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G = C23.382C24  order 128 = 27

99th central stem extension by C23 of C24

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

Aliases: C25.45C22, C24.574C23, C23.382C24, C22.1852+ (1+4), (C2×C42)⋊6C22, C24⋊3C4.9C2, (C22×C4).380D4, C23.611(C2×D4), (C22×C4).68C23, C23.8Q8⋊61C2, C23.4Q8⋊13C2, C23.145(C4○D4), C23.34D4⋊29C2, C23.11D4⋊24C2, (C23×C4).369C22, C22.262(C22×D4), C2.C42⋊24C22, C24.C22⋊61C2, C2.15(C22.29C24), C2.54(C22.19C24), C2.25(C22.45C24), C22.61(C22.D4), (C2×C4).347(C2×D4), (C2×C4⋊C4)⋊111C22, (C2×C42⋊C2)⋊28C2, C22.259(C2×C4○D4), (C22×C22⋊C4).24C2, C2.27(C2×C22.D4), (C2×C22⋊C4).150C22, SmallGroup(128,1214)

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

Derived series
C1 — C23 — C23.382C24
Chief series
C1 — C2 — C22 — C23 — C24 — C25 — C22×C22⋊C4 — C23.382C24
Lower central
C1 — C23 — C23.382C24
Upper central
C1 — C23 — C23.382C24
Jennings
C1 — C23 — C23.382C24

Subgroups: 724 in 336 conjugacy classes, 104 normal (18 characteristic)
C1, C2 [×3], C2 [×4], C2 [×8], C4 [×14], C22, C22 [×10], C22 [×48], C2×C4 [×4], C2×C4 [×46], C23, C23 [×10], C23 [×48], C42 [×4], C22⋊C4 [×28], C4⋊C4 [×8], C22×C4 [×2], C22×C4 [×14], C22×C4 [×8], C24, C24 [×2], C24 [×12], C2.C42 [×8], C2×C42 [×2], C2×C22⋊C4 [×16], C2×C22⋊C4 [×4], C2×C4⋊C4 [×2], C2×C4⋊C4 [×4], C42⋊C2 [×4], C23×C4 [×2], C25, C24⋊3C4 [×2], C23.34D4, C23.8Q8 [×2], C24.C22 [×4], C23.11D4 [×2], C23.4Q8 [×2], C22×C22⋊C4, C2×C42⋊C2, C23.382C24

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, C22.19C24, C22.29C24, C22.45C24 [×4], C23.382C24

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

Smallest permutation representation
►On 32 points
Generators in S32
(1 9)(2 10)(3 11)(4 12)(5 22)(6 23)(7 24)(8 21)(13 25)(14 26)(15 27)(16 28)(17 31)(18 32)(19 29)(20 30)

(1 27)(2 28)(3 25)(4 26)(5 30)(6 31)(7 32)(8 29)(9 15)(10 16)(11 13)(12 14)(17 23)(18 24)(19 21)(20 22)

(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)

(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)

(1 23 9 6)(2 7 10 24)(3 21 11 8)(4 5 12 22)(13 29 25 19)(14 20 26 30)(15 31 27 17)(16 18 28 32)

(1 27)(3 25)(5 7)(6 29)(8 31)(9 15)(11 13)(17 21)(18 20)(19 23)(22 24)(30 32)

(1 25)(2 14)(3 27)(4 16)(5 18)(6 29)(7 20)(8 31)(9 13)(10 26)(11 15)(12 28)(17 21)(19 23)(22 32)(24 30)
 
G:=sub<Sym(32)| (1,9)(2,10)(3,11)(4,12)(5,22)(6,23)(7,24)(8,21)(13,25)(14,26)(15,27)(16,28)(17,31)(18,32)(19,29)(20,30), (1,27)(2,28)(3,25)(4,26)(5,30)(6,31)(7,32)(8,29)(9,15)(10,16)(11,13)(12,14)(17,23)(18,24)(19,21)(20,22), (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), (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), (1,23,9,6)(2,7,10,24)(3,21,11,8)(4,5,12,22)(13,29,25,19)(14,20,26,30)(15,31,27,17)(16,18,28,32), (1,27)(3,25)(5,7)(6,29)(8,31)(9,15)(11,13)(17,21)(18,20)(19,23)(22,24)(30,32), (1,25)(2,14)(3,27)(4,16)(5,18)(6,29)(7,20)(8,31)(9,13)(10,26)(11,15)(12,28)(17,21)(19,23)(22,32)(24,30)>;
 
G:=Group( (1,9)(2,10)(3,11)(4,12)(5,22)(6,23)(7,24)(8,21)(13,25)(14,26)(15,27)(16,28)(17,31)(18,32)(19,29)(20,30), (1,27)(2,28)(3,25)(4,26)(5,30)(6,31)(7,32)(8,29)(9,15)(10,16)(11,13)(12,14)(17,23)(18,24)(19,21)(20,22), (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), (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), (1,23,9,6)(2,7,10,24)(3,21,11,8)(4,5,12,22)(13,29,25,19)(14,20,26,30)(15,31,27,17)(16,18,28,32), (1,27)(3,25)(5,7)(6,29)(8,31)(9,15)(11,13)(17,21)(18,20)(19,23)(22,24)(30,32), (1,25)(2,14)(3,27)(4,16)(5,18)(6,29)(7,20)(8,31)(9,13)(10,26)(11,15)(12,28)(17,21)(19,23)(22,32)(24,30) );
 
G=PermutationGroup([(1,9),(2,10),(3,11),(4,12),(5,22),(6,23),(7,24),(8,21),(13,25),(14,26),(15,27),(16,28),(17,31),(18,32),(19,29),(20,30)], [(1,27),(2,28),(3,25),(4,26),(5,30),(6,31),(7,32),(8,29),(9,15),(10,16),(11,13),(12,14),(17,23),(18,24),(19,21),(20,22)], [(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)], [(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)], [(1,23,9,6),(2,7,10,24),(3,21,11,8),(4,5,12,22),(13,29,25,19),(14,20,26,30),(15,31,27,17),(16,18,28,32)], [(1,27),(3,25),(5,7),(6,29),(8,31),(9,15),(11,13),(17,21),(18,20),(19,23),(22,24),(30,32)], [(1,25),(2,14),(3,27),(4,16),(5,18),(6,29),(7,20),(8,31),(9,13),(10,26),(11,15),(12,28),(17,21),(19,23),(22,32),(24,30)])
 

Matrix representation ►G ⊆ GL6(𝔽5)

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

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

38 conjugacy classes

class 1 2A···2G2H2I2J2K2L2M2N2O4A4B4C4D4E···4R4S4T4U4V
order12···22222222244444···44444
size11···12222444422224···48888

38 irreducible representations

dim111111111224
type+++++++++++
imageC1C2C2C2C2C2C2C2C2D4C4○D42+ (1+4)
kernelC23.382C24C24⋊3C4C23.34D4C23.8Q8C24.C22C23.11D4C23.4Q8C22×C22⋊C4C2×C42⋊C2C22×C4C23C22
# reps1212422114162

In GAP, Magma, Sage, TeX 

C_2^3._{382}C_2^4
 
% in TeX
 
G:=Group("C2^3.382C2^4");
 
// GroupNames label
 
G:=SmallGroup(128,1214);
 
// by ID
 
G=gap.SmallGroup(128,1214);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,2,448,253,120,758,723,675]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=f^2=g^2=1,d^2=c,e^2=a,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=b*d=d*b,b*e=e*b,b*f=f*b,b*g=g*b,c*d=d*c,f*e*f=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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