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G = C2×C22.56C24  order 128 = 27

Direct product of C2 and C22.56C24

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

Aliases: C2×C22.56C24, C23.59C24, C24.513C23, C42.580C23, C22.116C25, C22.852- 1+4, C22.1192+ 1+4, C4⋊D4⋊88C22, C4⋊C4.305C23, (C2×C4).106C24, C22⋊Q8⋊97C22, (C2×D4).310C23, C4.4D4⋊89C22, C22⋊C4.36C23, (C2×Q8).295C23, C42.C2⋊62C22, (C2×C42).961C22, (C23×C4).614C22, C2.35(C2×2- 1+4), C2.47(C2×2+ 1+4), (C22×C4).1214C23, (C22×D4).433C22, C22.D4⋊58C22, (C22×Q8).367C22, (C2×C4⋊D4)⋊71C2, (C2×C22⋊Q8)⋊80C2, (C2×C4.4D4)⋊58C2, (C2×C42.C2)⋊48C2, (C2×C4⋊C4).715C22, (C2×C22.D4)⋊63C2, (C2×C22⋊C4).387C22, SmallGroup(128,2259)

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

Derived series
C1 — C22 — C2×C22.56C24
Chief series
C1 — C2 — C22 — C23 — C24 — C23×C4 — C2×C22.D4 — C2×C22.56C24
Lower central
C1 — C22 — C2×C22.56C24
Upper central
C1 — C23 — C2×C22.56C24
Jennings
C1 — C22 — C2×C22.56C24

Generators and relations for C2×C22.56C24
 G = < a,b,c,d,e,f,g | a2=b2=c2=d2=e2=g2=1, f2=b, ab=ba, ac=ca, ad=da, ae=ea, af=fa, ag=ga, bc=cb, ede=bd=db, geg=be=eb, bf=fb, bg=gb, fdf-1=cd=dc, ce=ec, cf=fc, cg=gc, gdg=bcd, fef-1=bce, fg=gf >

Subgroups: 956 in 576 conjugacy classes, 388 normal (8 characteristic)
C1, C2, C2, C2, 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×C22⋊C4, C2×C4⋊C4, C4⋊D4, C22⋊Q8, C22.D4, C4.4D4, C42.C2, C23×C4, C22×D4, C22×Q8, C2×C4⋊D4, C2×C22⋊Q8, C2×C22.D4, C2×C4.4D4, C2×C42.C2, C22.56C24, C2×C22.56C24
Quotients: C1, C2, C22, C23, C24, 2+ 1+4, 2- 1+4, C25, C22.56C24, C2×2+ 1+4, C2×2- 1+4, C2×C22.56C24

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

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

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

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

38 conjugacy classes

class 1 2A···2G2H···2O4A···4V
order12···22···24···4
size11···14···44···4

38 irreducible representations

dim111111144
type++++++++-
imageC1C2C2C2C2C2C22+ 1+42- 1+4
kernelC2×C22.56C24C2×C4⋊D4C2×C22⋊Q8C2×C22.D4C2×C4.4D4C2×C42.C2C22.56C24C22C22
# reps1444211642

Matrix representation of C2×C22.56C24 ►in GL9(𝔽5)

400000000
040000000
004000000
000400000
000040000
000001000
000000100
000000010
000000001
,
100000000
040000000
004000000
000400000
000040000
000004000
000000400
000000040
000000004
,
100000000
010000000
001000000
000100000
000010000
000004000
000000400
000000040
000000004
,
100000000
000300000
033310000
020000000
002220000
000000330
000002003
000000002
000000030
,
100000000
000100000
044430000
010000000
000010000
000000100
000001000
000000004
000000040
,
400000000
001000000
040000000
011120000
004440000
000003000
000000300
000000420
000001002
,
400000000
010000000
001000000
000400000
044040000
000001000
000000400
000000210
000002004

G:=sub<GL(9,GF(5))| [4,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,4],[1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,4],[1,0,0,0,0,0,0,0,0,0,0,3,2,0,0,0,0,0,0,0,3,0,2,0,0,0,0,0,3,3,0,2,0,0,0,0,0,0,1,0,2,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,3,0,0,3,0,0,0,0,0,0,3,2,0],[1,0,0,0,0,0,0,0,0,0,0,4,1,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,1,4,0,0,0,0,0,0,0,0,3,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,4,0],[4,0,0,0,0,0,0,0,0,0,0,4,1,0,0,0,0,0,0,1,0,1,4,0,0,0,0,0,0,0,1,4,0,0,0,0,0,0,0,2,4,0,0,0,0,0,0,0,0,0,3,0,0,1,0,0,0,0,0,0,3,4,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,2],[4,0,0,0,0,0,0,0,0,0,1,0,0,4,0,0,0,0,0,0,1,0,4,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,0,0,0,0,4,2,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,4] >;
 

C2×C22.56C24 in GAP, Magma, Sage, TeX 

C_2\times C_2^2._{56}C_2^4
 
% in TeX
 
G:=Group("C2xC2^2.56C2^4");
 
// GroupNames label
 
G:=SmallGroup(128,2259);
 
// by ID
 
G=gap.SmallGroup(128,2259);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,2,-2,2,477,1430,1059,184,2915,570]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=d^2=e^2=g^2=1,f^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,b*c=c*b,e*d*e=b*d=d*b,g*e*g=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,g*d*g=b*c*d,f*e*f^-1=b*c*e,f*g=g*f>;
 
// generators/relations
 

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