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G = C23.308C24order 128 = 27

25th central stem extension by C23 of C24

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

Aliases: C25.34C22, C24.245C23, C23.308C24, C22.1252+ 1+4, C2.7D42, C42C22≀C2, (C2×D4)⋊29D4, C232(C2×D4), (D4×C23)⋊3C2, (C22×C4)⋊21D4, C232D46C2, C243C412C2, C224(C4⋊D4), (C23×C4)⋊20C22, (C2×C42)⋊18C22, (C22×D4)⋊3C22, C2.15(D45D4), C23.10D46C2, C23.7Q831C2, C23.297(C4○D4), (C22×C4).789C23, C22.188(C22×D4), C2.C4217C22, C24.3C2226C2, C2.9(C22.29C24), (C2×C4×D4)⋊20C2, (C2×C4)⋊2(C2×D4), (C2×C4⋊D4)⋊3C2, (C2×C22≀C2)⋊4C2, C2.13(C2×C4⋊D4), (C2×C4⋊C4)⋊109C22, C2.15(C2×C22≀C2), (C2×C22⋊C4)⋊11C22, C22.187(C2×C4○D4), SmallGroup(128,1140)

Series: Derived Chief Lower central Upper central Jennings

C1C23 — C23.308C24
C1C2C22C23C22×C4C23×C4C2×C4×D4 — C23.308C24
C1C23 — C23.308C24
C1C23 — C23.308C24
C1C23 — C23.308C24

Generators and relations for C23.308C24
 G = < a,b,c,d,e,f,g | a2=b2=c2=d2=f2=1, e2=g2=a, ab=ba, ac=ca, ede-1=gdg-1=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 >

Subgroups: 1396 in 616 conjugacy classes, 132 normal (26 characteristic)
C1, C2 [×3], C2 [×4], C2 [×14], C4 [×4], C4 [×10], C22 [×3], C22 [×8], C22 [×94], C2×C4 [×10], C2×C4 [×34], D4 [×56], C23, C23 [×16], C23 [×98], C42 [×2], C22⋊C4 [×28], C4⋊C4 [×6], C22×C4 [×5], C22×C4 [×8], C22×C4 [×10], C2×D4 [×12], C2×D4 [×76], C24 [×2], C24 [×4], C24 [×24], C2.C42 [×2], C2×C42, C2×C22⋊C4 [×2], C2×C22⋊C4 [×14], C2×C4⋊C4, C2×C4⋊C4 [×2], C4×D4 [×4], C22≀C2 [×8], C4⋊D4 [×8], C23×C4 [×2], C22×D4, C22×D4 [×8], C22×D4 [×12], C25 [×2], C243C4 [×2], C23.7Q8, C24.3C22 [×2], C232D4 [×2], C23.10D4 [×2], C2×C4×D4, C2×C22≀C2 [×2], C2×C4⋊D4 [×2], D4×C23, C23.308C24
Quotients: C1, C2 [×15], C22 [×35], D4 [×16], C23 [×15], C2×D4 [×24], C4○D4 [×2], C24, C22≀C2 [×4], C4⋊D4 [×4], C22×D4 [×4], C2×C4○D4, 2+ 1+4 [×2], C2×C22≀C2, C2×C4⋊D4, C22.29C24, D42 [×2], D45D4 [×2], C23.308C24

Smallest permutation representation of C23.308C24
On 32 points
Generators in S32
(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 18)(2 19)(3 20)(4 17)(5 10)(6 11)(7 12)(8 9)(13 29)(14 30)(15 31)(16 32)(21 27)(22 28)(23 25)(24 26)
(1 24)(2 21)(3 22)(4 23)(5 16)(6 13)(7 14)(8 15)(9 31)(10 32)(11 29)(12 30)(17 25)(18 26)(19 27)(20 28)
(1 25)(2 28)(3 27)(4 26)(5 8)(6 7)(9 10)(11 12)(13 14)(15 16)(17 24)(18 23)(19 22)(20 21)(29 30)(31 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 14)(2 8)(3 16)(4 6)(5 22)(7 24)(9 19)(10 28)(11 17)(12 26)(13 23)(15 21)(18 30)(20 32)(25 29)(27 31)
(1 19 3 17)(2 20 4 18)(5 29 7 31)(6 30 8 32)(9 16 11 14)(10 13 12 15)(21 28 23 26)(22 25 24 27)

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

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

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

38 conjugacy classes

class 1 2A···2G2H2I2J2K2L···2U4A4B4C4D4E···4L4M4N4O4P
order12···222222···244444···44444
size11···122224···422224···48888

38 irreducible representations

dim11111111112224
type+++++++++++++
imageC1C2C2C2C2C2C2C2C2C2D4D4C4○D42+ 1+4
kernelC23.308C24C243C4C23.7Q8C24.3C22C232D4C23.10D4C2×C4×D4C2×C22≀C2C2×C4⋊D4D4×C23C22×C4C2×D4C23C22
# reps121222122141242

Matrix representation of C23.308C24 in GL6(ℤ)

100000
010000
001000
000100
0000-10
00000-1
,
-100000
0-10000
00-1000
000-100
000010
000001
,
100000
010000
00-1000
000-100
000010
000001
,
100000
0-10000
00-1000
000100
000001
000010
,
-100000
0-10000
00-1000
000100
000001
0000-10
,
010000
100000
000100
001000
000010
000001
,
100000
010000
001000
000100
000001
0000-10

G:=sub<GL(6,Integers())| [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,-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,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,-1,0,0,0,0,1,0],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,-1,0,0,0,0,1,0] >;

C23.308C24 in GAP, Magma, Sage, TeX

C_2^3._{308}C_2^4
% in TeX

G:=Group("C2^3.308C2^4");
// GroupNames label

G:=SmallGroup(128,1140);
// by ID

G=gap.SmallGroup(128,1140);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,253,120,758,723,675,80]);
// Polycyclic

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