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G = C23.318C24order 128 = 27

35th central stem extension by C23 of C24

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

Aliases: C25.38C22, C23.318C24, C24.253C23, C22.1322+ 1+4, C2.10D42, C22⋊C434D4, (C2×D4).280D4, C243C415C2, (C2×C42)⋊20C22, (C23×C4)⋊10C22, C23.155(C2×D4), C2.21(D45D4), C23.4Q83C2, C23.11D45C2, C23.8Q830C2, C23.7Q838C2, C23.227(C4○D4), C23.10D412C2, C23.23D429C2, (C22×C4).792C23, C22.198(C22×D4), C24.C2233C2, C2.C4221C22, (C22×D4).121C22, C222(C22.D4), C2.10(C22.32C24), C2.23(C22.19C24), C2.12(C22.45C24), (C2×C4×D4)⋊23C2, (C2×C4⋊C4)⋊14C22, (C2×C4).309(C2×D4), (C2×C22≀C2).9C2, (C22×C22⋊C4)⋊20C2, (C2×C22⋊C4)⋊15C22, C22.197(C2×C4○D4), (C2×C22.D4)⋊6C2, C2.15(C2×C22.D4), SmallGroup(128,1150)

Series: Derived Chief Lower central Upper central Jennings

C1C23 — C23.318C24
C1C2C22C23C24C25C22×C22⋊C4 — C23.318C24
C1C23 — C23.318C24
C1C23 — C23.318C24
C1C23 — C23.318C24

Generators and relations for C23.318C24
 G = < a,b,c,d,e,f,g | a2=b2=c2=d2=f2=g2=1, e2=b, 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: 884 in 400 conjugacy classes, 112 normal (82 characteristic)
C1, C2 [×7], C2 [×10], C4 [×14], C22 [×7], C22 [×4], C22 [×58], C2×C4 [×6], C2×C4 [×42], D4 [×16], C23, C23 [×12], C23 [×54], C42 [×2], C22⋊C4 [×4], C22⋊C4 [×27], C4⋊C4 [×8], C22×C4 [×11], C22×C4 [×15], C2×D4 [×4], C2×D4 [×14], C24 [×4], C24 [×12], C2.C42 [×6], C2×C42, C2×C22⋊C4 [×16], C2×C22⋊C4 [×4], C2×C4⋊C4 [×5], C4×D4 [×4], C22≀C2 [×4], C22.D4 [×4], C23×C4 [×3], C22×D4 [×3], C25, C243C4, C23.7Q8, C23.8Q8, C23.23D4 [×2], C24.C22 [×2], C23.10D4 [×2], C23.11D4, C23.4Q8, C22×C22⋊C4, C2×C4×D4, C2×C22≀C2, C2×C22.D4, C23.318C24
Quotients: C1, C2 [×15], C22 [×35], D4 [×8], C23 [×15], C2×D4 [×12], C4○D4 [×6], C24, C22.D4 [×4], C22×D4 [×2], C2×C4○D4 [×3], 2+ 1+4 [×2], C2×C22.D4, C22.19C24, C22.32C24, D42, D45D4 [×2], C22.45C24, C23.318C24

Smallest permutation representation of C23.318C24
On 32 points
Generators in S32
(1 7)(2 8)(3 5)(4 6)(9 30)(10 31)(11 32)(12 29)(13 21)(14 22)(15 23)(16 24)(17 27)(18 28)(19 25)(20 26)
(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 15)(2 16)(3 13)(4 14)(5 21)(6 22)(7 23)(8 24)(9 27)(10 28)(11 25)(12 26)(17 30)(18 31)(19 32)(20 29)
(1 28)(2 19)(3 26)(4 17)(5 20)(6 27)(7 18)(8 25)(9 22)(10 15)(11 24)(12 13)(14 30)(16 32)(21 29)(23 31)
(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 3)(2 14)(4 16)(5 7)(6 24)(8 22)(9 27)(11 25)(13 15)(17 30)(19 32)(21 23)
(1 13)(2 14)(3 15)(4 16)(5 23)(6 24)(7 21)(8 22)(9 19)(10 20)(11 17)(12 18)(25 30)(26 31)(27 32)(28 29)

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

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

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

38 conjugacy classes

class 1 2A···2G2H2I2J2K2L···2Q4A4B4C4D4E···4P4Q4R4S4T
order12···222222···244444···44444
size11···122224···422224···48888

38 irreducible representations

dim11111111111112224
type++++++++++++++++
imageC1C2C2C2C2C2C2C2C2C2C2C2C2D4D4C4○D42+ 1+4
kernelC23.318C24C243C4C23.7Q8C23.8Q8C23.23D4C24.C22C23.10D4C23.11D4C23.4Q8C22×C22⋊C4C2×C4×D4C2×C22≀C2C2×C22.D4C22⋊C4C2×D4C23C22
# reps111122211111144122

Matrix representation of C23.318C24 in GL6(𝔽5)

400000
040000
004000
000400
000010
000001
,
400000
040000
001000
000100
000010
000001
,
100000
010000
001000
000100
000040
000004
,
440000
010000
004400
000100
000010
000001
,
200000
130000
001000
003400
000001
000010
,
400000
210000
001000
000100
000010
000004
,
400000
210000
001000
003400
000040
000004

G:=sub<GL(6,GF(5))| [4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[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],[4,0,0,0,0,0,4,1,0,0,0,0,0,0,4,0,0,0,0,0,4,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[2,1,0,0,0,0,0,3,0,0,0,0,0,0,1,3,0,0,0,0,0,4,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[4,2,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,4],[4,2,0,0,0,0,0,1,0,0,0,0,0,0,1,3,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4] >;

C23.318C24 in GAP, Magma, Sage, TeX

C_2^3._{318}C_2^4
% in TeX

G:=Group("C2^3.318C2^4");
// GroupNames label

G:=SmallGroup(128,1150);
// by ID

G=gap.SmallGroup(128,1150);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,253,232,758,723,675]);
// Polycyclic

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