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G = C42.259C23  order 128 = 27

120th non-split extension by C42 of C23 acting via C23/C2=C22

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

Aliases: C42.259C23, C4⋊C8⋊49C22, (C4×C8)⋊53C22, C24.77(C2×C4), C8⋊C4⋊55C22, (C4×M4(2))⋊30C2, (C2×C8).398C23, (C2×C4).640C24, C42.203(C2×C4), C4⋊M4(2)⋊32C2, C42⋊C2.29C4, C23.99(C22×C4), C4.66(C42⋊C2), C2.10(Q8○M4(2)), C22⋊C8.137C22, C24.4C4.23C2, (C2×C42).754C22, C22.168(C23×C4), (C23×C4).522C22, (C22×C4).911C23, C42.7C22⋊19C2, C42⋊C2.289C22, C22.38(C42⋊C2), (C2×M4(2)).342C22, (C2×C4⋊C4).68C4, C4⋊C4.216(C2×C4), C4.291(C2×C4○D4), C22⋊C4.67(C2×C4), (C2×C22⋊C4).45C4, (C2×C4).678(C4○D4), (C2×C4).256(C22×C4), (C22×C4).334(C2×C4), C2.40(C2×C42⋊C2), (C2×C42⋊C2).57C2, SmallGroup(128,1653)

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

Derived series
C1 — C22 — C42.259C23
Chief series
C1 — C2 — C4 — C2×C4 — C22×C4 — C2×C42 — C2×C42⋊C2 — C42.259C23
Lower central
C1 — C22 — C42.259C23
Upper central
C1 — C2×C4 — C42.259C23
Jennings
C1 — C2 — C2 — C2×C4 — C42.259C23

Subgroups: 284 in 196 conjugacy classes, 132 normal (14 characteristic)
C1, C2, C2 [×2], C2 [×4], C4 [×4], C4 [×10], C22, C22 [×2], C22 [×12], C8 [×8], C2×C4 [×2], C2×C4 [×14], C2×C4 [×14], C23, C23 [×2], C23 [×4], C42 [×8], C22⋊C4 [×8], C4⋊C4 [×8], C2×C8 [×8], M4(2) [×8], C22×C4 [×2], C22×C4 [×8], C22×C4 [×4], C24, C4×C8 [×4], C8⋊C4 [×4], C22⋊C8 [×8], C4⋊C8 [×8], C2×C42 [×2], C2×C22⋊C4 [×2], C2×C4⋊C4 [×2], C42⋊C2 [×8], C2×M4(2) [×4], C23×C4, C4×M4(2) [×2], C24.4C4 [×2], C4⋊M4(2) [×2], C42.7C22 [×8], C2×C42⋊C2, C42.259C23

Quotients:
C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], C23 [×15], C22×C4 [×14], C4○D4 [×4], C24, C42⋊C2 [×4], C23×C4, C2×C4○D4 [×2], C2×C42⋊C2, Q8○M4(2) [×2], C42.259C23

Generators and relations
 G = < a,b,c,d,e | a4=b4=d2=e2=1, c2=b, ab=ba, cac-1=a-1b2, dad=ab2, ae=ea, bc=cb, bd=db, be=eb, dcd=a2b2c, ece=b2c, de=ed >

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

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

(2 32)(4 26)(6 28)(8 30)(9 22)(10 14)(11 24)(12 16)(13 18)(15 20)(17 21)(19 23)

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

Matrix representation ►G ⊆ GL6(𝔽17)

1300000
1340000
0001600
001000
0000016
000010
,
100000
010000
004000
000400
000040
000004
,
490000
4130000
000010
000001
004000
000400
,
100000
1160000
001000
0001600
000010
0000016
,
100000
010000
001000
000100
0000160
0000016

G:=sub<GL(6,GF(17))| [13,13,0,0,0,0,0,4,0,0,0,0,0,0,0,1,0,0,0,0,16,0,0,0,0,0,0,0,0,1,0,0,0,0,16,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[4,4,0,0,0,0,9,13,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,1,0,0,0,0,0,0,1,0,0],[1,1,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,16],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16] >;
 

44 conjugacy classes

class 1 2A2B2C2D2E2F2G4A4B4C4D4E···4N4O···4T8A···8P
order1222222244444···44···48···8
size1111224411112···24···44···4

44 irreducible representations

dim11111111124
type++++++
imageC1C2C2C2C2C2C4C4C4C4○D4Q8○M4(2)
kernelC42.259C23C4×M4(2)C24.4C4C4⋊M4(2)C42.7C22C2×C42⋊C2C2×C22⋊C4C2×C4⋊C4C42⋊C2C2×C4C2
# reps12228144884

In GAP, Magma, Sage, TeX 

C_4^2._{259}C_2^3
 
% in TeX
 
G:=Group("C4^2.259C2^3");
 
// GroupNames label
 
G:=SmallGroup(128,1653);
 
// by ID
 
G=gap.SmallGroup(128,1653);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,-2,224,253,723,100,2019,124]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^4=b^4=d^2=e^2=1,c^2=b,a*b=b*a,c*a*c^-1=a^-1*b^2,d*a*d=a*b^2,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,d*c*d=a^2*b^2*c,e*c*e=b^2*c,d*e=e*d>;
 
// generators/relations
 

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