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G = C23.597C24order 128 = 27

314th central stem extension by C23 of C24

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

Aliases: C25.60C22, C23.597C24, C24.404C23, C22.3712+ 1+4, (C2×D4).140D4, C243C426C2, (C23×C4)⋊13C22, (C2×C42)⋊33C22, C23.212(C2×D4), C2.102(D45D4), C23.Q861C2, C23.175(C4○D4), C23.23D489C2, C23.11D486C2, C23.10D485C2, C2.45(C233D4), (C22×C4).183C23, C22.406(C22×D4), C2.C4239C22, C24.3C2280C2, (C22×D4).234C22, C24.C22129C2, C2.67(C22.32C24), C2.15(C22.54C24), C2.79(C22.45C24), (C2×C4).99(C2×D4), (C2×C4⋊C4)⋊35C22, (C2×C22≀C2).15C2, (C2×C22⋊C4)⋊31C22, C22.459(C2×C4○D4), (C2×C22.D4)⋊36C2, SmallGroup(128,1429)

Series: Derived Chief Lower central Upper central Jennings

C1C23 — C23.597C24
C1C2C22C23C24C25C243C4 — C23.597C24
C1C23 — C23.597C24
C1C23 — C23.597C24
C1C23 — C23.597C24

Generators and relations for C23.597C24
 G = < a,b,c,d,e,f,g | a2=b2=c2=f2=g2=1, d2=e2=ba=ab, ac=ca, ede-1=ad=da, geg=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, gdg=abd, fg=gf >

Subgroups: 820 in 340 conjugacy classes, 96 normal (22 characteristic)
C1, C2 [×3], C2 [×4], C2 [×8], C4 [×12], C22 [×3], C22 [×4], C22 [×56], C2×C4 [×2], C2×C4 [×36], D4 [×12], C23, C23 [×8], C23 [×52], C42, C22⋊C4 [×26], C4⋊C4 [×7], C22×C4, C22×C4 [×10], C22×C4 [×4], C2×D4 [×4], C2×D4 [×12], C24 [×2], C24 [×2], C24 [×10], C2.C42 [×6], C2×C42, C2×C22⋊C4 [×2], C2×C22⋊C4 [×16], C2×C4⋊C4, C2×C4⋊C4 [×4], C22≀C2 [×4], C22.D4 [×4], C23×C4, C22×D4, C22×D4 [×2], C25, C243C4 [×2], C23.23D4 [×2], C24.C22 [×2], C24.3C22, C23.10D4 [×2], C23.Q8 [×2], C23.11D4 [×2], C2×C22≀C2, C2×C22.D4, C23.597C24
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], C2×D4 [×6], C4○D4 [×4], C24, C22×D4, C2×C4○D4 [×2], 2+ 1+4 [×4], C233D4, C22.32C24 [×2], D45D4 [×2], C22.45C24, C22.54C24, C23.597C24

Smallest permutation representation of C23.597C24
On 32 points
Generators in S32
(1 9)(2 10)(3 11)(4 12)(5 21)(6 22)(7 23)(8 24)(13 27)(14 28)(15 25)(16 26)(17 32)(18 29)(19 30)(20 31)
(1 11)(2 12)(3 9)(4 10)(5 23)(6 24)(7 21)(8 22)(13 25)(14 26)(15 27)(16 28)(17 30)(18 31)(19 32)(20 29)
(1 20)(2 17)(3 18)(4 19)(5 14)(6 15)(7 16)(8 13)(9 31)(10 32)(11 29)(12 30)(21 28)(22 25)(23 26)(24 27)
(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 27 3 25)(2 14 4 16)(5 19 7 17)(6 31 8 29)(9 13 11 15)(10 28 12 26)(18 22 20 24)(21 30 23 32)
(1 29)(2 17)(3 31)(4 19)(6 24)(8 22)(9 18)(10 32)(11 20)(12 30)(13 25)(15 27)
(1 20)(2 19)(3 18)(4 17)(5 26)(6 25)(7 28)(8 27)(9 31)(10 30)(11 29)(12 32)(13 24)(14 23)(15 22)(16 21)

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

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

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

32 conjugacy classes

class 1 2A···2G2H···2O4A···4J4K···4P
order12···22···24···44···4
size11···14···44···48···8

32 irreducible representations

dim1111111111224
type++++++++++++
imageC1C2C2C2C2C2C2C2C2C2D4C4○D42+ 1+4
kernelC23.597C24C243C4C23.23D4C24.C22C24.3C22C23.10D4C23.Q8C23.11D4C2×C22≀C2C2×C22.D4C2×D4C23C22
# reps1222122211484

Matrix representation of C23.597C24 in GL6(𝔽5)

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

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

C23.597C24 in GAP, Magma, Sage, TeX

C_2^3._{597}C_2^4
% in TeX

G:=Group("C2^3.597C2^4");
// GroupNames label

G:=SmallGroup(128,1429);
// by ID

G=gap.SmallGroup(128,1429);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,336,253,344,758,723,1571,346]);
// Polycyclic

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

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