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G = C2×C22.50C24  order 128 = 27

Direct product of C2 and C22.50C24

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

Aliases: C2×C22.50C24, C22.63C25, C42.561C23, C23.276C24, C24.497C23, C22.832- 1+4, C4⋊Q8⋊86C22, (C2×C4).62C24, (C4×Q8)⋊96C22, C4⋊C4.472C23, C22⋊Q8⋊90C22, (C4×D4).353C22, (C2×D4).458C23, C22⋊C4.87C23, (C2×Q8).433C23, C42⋊C2⋊98C22, C42⋊2C2⋊31C22, (C2×C42).932C22, (C23×C4).601C22, C2.15(C2×2- 1+4), (C22×C4).1199C23, C4.4D4.170C22, (C22×D4).593C22, (C22×Q8).493C22, (C2×C4×Q8)⋊54C2, (C2×C4⋊Q8)⋊54C2, (C2×C4×D4).89C2, C4.174(C2×C4○D4), (C2×C22⋊Q8)⋊74C2, C2.35(C22×C4○D4), (C2×C42⋊2C2)⋊37C2, (C2×C42⋊C2)⋊63C2, (C2×C4).907(C4○D4), (C2×C4⋊C4).957C22, (C2×C4.4D4).43C2, C22.161(C2×C4○D4), (C2×C22⋊C4).543C22, SmallGroup(128,2206)

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

Derived series
C1 — C22 — C2×C22.50C24
Chief series
C1 — C2 — C22 — C23 — C22×C4 — C23×C4 — C2×C42⋊C2 — C2×C22.50C24
Lower central
C1 — C22 — C2×C22.50C24
Upper central
C1 — C23 — C2×C22.50C24
Jennings
C1 — C22 — C2×C22.50C24

Generators and relations for C2×C22.50C24
 G = < a,b,c,d,e,f,g | a2=b2=c2=d2=1, e2=cb=bc, f2=g2=b, ab=ba, ac=ca, ad=da, ae=ea, af=fa, ag=ga, ede-1=bd=db, geg-1=be=eb, bf=fb, bg=gb, fdf-1=cd=dc, ce=ec, cf=fc, cg=gc, dg=gd, ef=fe, fg=gf >

Subgroups: 732 in 556 conjugacy classes, 404 normal (18 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C2×C4, C2×C4, D4, Q8, C23, C23, C23, C42, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C2×Q8, C24, C2×C42, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C2×C4⋊C4, C42⋊C2, C4×D4, C4×Q8, C22⋊Q8, C4.4D4, C42⋊2C2, C4⋊Q8, C23×C4, C22×D4, C22×Q8, C22×Q8, C2×C42⋊C2, C2×C4×D4, C2×C4×Q8, C2×C4×Q8, C2×C22⋊Q8, C2×C4.4D4, C2×C42⋊2C2, C2×C4⋊Q8, C22.50C24, C2×C22.50C24
Quotients: C1, C2, C22, C23, C4○D4, C24, C2×C4○D4, 2- 1+4, C25, C22.50C24, C22×C4○D4, C2×2- 1+4, C2×C22.50C24

Smallest permutation representation of C2×C22.50C24
►On 64 points
Generators in S64
(1 49)(2 50)(3 51)(4 52)(5 16)(6 13)(7 14)(8 15)(9 35)(10 36)(11 33)(12 34)(17 47)(18 48)(19 45)(20 46)(21 43)(22 44)(23 41)(24 42)(25 59)(26 60)(27 57)(28 58)(29 55)(30 56)(31 53)(32 54)(37 63)(38 64)(39 61)(40 62)

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

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

(2 14)(4 16)(5 52)(7 50)(10 32)(12 30)(17 37)(18 20)(19 39)(21 25)(22 24)(23 27)(26 28)(34 56)(36 54)(38 40)(41 57)(42 44)(43 59)(45 61)(46 48)(47 63)(58 60)(62 64)

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

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

50 conjugacy classes

class 1 2A···2G2H2I2J2K4A···4X4Y···4AL
order12···222224···44···4
size11···144442···24···4

50 irreducible representations

dim11111111124
type+++++++++-
imageC1C2C2C2C2C2C2C2C2C4○D42- 1+4
kernelC2×C22.50C24C2×C42⋊C2C2×C4×D4C2×C4×Q8C2×C22⋊Q8C2×C4.4D4C2×C42⋊2C2C2×C4⋊Q8C22.50C24C2×C4C22
# reps1213224116162

Matrix representation of C2×C22.50C24 ►in GL6(𝔽5)

400000
040000
001000
000100
000010
000001
,
400000
040000
004000
000400
000010
000001
,
100000
010000
001000
000100
000040
000004
,
100000
040000
001000
004400
000010
000024
,
010000
400000
001200
004400
000020
000002
,
200000
020000
002000
000200
000032
000012
,
200000
030000
002000
003300
000010
000001

G:=sub<GL(6,GF(5))| [4,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[1,0,0,0,0,0,0,4,0,0,0,0,0,0,1,4,0,0,0,0,0,4,0,0,0,0,0,0,1,2,0,0,0,0,0,4],[0,4,0,0,0,0,1,0,0,0,0,0,0,0,1,4,0,0,0,0,2,4,0,0,0,0,0,0,2,0,0,0,0,0,0,2],[2,0,0,0,0,0,0,2,0,0,0,0,0,0,2,0,0,0,0,0,0,2,0,0,0,0,0,0,3,1,0,0,0,0,2,2],[2,0,0,0,0,0,0,3,0,0,0,0,0,0,2,3,0,0,0,0,0,3,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;
 

C2×C22.50C24 in GAP, Magma, Sage, TeX 

C_2\times C_2^2._{50}C_2^4
 
% in TeX
 
G:=Group("C2xC2^2.50C2^4");
 
// GroupNames label
 
G:=SmallGroup(128,2206);
 
// by ID
 
G=gap.SmallGroup(128,2206);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,2,-2,2,477,680,1430,184,570,136]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=d^2=1,e^2=c*b=b*c,f^2=g^2=b,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,a*g=g*a,e*d*e^-1=b*d=d*b,g*e*g^-1=b*e=e*b,b*f=f*b,b*g=g*b,f*d*f^-1=c*d=d*c,c*e=e*c,c*f=f*c,c*g=g*c,d*g=g*d,e*f=f*e,f*g=g*f>;
 
// generators/relations
 

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