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G = C23.572C24  order 128 = 27

289th central stem extension by C23 of C24

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

Aliases: C24.57C23, C23.572C24, C22.2582- (1+4), C22.3462+ (1+4), C2.36(D42), C22⋊C4⋊11D4, C23.59(C2×D4), C2.55(D4⋊6D4), C2.84(D4⋊5D4), C23.Q8⋊47C2, C23.4Q8⋊40C2, C23.8Q8⋊93C2, C23.7Q8⋊82C2, C23.10D4⋊72C2, C23.23D4⋊78C2, (C22×C4).861C23, (C23×C4).442C22, (C2×C42).632C22, C22.381(C22×D4), (C22×D4).213C22, (C22×Q8).172C22, C23.78C23⋊36C2, C24.C22⋊116C2, C23.65C23⋊113C2, C2.C42.283C22, C2.7(C22.56C24), C2.37(C22.31C24), C2.51(C23.38C23), C2.67(C22.36C24), (C2×C4).412(C2×D4), (C2×C22⋊Q8)⋊32C2, (C2×C4.4D4)⋊25C2, (C2×C4⋊D4).42C2, (C2×C4).416(C4○D4), (C2×C4⋊C4).390C22, C22.438(C2×C4○D4), (C2×C22.D4)⋊29C2, (C2×C22⋊C4).243C22, SmallGroup(128,1404)

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

Derived series
C1 — C23 — C23.572C24
Chief series
C1 — C2 — C22 — C23 — C24 — C23×C4 — C2×C22.D4 — C23.572C24
Lower central
C1 — C23 — C23.572C24
Upper central
C1 — C23 — C23.572C24
Jennings
C1 — C23 — C23.572C24

Subgroups: 644 in 309 conjugacy classes, 104 normal (82 characteristic)
C1, C2 [×7], C2 [×5], C4 [×17], C22 [×7], C22 [×27], C2×C4 [×10], C2×C4 [×39], D4 [×12], Q8 [×4], C23, C23 [×4], C23 [×19], C42 [×2], C22⋊C4 [×8], C22⋊C4 [×18], C4⋊C4 [×16], C22×C4 [×12], C22×C4 [×10], C2×D4 [×15], C2×Q8 [×5], C24 [×3], C2.C42 [×6], C2×C42, C2×C22⋊C4 [×13], C2×C4⋊C4 [×9], C4⋊D4 [×4], C22⋊Q8 [×4], C22.D4 [×4], C4.4D4 [×4], C23×C4 [×2], C22×D4 [×3], C22×Q8, C23.7Q8, C23.8Q8, C23.23D4, C24.C22, C23.65C23, C23.10D4 [×3], C23.78C23, C23.Q8, C23.4Q8, C2×C4⋊D4, C2×C22⋊Q8, C2×C22.D4, C2×C4.4D4, C23.572C24

Quotients:
C1, C2 [×15], C22 [×35], D4 [×8], C23 [×15], C2×D4 [×12], C4○D4 [×2], C24, C22×D4 [×2], C2×C4○D4, 2+ (1+4) [×2], 2- (1+4) [×2], C23.38C23, C22.31C24, C22.36C24, D42, D4⋊5D4, D4⋊6D4, C22.56C24, C23.572C24

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

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

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

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

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

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

Matrix representation ►G ⊆ GL6(𝔽5)

100000
010000
001000
000100
000040
000004
,
100000
010000
004000
000400
000010
000001
,
400000
040000
001000
000100
000010
000001
,
400000
040000
004000
001100
000031
000002
,
400000
010000
001000
000100
000024
000033
,
040000
400000
004300
000100
000020
000002
,
100000
010000
001200
000400
000040
000011

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

32 conjugacy classes

class 1 2A···2G2H2I2J2K2L4A···4N4O···4S
order12···2222224···44···4
size11···1444484···48···8

32 irreducible representations

dim111111111111112244
type++++++++++++++++-
imageC1C2C2C2C2C2C2C2C2C2C2C2C2C2D4C4○D42+ (1+4)2- (1+4)
kernelC23.572C24C23.7Q8C23.8Q8C23.23D4C24.C22C23.65C23C23.10D4C23.78C23C23.Q8C23.4Q8C2×C4⋊D4C2×C22⋊Q8C2×C22.D4C2×C4.4D4C22⋊C4C2×C4C22C22
# reps111111311111118422

In GAP, Magma, Sage, TeX 

C_2^3._{572}C_2^4
 
% in TeX
 
G:=Group("C2^3.572C2^4");
 
// GroupNames label
 
G:=SmallGroup(128,1404);
 
// by ID
 
G=gap.SmallGroup(128,1404);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,2,112,253,758,723,100,1571,346]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=e^2=g^2=1,d^2=f^2=a,a*b=b*a,a*c=c*a,e*d*e=a*d=d*a,g*e*g=a*e=e*a,a*f=f*a,a*g=g*a,b*c=c*b,f*d*f^-1=b*d=d*b,b*e=e*b,b*f=f*b,b*g=g*b,c*d=d*c,f*e*f^-1=c*e=e*c,c*f=f*c,c*g=g*c,g*d*g=a*b*d,f*g=g*f>;
 
// generators/relations
 

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