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G = C23.203C24  order 128 = 27

56th central extension by C23 of C24

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

Aliases: C25.21C22, C23.203C24, C24.194C23, C22.422+ (1+4), C22≀C2⋊7C4, C24⋊10(C2×C4), (C22×C4)⋊11D4, C24⋊3C4⋊6C2, (C23×C4)⋊5C22, C22.42(C4×D4), (C2×C42)⋊14C22, C23.606(C2×D4), C23.8Q8⋊5C2, C23.23D4⋊5C2, C2.3(C23⋊3D4), C23.84(C22×C4), C22.94(C23×C4), C22.91(C22×D4), C23.223(C4○D4), (C22×C4).468C23, C24.C22⋊4C2, C2.C42⋊10C22, C24.3C22⋊13C2, C2.4(C22.29C24), C2.3(C22.32C24), (C22×D4).104C22, C2.13(C22.11C24), (C2×C4×D4)⋊5C2, C2.20(C2×C4×D4), (C2×D4)⋊13(C2×C4), (C2×C4⋊C4)⋊7C22, C22⋊C4⋊8(C2×C4), (C4×C22⋊C4)⋊31C2, (C2×C4).675(C2×D4), (C2×C22≀C2).4C2, (C22×C22⋊C4)⋊7C2, (C2×C4).24(C22×C4), C22.88(C2×C4○D4), (C2×C22⋊C4)⋊72C22, SmallGroup(128,1053)

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

Derived series
C1 — C22 — C23.203C24
Chief series
C1 — C2 — C22 — C23 — C24 — C25 — C22×C22⋊C4 — C23.203C24
Lower central
C1 — C22 — C23.203C24
Upper central
C1 — C23 — C23.203C24
Jennings
C1 — C23 — C23.203C24

Subgroups: 876 in 404 conjugacy classes, 148 normal (34 characteristic)
C1, C2 [×3], C2 [×4], C2 [×10], C4 [×16], C22 [×3], C22 [×8], C22 [×58], C2×C4 [×10], C2×C4 [×40], D4 [×16], C23, C23 [×12], C23 [×54], C42 [×4], C22⋊C4 [×12], C22⋊C4 [×20], C4⋊C4 [×4], C22×C4 [×5], C22×C4 [×10], C22×C4 [×10], C2×D4 [×12], C2×D4 [×8], C24 [×2], C24 [×6], C24 [×8], C2.C42 [×6], C2×C42, C2×C42 [×2], C2×C22⋊C4 [×4], C2×C22⋊C4 [×12], C2×C22⋊C4 [×4], C2×C4⋊C4, C2×C4⋊C4 [×2], C4×D4 [×4], C22≀C2 [×8], C23×C4 [×3], C22×D4, C22×D4 [×2], C25, C4×C22⋊C4, C24⋊3C4, C23.8Q8, C23.23D4, C23.23D4 [×2], C24.C22 [×4], C24.3C22 [×2], C22×C22⋊C4, C2×C4×D4, C2×C22≀C2, C23.203C24

Quotients:
C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], D4 [×4], C23 [×15], C22×C4 [×14], C2×D4 [×6], C4○D4 [×2], C24, C4×D4 [×4], C23×C4, C22×D4, C2×C4○D4, 2+ (1+4) [×4], C2×C4×D4, C22.11C24 [×2], C23⋊3D4, C22.29C24, C22.32C24 [×2], C23.203C24

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

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

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

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

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

Matrix representation ►G ⊆ GL8(𝔽5)

10000000
01000000
00100000
00010000
00004000
00000400
00000040
00000004
,
40000000
04000000
00400000
00040000
00001000
00000100
00000010
00000001
,
40000000
04000000
00100000
00010000
00001000
00000100
00000010
00000001
,
01000000
40000000
00040000
00400000
00000010
00000001
00001000
00000100
,
30000000
03000000
00400000
00040000
00000100
00001000
00000004
00000040
,
40000000
01000000
00400000
00010000
00001000
00000400
00000010
00000004
,
10000000
01000000
00100000
00010000
00001000
00000100
00000040
00000004

G:=sub<GL(8,GF(5))| [1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4],[4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0],[3,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0],[4,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4] >;
 

44 conjugacy classes

class 1 2A···2G2H2I2J2K2L···2Q4A···4H4I···4Z
order12···222222···24···44···4
size11···122224···42···24···4

44 irreducible representations

dim11111111111224
type++++++++++++
imageC1C2C2C2C2C2C2C2C2C2C4D4C4○D42+ (1+4)
kernelC23.203C24C4×C22⋊C4C24⋊3C4C23.8Q8C23.23D4C24.C22C24.3C22C22×C22⋊C4C2×C4×D4C2×C22≀C2C22≀C2C22×C4C23C22
# reps111134211116444

In GAP, Magma, Sage, TeX 

C_2^3._{203}C_2^4
 
% in TeX
 
G:=Group("C2^3.203C2^4");
 
// GroupNames label
 
G:=SmallGroup(128,1053);
 
// by ID
 
G=gap.SmallGroup(128,1053);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,2,448,253,456,758,219,675]);
 
// 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=c,a*b=b*a,a*c=c*a,e*d*e^-1=g*d*g=a*d=d*a,f*e*f=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,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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