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G = C23.311C24order 128 = 27

28th central stem extension by C23 of C24

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

Aliases: C25.36C22, C24.248C23, C23.311C24, C22.1272+ 1+4, (C22×C4)⋊22D4, C243C414C2, (C2×C42)⋊19C22, C22⋊C4.123D4, C23.421(C2×D4), C2.17(D45D4), C23.Q81C2, C23.10D48C2, C23.8Q827C2, C23.138(C4○D4), C23.23D426C2, C22.72(C4⋊D4), (C22×C4).503C23, (C23×C4).329C22, C22.191(C22×D4), C24.3C2227C2, C24.C2229C2, C2.C4219C22, (C22×D4).117C22, C2.10(C22.29C24), C2.19(C22.19C24), C2.10(C22.45C24), (C2×C4⋊C4)⋊12C22, (C2×C4).305(C2×D4), C2.15(C2×C4⋊D4), (C2×C22≀C2).7C2, (C2×C42⋊C2)⋊18C2, (C2×C22⋊C4)⋊13C22, (C22×C22⋊C4)⋊18C2, C22.190(C2×C4○D4), (C2×C22.D4)⋊4C2, SmallGroup(128,1143)

Series: Derived Chief Lower central Upper central Jennings

C1C23 — C23.311C24
C1C2C22C23C24C25C22×C22⋊C4 — C23.311C24
C1C23 — C23.311C24
C1C23 — C23.311C24
C1C23 — C23.311C24

Generators and relations for C23.311C24
 G = < a,b,c,d,e,f,g | a2=b2=c2=f2=g2=1, d2=b, e2=a, ab=ba, ac=ca, ede-1=gdg=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: 868 in 393 conjugacy classes, 112 normal (34 characteristic)
C1, C2 [×3], C2 [×4], C2 [×9], C4 [×15], C22 [×3], C22 [×8], C22 [×55], C2×C4 [×8], C2×C4 [×41], D4 [×12], C23, C23 [×10], C23 [×55], C42 [×4], C22⋊C4 [×4], C22⋊C4 [×28], C4⋊C4 [×10], C22×C4 [×5], C22×C4 [×10], C22×C4 [×8], C2×D4 [×16], C24 [×2], C24 [×2], C24 [×12], C2.C42 [×4], C2×C42 [×2], C2×C22⋊C4 [×5], C2×C22⋊C4 [×12], C2×C22⋊C4 [×4], C2×C4⋊C4 [×2], C2×C4⋊C4 [×4], C42⋊C2 [×4], C22≀C2 [×4], C22.D4 [×4], C23×C4 [×2], C22×D4, C22×D4 [×2], C25, C243C4, C23.8Q8, C23.23D4, C24.C22 [×2], C24.3C22 [×2], C23.10D4 [×2], C23.Q8 [×2], C22×C22⋊C4, C2×C42⋊C2, C2×C22≀C2, C2×C22.D4, C23.311C24
Quotients: C1, C2 [×15], C22 [×35], D4 [×8], C23 [×15], C2×D4 [×12], C4○D4 [×6], C24, C4⋊D4 [×4], C22×D4 [×2], C2×C4○D4 [×3], 2+ 1+4 [×2], C2×C4⋊D4, C22.19C24, C22.29C24, D45D4 [×2], C22.45C24 [×2], C23.311C24

Smallest permutation representation of C23.311C24
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 7 23 11)(2 12 24 8)(3 5 21 9)(4 10 22 6)(13 29 25 20)(14 17 26 30)(15 31 27 18)(16 19 28 32)
(1 3)(5 18)(6 17)(7 20)(8 19)(9 31)(10 30)(11 29)(12 32)(13 15)(21 23)(25 27)
(1 25)(2 14)(3 27)(4 16)(5 18)(6 32)(7 20)(8 30)(9 31)(10 19)(11 29)(12 17)(13 23)(15 21)(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), (1,7,23,11)(2,12,24,8)(3,5,21,9)(4,10,22,6)(13,29,25,20)(14,17,26,30)(15,31,27,18)(16,19,28,32), (1,3)(5,18)(6,17)(7,20)(8,19)(9,31)(10,30)(11,29)(12,32)(13,15)(21,23)(25,27), (1,25)(2,14)(3,27)(4,16)(5,18)(6,32)(7,20)(8,30)(9,31)(10,19)(11,29)(12,17)(13,23)(15,21)(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), (1,7,23,11)(2,12,24,8)(3,5,21,9)(4,10,22,6)(13,29,25,20)(14,17,26,30)(15,31,27,18)(16,19,28,32), (1,3)(5,18)(6,17)(7,20)(8,19)(9,31)(10,30)(11,29)(12,32)(13,15)(21,23)(25,27), (1,25)(2,14)(3,27)(4,16)(5,18)(6,32)(7,20)(8,30)(9,31)(10,19)(11,29)(12,17)(13,23)(15,21)(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)], [(1,7,23,11),(2,12,24,8),(3,5,21,9),(4,10,22,6),(13,29,25,20),(14,17,26,30),(15,31,27,18),(16,19,28,32)], [(1,3),(5,18),(6,17),(7,20),(8,19),(9,31),(10,30),(11,29),(12,32),(13,15),(21,23),(25,27)], [(1,25),(2,14),(3,27),(4,16),(5,18),(6,32),(7,20),(8,30),(9,31),(10,19),(11,29),(12,17),(13,23),(15,21),(22,28),(24,26)])

38 conjugacy classes

class 1 2A···2G2H2I2J2K2L2M2N2O2P4A4B4C4D4E···4R4S4T4U
order12···222222222244444···4444
size11···122224444822224···4888

38 irreducible representations

dim1111111111112224
type+++++++++++++++
imageC1C2C2C2C2C2C2C2C2C2C2C2D4D4C4○D42+ 1+4
kernelC23.311C24C243C4C23.8Q8C23.23D4C24.C22C24.3C22C23.10D4C23.Q8C22×C22⋊C4C2×C42⋊C2C2×C22≀C2C2×C22.D4C22⋊C4C22×C4C23C22
# reps11112222111144122

Matrix representation of C23.311C24 in GL6(𝔽5)

400000
040000
001000
000100
000010
000001
,
100000
010000
004000
000400
000010
000001
,
100000
010000
001000
000100
000040
000004
,
010000
100000
000100
004000
000010
000001
,
200000
030000
004000
000400
000043
000001
,
100000
010000
004000
000100
000010
000044
,
100000
040000
004000
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],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[2,0,0,0,0,0,0,3,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,3,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1,4,0,0,0,0,0,4],[1,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4] >;

C23.311C24 in GAP, Magma, Sage, TeX

C_2^3._{311}C_2^4
% in TeX

G:=Group("C2^3.311C2^4");
// GroupNames label

G:=SmallGroup(128,1143);
// by ID

G=gap.SmallGroup(128,1143);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,224,253,120,758,723,675]);
// Polycyclic

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