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G = C23.398C24order 128 = 27

115th central stem extension by C23 of C24

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

Aliases: C24.12C23, C23.398C24, C22.1492- 1+4, C22.1972+ 1+4, (C22×C4).385D4, C23.368(C2×D4), (C2×C42).47C22, C23.4Q815C2, C23.7Q858C2, C23.310(C4○D4), C23.11D430C2, (C23×C4).382C22, (C22×C4).523C23, C22.274(C22×D4), C24.C2266C2, C23.23D4.27C2, (C22×D4).148C22, C23.63C2365C2, C23.83C2325C2, C23.81C2325C2, C2.29(C22.45C24), C2.C42.150C22, C2.23(C22.26C24), C22.16(C22.D4), C2.11(C22.31C24), C2.41(C22.46C24), (C2×C4).1192(C2×D4), (C2×C42⋊C2)⋊30C2, (C2×C4).376(C4○D4), (C2×C4⋊C4).267C22, C22.275(C2×C4○D4), (C2×C2.C42)⋊34C2, (C2×C22⋊C4).46C22, C2.33(C2×C22.D4), (C2×C22.D4).16C2, SmallGroup(128,1230)

Series: Derived Chief Lower central Upper central Jennings

C1C23 — C23.398C24
C1C2C22C23C24C23×C4C2×C2.C42 — C23.398C24
C1C23 — C23.398C24
C1C23 — C23.398C24
C1C23 — C23.398C24

Generators and relations for C23.398C24
 G = < a,b,c,d,e,f,g | a2=b2=c2=g2=1, d2=c, e2=a, f2=b, ab=ba, ac=ca, ede-1=gdg=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: 516 in 273 conjugacy classes, 104 normal (34 characteristic)
C1, C2 [×3], C2 [×4], C2 [×5], C4 [×17], C22 [×3], C22 [×8], C22 [×19], C2×C4 [×8], C2×C4 [×51], D4 [×4], C23, C23 [×6], C23 [×11], C42 [×4], C22⋊C4 [×16], C4⋊C4 [×14], C22×C4 [×5], C22×C4 [×12], C22×C4 [×14], C2×D4 [×6], C24 [×2], C2.C42 [×2], C2.C42 [×10], C2×C42 [×2], C2×C22⋊C4 [×3], C2×C22⋊C4 [×6], C2×C4⋊C4 [×4], C2×C4⋊C4 [×4], C42⋊C2 [×4], C22.D4 [×4], C23×C4, C23×C4 [×2], C22×D4, C2×C2.C42, C23.7Q8 [×2], C23.23D4 [×2], C23.63C23 [×2], C24.C22 [×2], C23.11D4, C23.81C23, C23.4Q8, C23.83C23, C2×C42⋊C2, C2×C22.D4, C23.398C24
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- 1+4, C2×C22.D4, C22.26C24, C22.31C24, C22.45C24 [×2], C22.46C24 [×2], C23.398C24

Smallest permutation representation of C23.398C24
On 64 points
Generators in S64
(1 23)(2 24)(3 21)(4 22)(5 46)(6 47)(7 48)(8 45)(9 58)(10 59)(11 60)(12 57)(13 54)(14 55)(15 56)(16 53)(17 34)(18 35)(19 36)(20 33)(25 41)(26 42)(27 43)(28 44)(29 37)(30 38)(31 39)(32 40)(49 64)(50 61)(51 62)(52 63)
(1 57)(2 58)(3 59)(4 60)(5 42)(6 43)(7 44)(8 41)(9 24)(10 21)(11 22)(12 23)(13 37)(14 38)(15 39)(16 40)(17 49)(18 50)(19 51)(20 52)(25 45)(26 46)(27 47)(28 48)(29 54)(30 55)(31 56)(32 53)(33 63)(34 64)(35 61)(36 62)
(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 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 23 26)(2 27 24 43)(3 44 21 28)(4 25 22 41)(5 12 46 57)(6 58 47 9)(7 10 48 59)(8 60 45 11)(13 61 54 50)(14 51 55 62)(15 63 56 52)(16 49 53 64)(17 32 34 40)(18 37 35 29)(19 30 36 38)(20 39 33 31)
(1 47 57 27)(2 28 58 48)(3 45 59 25)(4 26 60 46)(5 22 42 11)(6 12 43 23)(7 24 44 9)(8 10 41 21)(13 17 37 49)(14 50 38 18)(15 19 39 51)(16 52 40 20)(29 64 54 34)(30 35 55 61)(31 62 56 36)(32 33 53 63)
(1 18)(2 36)(3 20)(4 34)(5 13)(6 55)(7 15)(8 53)(9 51)(10 63)(11 49)(12 61)(14 47)(16 45)(17 22)(19 24)(21 33)(23 35)(25 40)(26 29)(27 38)(28 31)(30 43)(32 41)(37 42)(39 44)(46 54)(48 56)(50 57)(52 59)(58 62)(60 64)

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

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

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

38 conjugacy classes

class 1 2A···2G2H2I2J2K2L4A4B4C4D4E···4V4W4X4Y
order12···22222244444···4444
size11···12222822224···4888

38 irreducible representations

dim11111111111122244
type++++++++++++++-
imageC1C2C2C2C2C2C2C2C2C2C2C2D4C4○D4C4○D42+ 1+42- 1+4
kernelC23.398C24C2×C2.C42C23.7Q8C23.23D4C23.63C23C24.C22C23.11D4C23.81C23C23.4Q8C23.83C23C2×C42⋊C2C2×C22.D4C22×C4C2×C4C23C22C22
# reps11222211111148811

Matrix representation of C23.398C24 in GL6(𝔽5)

400000
040000
004000
000400
000010
000001
,
400000
040000
001000
000100
000010
000001
,
100000
010000
001000
000100
000040
000004
,
400000
010000
000200
003000
000020
000002
,
030000
300000
000200
002000
000002
000030
,
030000
300000
001000
000100
000001
000010
,
010000
100000
000100
001000
000010
000001

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],[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],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[4,0,0,0,0,0,0,1,0,0,0,0,0,0,0,3,0,0,0,0,2,0,0,0,0,0,0,0,2,0,0,0,0,0,0,2],[0,3,0,0,0,0,3,0,0,0,0,0,0,0,0,2,0,0,0,0,2,0,0,0,0,0,0,0,0,3,0,0,0,0,2,0],[0,3,0,0,0,0,3,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;

C23.398C24 in GAP, Magma, Sage, TeX

C_2^3._{398}C_2^4
% in TeX

G:=Group("C2^3.398C2^4");
// GroupNames label

G:=SmallGroup(128,1230);
// by ID

G=gap.SmallGroup(128,1230);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,448,253,120,758,723,184,675]);
// Polycyclic

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