1+42- 1+4C2.C25
kernelC22.143C25C23.36C23C22.26C24C23.37C23C23.38C23C22.33C24C22.34C24C22.36C24D4⋊6D4Q8⋊5D4D4×Q8Q8⋊6D4D4⋊3Q8Q8⋊3Q8C22.53C24C22.56C24C4C4C2
# reps1111242422112134222

Matrix representation of C22.143C25 ►in GL8(𝔽5)

10000000
01000000
00100000
00010000
00004000
00000400
00000040
00000004
,
40000000
04000000
00400000
00040000
00004000
00000400
00000040
00000004
,
02000000
30000000
00020000
00300000
00004000
00003100
00000010
00001444
,
40300000
04030000
10100000
01010000
00000010
00000403
00004000
00000101
,
03000000
30000000
02020000
20200000
00004100
00003100
00000112
00001444
,
01000000
10000000
00010000
00100000
00002300
00004300
00000331
00003222
,
40000000
04000000
00400000
00040000
00004100
00003100
00000443
00001011

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

C22.143C25 in GAP, Magma, Sage, TeX 

C_2^2._{143}C_2^5
 
% in TeX
 
G:=Group("C2^2.143C2^5");
 
// GroupNames label
 
G:=SmallGroup(128,2286);
 
// by ID
 
G=gap.SmallGroup(128,2286);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,2,-2,2,477,456,1430,723,352,2019,570,1684,102]);
 
// 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=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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