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G = C23.400C24order 128 = 27

117th central stem extension by C23 of C24

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

Aliases: C24.13C23, C23.400C24, C22.1982+ 1+4, C428C430C2, C429C422C2, C23.184(C2×D4), (C22×C4).387D4, (C2×C42).49C22, C23.10D437C2, (C23×C4).384C22, C22.276(C22×D4), C24.C2267C2, (C22×C4).1483C23, C24.3C2249C2, C4.51(C22.D4), (C22×D4).149C22, C2.17(C22.29C24), C2.C42.152C22, C2.25(C22.26C24), C2.36(C22.47C24), C2.16(C22.49C24), (C4×C22⋊C4)⋊75C2, (C2×C4).831(C2×D4), (C2×C4⋊D4).33C2, (C2×C42⋊C2)⋊31C2, (C2×C4).127(C4○D4), (C2×C4⋊C4).269C22, C22.277(C2×C4○D4), C2.35(C2×C22.D4), (C2×C22⋊C4).160C22, SmallGroup(128,1232)

Series: Derived Chief Lower central Upper central Jennings

C1C23 — C23.400C24
C1C2C22C23C22×C4C2×C42C4×C22⋊C4 — C23.400C24
C1C23 — C23.400C24
C1C23 — C23.400C24
C1C23 — C23.400C24

Generators and relations for C23.400C24
 G = < a,b,c,d,e,f,g | a2=b2=c2=e2=1, d2=ca=ac, f2=b, g2=a, ab=ba, ede=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: 580 in 284 conjugacy classes, 104 normal (26 characteristic)
C1, C2 [×3], C2 [×4], C2 [×4], C4 [×4], C4 [×14], C22 [×3], C22 [×4], C22 [×24], C2×C4 [×12], C2×C4 [×38], D4 [×12], C23, C23 [×2], C23 [×20], C42 [×8], C22⋊C4 [×24], C4⋊C4 [×10], C22×C4 [×2], C22×C4 [×14], C22×C4 [×4], C2×D4 [×14], C24, C24 [×2], C2.C42 [×6], C2×C42 [×2], C2×C42 [×2], C2×C22⋊C4 [×14], C2×C4⋊C4 [×3], C2×C4⋊C4 [×4], C42⋊C2 [×4], C4⋊D4 [×4], C23×C4, C22×D4, C22×D4 [×2], C4×C22⋊C4, C428C4, C429C4, C24.C22 [×4], C24.3C22 [×2], C23.10D4 [×4], C2×C42⋊C2, C2×C4⋊D4, C23.400C24
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], C2×D4 [×6], C4○D4 [×8], C24, C22.D4 [×4], C22×D4, C2×C4○D4 [×4], 2+ 1+4 [×2], C2×C22.D4, C22.26C24, C22.29C24, C22.47C24 [×2], C22.49C24 [×2], C23.400C24

Smallest permutation representation of C23.400C24
On 64 points
Generators in S64
(1 36)(2 33)(3 34)(4 35)(5 63)(6 64)(7 61)(8 62)(9 16)(10 13)(11 14)(12 15)(17 24)(18 21)(19 22)(20 23)(25 31)(26 32)(27 29)(28 30)(37 43)(38 44)(39 41)(40 42)(45 60)(46 57)(47 58)(48 59)(49 54)(50 55)(51 56)(52 53)
(1 7)(2 8)(3 5)(4 6)(9 25)(10 26)(11 27)(12 28)(13 32)(14 29)(15 30)(16 31)(17 50)(18 51)(19 52)(20 49)(21 56)(22 53)(23 54)(24 55)(33 62)(34 63)(35 64)(36 61)(37 48)(38 45)(39 46)(40 47)(41 57)(42 58)(43 59)(44 60)
(1 34)(2 35)(3 36)(4 33)(5 61)(6 62)(7 63)(8 64)(9 14)(10 15)(11 16)(12 13)(17 22)(18 23)(19 24)(20 21)(25 29)(26 30)(27 31)(28 32)(37 41)(38 42)(39 43)(40 44)(45 58)(46 59)(47 60)(48 57)(49 56)(50 53)(51 54)(52 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 41)(2 40)(3 43)(4 38)(5 59)(6 45)(7 57)(8 47)(9 54)(10 50)(11 56)(12 52)(13 55)(14 51)(15 53)(16 49)(17 26)(18 29)(19 28)(20 31)(21 27)(22 30)(23 25)(24 32)(33 42)(34 37)(35 44)(36 39)(46 61)(48 63)(58 62)(60 64)
(1 47 7 40)(2 37 8 48)(3 45 5 38)(4 39 6 46)(9 52 25 19)(10 20 26 49)(11 50 27 17)(12 18 28 51)(13 23 32 54)(14 55 29 24)(15 21 30 56)(16 53 31 22)(33 43 62 59)(34 60 63 44)(35 41 64 57)(36 58 61 42)
(1 17 36 24)(2 21 33 18)(3 19 34 22)(4 23 35 20)(5 52 63 53)(6 54 64 49)(7 50 61 55)(8 56 62 51)(9 60 16 45)(10 46 13 57)(11 58 14 47)(12 48 15 59)(25 44 31 38)(26 39 32 41)(27 42 29 40)(28 37 30 43)

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

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

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

38 conjugacy classes

class 1 2A···2G2H2I2J2K4A···4H4I···4X4Y4Z
order12···222224···44···444
size11···144882···24···488

38 irreducible representations

dim111111111224
type+++++++++++
imageC1C2C2C2C2C2C2C2C2D4C4○D42+ 1+4
kernelC23.400C24C4×C22⋊C4C428C4C429C4C24.C22C24.3C22C23.10D4C2×C42⋊C2C2×C4⋊D4C22×C4C2×C4C22
# reps1111424114162

Matrix representation of C23.400C24 in GL6(𝔽5)

400000
040000
001000
000100
000010
000001
,
100000
010000
004000
000400
000040
000004
,
100000
010000
001000
000100
000040
000004
,
240000
030000
001200
000400
000003
000030
,
240000
330000
004000
000400
000001
000010
,
400000
040000
004300
001100
000020
000003
,
430000
110000
004000
000400
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],[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],[2,0,0,0,0,0,4,3,0,0,0,0,0,0,1,0,0,0,0,0,2,4,0,0,0,0,0,0,0,3,0,0,0,0,3,0],[2,3,0,0,0,0,4,3,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,1,0,0,0,0,3,1,0,0,0,0,0,0,2,0,0,0,0,0,0,3],[4,1,0,0,0,0,3,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;

C23.400C24 in GAP, Magma, Sage, TeX

C_2^3._{400}C_2^4
% in TeX

G:=Group("C2^3.400C2^4");
// GroupNames label

G:=SmallGroup(128,1232);
// by ID

G=gap.SmallGroup(128,1232);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,560,253,758,723,184,675,80]);
// Polycyclic

G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=e^2=1,d^2=c*a=a*c,f^2=b,g^2=a,a*b=b*a,e*d*e=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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