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G = C2×C22.57C24  order 128 = 27

Direct product of C2 and C22.57C24

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

Aliases: C2×C22.57C24, C42.581C23, C22.117C25, C24.514C23, C23.279C24, C22.862- 1+4, C22.1202+ 1+4, C4⋊Q8⋊97C22, C4⋊C4.306C23, (C2×C4).107C24, C22⋊Q8⋊98C22, (C2×D4).311C23, C22⋊C4.37C23, (C2×Q8).296C23, C42.C2⋊63C22, C42⋊2C2⋊41C22, (C2×C42).962C22, (C23×C4).615C22, C2.48(C2×2+ 1+4), C2.36(C2×2- 1+4), (C22×C4).1215C23, C4.4D4.179C22, (C22×D4).434C22, (C22×Q8).368C22, C22.D4.36C22, (C2×C4⋊Q8)⋊59C2, (C2×C22⋊Q8)⋊81C2, (C2×C42.C2)⋊49C2, (C2×C42⋊2C2)⋊40C2, (C2×C4⋊C4).716C22, (C2×C4.4D4).44C2, (C2×C22⋊C4).388C22, (C2×C22.D4).34C2, SmallGroup(128,2260)

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

Derived series
C1 — C22 — C2×C22.57C24
Chief series
C1 — C2 — C22 — C23 — C24 — C23×C4 — C2×C22.D4 — C2×C22.57C24
Lower central
C1 — C22 — C2×C22.57C24
Upper central
C1 — C23 — C2×C22.57C24
Jennings
C1 — C22 — C2×C22.57C24

Generators and relations for C2×C22.57C24
 G = < a,b,c,d,e,f,g | a2=b2=c2=g2=1, d2=e2=f2=b, ab=ba, ac=ca, ad=da, ae=ea, af=fa, ag=ga, bc=cb, ede-1=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: 732 in 512 conjugacy classes, 388 normal (10 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×C42, C2×C22⋊C4, C2×C4⋊C4, C22⋊Q8, C22.D4, C4.4D4, C42.C2, C42⋊2C2, C4⋊Q8, C23×C4, C22×D4, C22×Q8, C22×Q8, C2×C22⋊Q8, C2×C22.D4, C2×C4.4D4, C2×C42.C2, C2×C42⋊2C2, C2×C4⋊Q8, C22.57C24, C2×C22.57C24
Quotients: C1, C2, C22, C23, C24, 2+ 1+4, 2- 1+4, C25, C22.57C24, C2×2+ 1+4, C2×2- 1+4, C2×C22.57C24

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

(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 20)(3 17)(4 18)(5 50)(6 51)(7 52)(8 49)(9 29)(10 30)(11 31)(12 32)(13 40)(14 37)(15 38)(16 39)(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)

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

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

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

38 conjugacy classes

class 1 2A···2G2H2I2J2K4A···4Z
order12···222224···4
size11···144444···4

38 irreducible representations

dim1111111144
type+++++++++-
imageC1C2C2C2C2C2C2C22+ 1+42- 1+4
kernelC2×C22.57C24C2×C22⋊Q8C2×C22.D4C2×C4.4D4C2×C42.C2C2×C42⋊2C2C2×C4⋊Q8C22.57C24C22C22
# reps14212421624

Matrix representation of C2×C22.57C24 ►in GL12(𝔽5)

400000000000
040000000000
004000000000
000400000000
000010000000
000001000000
000000100000
000000010000
000000004000
000000000400
000000000040
000000000004
,
400000000000
040000000000
004000000000
000400000000
000040000000
000004000000
000000400000
000000040000
000000004000
000000000400
000000000040
000000000004
,
400000000000
040000000000
004000000000
000400000000
000040000000
000004000000
000000400000
000000040000
000000001000
000000000100
000000000010
000000000001
,
200000000000
030000000000
003000000000
023200000000
000020000000
000003000000
000000300000
000003220000
000000000020
000000000003
000000002000
000000000300
,
004000000000
414200000000
100000000000
140400000000
000000100000
000011430000
000040000000
000011040000
000000000010
000000000001
000000004000
000000000400
,
040000000000
100000000000
414200000000
404100000000
000001000000
000040000000
000011430000
000040110000
000000000100
000000004000
000000000004
000000000010
,
100000000000
010000000000
004000000000
140400000000
000010000000
000001000000
000000400000
000011040000
000000001000
000000000100
000000000040
000000000004

G:=sub<GL(12,GF(5))| [4,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,4],[4,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,4],[4,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1],[2,0,0,0,0,0,0,0,0,0,0,0,0,3,0,2,0,0,0,0,0,0,0,0,0,0,3,3,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,3,0,3,0,0,0,0,0,0,0,0,0,0,3,2,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0],[0,4,1,1,0,0,0,0,0,0,0,0,0,1,0,4,0,0,0,0,0,0,0,0,4,4,0,0,0,0,0,0,0,0,0,0,0,2,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,1,4,1,0,0,0,0,0,0,0,0,0,1,0,1,0,0,0,0,0,0,0,0,1,4,0,0,0,0,0,0,0,0,0,0,0,3,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0],[0,1,4,4,0,0,0,0,0,0,0,0,4,0,1,0,0,0,0,0,0,0,0,0,0,0,4,4,0,0,0,0,0,0,0,0,0,0,2,1,0,0,0,0,0,0,0,0,0,0,0,0,0,4,1,4,0,0,0,0,0,0,0,0,1,0,1,0,0,0,0,0,0,0,0,0,0,0,4,1,0,0,0,0,0,0,0,0,0,0,3,1,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,4,0],[1,0,0,1,0,0,0,0,0,0,0,0,0,1,0,4,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,1,0,1,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,4] >;
 

C2×C22.57C24 in GAP, Magma, Sage, TeX 

C_2\times C_2^2._{57}C_2^4
 
% in TeX
 
G:=Group("C2xC2^2.57C2^4");
 
// GroupNames label
 
G:=SmallGroup(128,2260);
 
// by ID
 
G=gap.SmallGroup(128,2260);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,2,-2,2,224,477,232,1430,1059,184,2915,570]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=g^2=1,d^2=e^2=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^-1=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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