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G = C23.412C24order 128 = 27

129th central stem extension by C23 of C24

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

Aliases: C24.17C23, C23.412C24, C22.1572- 1+4, C22.2072+ 1+4, C4⋊C424D4, C429C425C2, C43(C422C2), C2.46(D46D4), C2.21(Q86D4), (C2×C42).52C22, C23.Q823C2, C23.11D433C2, C22.282(C22×D4), (C22×C4).1485C23, C24.C2269C2, (C22×D4).153C22, C23.65C2375C2, C24.3C22.38C2, C2.C42.489C22, C2.27(C22.26C24), C2.26(C22.50C24), C2.35(C22.36C24), C2.41(C22.47C24), (C4×C4⋊C4)⋊75C2, (C2×C4).67(C2×D4), (C2×C422C2)⋊7C2, (C2×C4).133(C4○D4), (C2×C4⋊C4).278C22, C2.19(C2×C422C2), C22.289(C2×C4○D4), (C2×C22⋊C4).162C22, SmallGroup(128,1244)

Series: Derived Chief Lower central Upper central Jennings

C1C23 — C23.412C24
C1C2C22C23C22×C4C2×C42C4×C4⋊C4 — C23.412C24
C1C23 — C23.412C24
C1C23 — C23.412C24
C1C23 — C23.412C24

Generators and relations for C23.412C24
 G = < a,b,c,d,e,f,g | a2=b2=c2=1, d2=abc, e2=f2=b, g2=a, ab=ba, ac=ca, ede-1=gdg-1=ad=da, 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, eg=ge, fg=gf >

Subgroups: 468 in 248 conjugacy classes, 104 normal (42 characteristic)
C1, C2 [×7], C2 [×2], C4 [×4], C4 [×16], C22 [×7], C22 [×14], C2×C4 [×14], C2×C4 [×32], D4 [×4], C23, C23 [×14], C42 [×10], C22⋊C4 [×20], C4⋊C4 [×4], C4⋊C4 [×18], C22×C4 [×7], C22×C4 [×6], C2×D4 [×6], C24 [×2], C2.C42 [×2], C2.C42 [×4], C2×C42 [×3], C2×C42 [×2], C2×C22⋊C4 [×12], C2×C4⋊C4 [×5], C2×C4⋊C4 [×6], C422C2 [×8], C22×D4, C4×C4⋊C4 [×2], C429C4, C24.C22 [×2], C23.65C23, C24.3C22 [×3], C23.Q8 [×2], C23.11D4 [×2], C2×C422C2 [×2], C23.412C24
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], C2×D4 [×6], C4○D4 [×8], C24, C422C2 [×4], C22×D4, C2×C4○D4 [×4], 2+ 1+4, 2- 1+4, C2×C422C2, C22.26C24, C22.36C24, D46D4, Q86D4, C22.47C24, C22.50C24, C23.412C24

Smallest permutation representation of C23.412C24
On 64 points
Generators in S64
(1 60)(2 57)(3 58)(4 59)(5 32)(6 29)(7 30)(8 31)(9 49)(10 50)(11 51)(12 52)(13 45)(14 46)(15 47)(16 48)(17 44)(18 41)(19 42)(20 43)(21 38)(22 39)(23 40)(24 37)(25 33)(26 34)(27 35)(28 36)(53 61)(54 62)(55 63)(56 64)
(1 32)(2 29)(3 30)(4 31)(5 60)(6 57)(7 58)(8 59)(9 39)(10 40)(11 37)(12 38)(13 41)(14 42)(15 43)(16 44)(17 48)(18 45)(19 46)(20 47)(21 52)(22 49)(23 50)(24 51)(25 55)(26 56)(27 53)(28 54)(33 63)(34 64)(35 61)(36 62)
(1 7)(2 8)(3 5)(4 6)(9 24)(10 21)(11 22)(12 23)(13 20)(14 17)(15 18)(16 19)(25 61)(26 62)(27 63)(28 64)(29 59)(30 60)(31 57)(32 58)(33 53)(34 54)(35 55)(36 56)(37 49)(38 50)(39 51)(40 52)(41 47)(42 48)(43 45)(44 46)
(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 42 32 14)(2 20 29 47)(3 44 30 16)(4 18 31 45)(5 46 60 19)(6 15 57 43)(7 48 58 17)(8 13 59 41)(9 63 39 33)(10 56 40 26)(11 61 37 35)(12 54 38 28)(21 36 52 62)(22 25 49 55)(23 34 50 64)(24 27 51 53)
(1 52 32 21)(2 22 29 49)(3 50 30 23)(4 24 31 51)(5 38 60 12)(6 9 57 39)(7 40 58 10)(8 11 59 37)(13 25 41 55)(14 56 42 26)(15 27 43 53)(16 54 44 28)(17 36 48 62)(18 63 45 33)(19 34 46 64)(20 61 47 35)
(1 53 60 61)(2 62 57 54)(3 55 58 63)(4 64 59 56)(5 35 32 27)(6 28 29 36)(7 33 30 25)(8 26 31 34)(9 16 49 48)(10 45 50 13)(11 14 51 46)(12 47 52 15)(17 39 44 22)(18 23 41 40)(19 37 42 24)(20 21 43 38)

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

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

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

38 conjugacy classes

class 1 2A···2G2H2I4A···4H4I···4Z4AA4AB
order12···2224···44···444
size11···1882···24···488

38 irreducible representations

dim1111111112244
type+++++++++++-
imageC1C2C2C2C2C2C2C2C2D4C4○D42+ 1+42- 1+4
kernelC23.412C24C4×C4⋊C4C429C4C24.C22C23.65C23C24.3C22C23.Q8C23.11D4C2×C422C2C4⋊C4C2×C4C22C22
# reps12121322241611

Matrix representation of C23.412C24 in GL6(𝔽5)

400000
040000
004000
000400
000010
000001
,
100000
010000
004000
000400
000040
000004
,
100000
010000
001000
000100
000040
000004
,
300000
020000
000100
001000
000044
000001
,
020000
300000
000400
001000
000022
000003
,
400000
040000
000400
001000
000030
000042
,
010000
400000
000100
004000
000040
000004

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

C23.412C24 in GAP, Magma, Sage, TeX

C_2^3._{412}C_2^4
% in TeX

G:=Group("C2^3.412C2^4");
// GroupNames label

G:=SmallGroup(128,1244);
// by ID

G=gap.SmallGroup(128,1244);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,784,253,232,758,723,184,675,80]);
// Polycyclic

G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=1,d^2=a*b*c,e^2=f^2=b,g^2=a,a*b=b*a,a*c=c*a,e*d*e^-1=g*d*g^-1=a*d=d*a,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,e*g=g*e,f*g=g*f>;
// generators/relations

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