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G = C23.578C24  order 128 = 27

295th central stem extension by C23 of C24

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

Aliases: C25.57C22, C23.578C24, C24.386C23, C22.3522+ (1+4), (C2×D4)⋊15D4, C2.42(D42), C22⋊C4⋊14D4, C24⋊3C4⋊23C2, C23⋊2D4⋊38C2, (C2×C42)⋊29C22, (C23×C4)⋊25C22, C23.204(C2×D4), C2.87(D4⋊5D4), (C22×D4)⋊13C22, C23.Q8⋊51C2, C23.167(C4○D4), C23.10D4⋊75C2, C23.23D4⋊81C2, C23.11D4⋊75C2, C2.39(C23⋊3D4), (C22×C4).176C23, C22.387(C22×D4), C2.C42⋊35C22, C24.3C22⋊73C2, C24.C22⋊119C2, C2.55(C22.29C24), C2.59(C22.32C24), C2.10(C22.54C24), (C2×C4).86(C2×D4), (C2×C4⋊D4)⋊33C2, (C2×C4⋊C4)⋊31C22, (C2×C22≀C2)⋊14C2, (C2×C22⋊C4)⋊28C22, C22.442(C2×C4○D4), (C2×C22.D4)⋊31C2, SmallGroup(128,1410)

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

Derived series
C1 — C23 — C23.578C24
Chief series
C1 — C2 — C22 — C23 — C24 — C25 — C2×C22≀C2 — C23.578C24
Lower central
C1 — C23 — C23.578C24
Upper central
C1 — C23 — C23.578C24
Jennings
C1 — C23 — C23.578C24

Subgroups: 980 in 401 conjugacy classes, 104 normal (82 characteristic)
C1, C2 [×7], C2 [×9], C4 [×13], C22 [×7], C22 [×63], C2×C4 [×6], C2×C4 [×31], D4 [×28], C23, C23 [×8], C23 [×59], C42, C22⋊C4 [×4], C22⋊C4 [×25], C4⋊C4 [×7], C22×C4 [×10], C22×C4 [×5], C2×D4 [×4], C2×D4 [×30], C24 [×5], C24 [×10], C2.C42 [×4], C2×C42, C2×C22⋊C4 [×17], C2×C4⋊C4 [×4], C22≀C2 [×8], C4⋊D4 [×4], C22.D4 [×4], C23×C4, C22×D4 [×7], C25, C24⋊3C4, C23.23D4, C24.C22, C24.3C22 [×2], C23⋊2D4 [×2], C23.10D4 [×2], C23.Q8, C23.11D4, C2×C22≀C2 [×2], C2×C4⋊D4, C2×C22.D4, C23.578C24

Quotients:
C1, C2 [×15], C22 [×35], D4 [×8], C23 [×15], C2×D4 [×12], C4○D4 [×2], C24, C22×D4 [×2], C2×C4○D4, 2+ (1+4) [×4], C23⋊3D4, C22.29C24, C22.32C24, D42, D4⋊5D4 [×2], C22.54C24, C23.578C24

Generators and relations
 G = < a,b,c,d,e,f,g | a2=b2=c2=e2=f2=g2=1, d2=b, ab=ba, ac=ca, ede=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 >

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

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

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

(1 25)(2 28)(3 27)(4 26)(5 7)(9 11)(13 23)(14 22)(15 21)(16 24)(18 20)(29 31)

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

Matrix representation ►G ⊆ GL6(𝔽5)

400000
040000
001000
000100
000010
000001
,
100000
010000
001000
000100
000040
000004
,
100000
010000
004000
000400
000010
000001
,
020000
300000
001000
000100
000033
000002
,
010000
100000
000100
001000
000040
000004
,
100000
010000
004000
000100
000040
000021
,
100000
040000
004000
000400
000010
000034

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

32 conjugacy classes

class 1 2A···2G2H···2O2P4A···4J4K···4O
order12···22···224···44···4
size11···14···484···48···8

32 irreducible representations

dim1111111111112224
type+++++++++++++++
imageC1C2C2C2C2C2C2C2C2C2C2C2D4D4C4○D42+ (1+4)
kernelC23.578C24C24⋊3C4C23.23D4C24.C22C24.3C22C23⋊2D4C23.10D4C23.Q8C23.11D4C2×C22≀C2C2×C4⋊D4C2×C22.D4C22⋊C4C2×D4C23C22
# reps1111222112114444

In GAP, Magma, Sage, TeX 

C_2^3._{578}C_2^4
 
% in TeX
 
G:=Group("C2^3.578C2^4");
 
// GroupNames label
 
G:=SmallGroup(128,1410);
 
// by ID
 
G=gap.SmallGroup(128,1410);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,2,224,253,758,723,1571,346]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=e^2=f^2=g^2=1,d^2=b,a*b=b*a,a*c=c*a,e*d*e=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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