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G = C23.380C24order 128 = 27

97th central stem extension by C23 of C24

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

Aliases: C25.44C22, C24.297C23, C23.380C24, C22.1832+ 1+4, (C2×C42)⋊5C22, C243C4.8C2, C22⋊C4.135D4, C23.181(C2×D4), C2.59(D45D4), (C22×C4).66C23, (C23×C4).94C22, C23.7Q851C2, C23.Q821C2, C23.8Q859C2, C222(C422C2), C23.144(C4○D4), C23.11D422C2, C22.260(C22×D4), C2.C4255C22, C24.C2259C2, C2.18(C22.32C24), C2.52(C22.19C24), C2.24(C22.45C24), (C4×C22⋊C4)⋊13C2, (C2×C4⋊C4)⋊19C22, (C2×C4).904(C2×D4), (C2×C422C2)⋊5C2, C2.10(C2×C422C2), C22.257(C2×C4○D4), (C22×C22⋊C4).23C2, (C2×C22⋊C4).148C22, SmallGroup(128,1212)

Series: Derived Chief Lower central Upper central Jennings

C1C23 — C23.380C24
C1C2C22C23C22×C4C2×C42C4×C22⋊C4 — C23.380C24
C1C23 — C23.380C24
C1C23 — C23.380C24
C1C23 — C23.380C24

Generators and relations for C23.380C24
 G = < a,b,c,d,e,f,g | a2=b2=c2=d2=1, e2=g2=ba=ab, f2=a, 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: 724 in 332 conjugacy classes, 104 normal (82 characteristic)
C1, C2 [×7], C2 [×8], C4 [×14], C22 [×7], C22 [×4], C22 [×48], C2×C4 [×4], C2×C4 [×42], C23, C23 [×10], C23 [×48], C42 [×3], C22⋊C4 [×4], C22⋊C4 [×26], C4⋊C4 [×9], C22×C4 [×12], C22×C4 [×10], C24 [×3], C24 [×12], C2.C42 [×8], C2×C42 [×2], C2×C22⋊C4 [×16], C2×C22⋊C4 [×4], C2×C4⋊C4 [×6], C422C2 [×4], C23×C4 [×2], C25, C4×C22⋊C4, C243C4 [×2], C23.7Q8, C23.8Q8 [×2], C24.C22 [×3], C23.Q8, C23.11D4 [×3], C22×C22⋊C4, C2×C422C2, C23.380C24
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], C22.19C24, C2×C422C2, C22.32C24, D45D4 [×2], C22.45C24 [×2], C23.380C24

Smallest permutation representation of C23.380C24
On 32 points
Generators in S32
(1 20)(2 17)(3 18)(4 19)(5 13)(6 14)(7 15)(8 16)(9 32)(10 29)(11 30)(12 31)(21 28)(22 25)(23 26)(24 27)
(1 18)(2 19)(3 20)(4 17)(5 15)(6 16)(7 13)(8 14)(9 30)(10 31)(11 32)(12 29)(21 26)(22 27)(23 28)(24 25)
(1 22)(2 23)(3 24)(4 21)(5 29)(6 30)(7 31)(8 32)(9 16)(10 13)(11 14)(12 15)(17 26)(18 27)(19 28)(20 25)
(1 22)(2 26)(3 24)(4 28)(5 31)(6 9)(7 29)(8 11)(10 15)(12 13)(14 32)(16 30)(17 23)(18 27)(19 21)(20 25)
(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 6 20 14)(2 31 17 12)(3 8 18 16)(4 29 19 10)(5 28 13 21)(7 26 15 23)(9 24 32 27)(11 22 30 25)
(1 2 3 4)(5 30 7 32)(6 31 8 29)(9 13 11 15)(10 14 12 16)(17 18 19 20)(21 22 23 24)(25 26 27 28)

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

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

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

38 conjugacy classes

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

38 irreducible representations

dim1111111111224
type++++++++++++
imageC1C2C2C2C2C2C2C2C2C2D4C4○D42+ 1+4
kernelC23.380C24C4×C22⋊C4C243C4C23.7Q8C23.8Q8C24.C22C23.Q8C23.11D4C22×C22⋊C4C2×C422C2C22⋊C4C23C22
# reps11212313114162

Matrix representation of C23.380C24 in GL6(𝔽5)

400000
040000
001000
000100
000040
000004
,
100000
010000
001000
000100
000040
000004
,
100000
010000
004000
000400
000010
000001
,
100000
440000
004000
000400
000010
000004
,
430000
110000
001000
000400
000001
000010
,
300000
030000
000100
001000
000002
000020
,
430000
110000
001000
000100
000001
000010

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

C23.380C24 in GAP, Magma, Sage, TeX

C_2^3._{380}C_2^4
% in TeX

G:=Group("C2^3.380C2^4");
// GroupNames label

G:=SmallGroup(128,1212);
// by ID

G=gap.SmallGroup(128,1212);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,253,344,758,723,100,675,192]);
// Polycyclic

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