Copied to
clipboard

G = C42.352C23order 128 = 27

213rd non-split extension by C42 of C23 acting via C23/C2=C22

p-group, metabelian, nilpotent (class 3), monomial

Aliases: C42.352C23, (C4×D8)⋊22C2, C4⋊Q88C22, D8⋊C49C2, C4⋊C814C22, (C4×C8)⋊15C22, C4⋊C4.345D4, D42Q87C2, (C4×SD16)⋊5C2, (C4×D4)⋊9C22, (C4×Q8)⋊9C22, C8⋊C45C22, D4⋊Q824C2, D4.9(C4○D4), C22⋊SD166C2, C22⋊D8.4C2, C2.17(D4○D8), C4⋊C4.71C23, (C2×C8).45C23, C4.Q867C22, C2.D825C22, D4.2D420C2, (C2×C4).316C24, C22⋊C4.146D4, C4.4D47C22, C23.255(C2×D4), SD16⋊C412C2, (C2×Q8).80C23, D4⋊C480C22, C2.26(D4○SD16), Q8⋊C425C22, (C2×D4).407C23, (C2×D8).125C22, C4⋊D4.26C22, C22⋊C8.29C22, C22.11C2411C2, C22⋊Q8.26C22, C23.20D419C2, C23.19D419C2, (C22×C4).289C23, C42.7C221C2, (C2×SD16).17C22, C22.576(C22×D4), C22.36C241C2, (C22×D4).361C22, C42⋊C2.127C22, C2.117(C22.19C24), C4.201(C2×C4○D4), (C2×C4).500(C2×D4), SmallGroup(128,1850)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C42.352C23
C1C2C4C2×C4C42C4×D4C22.11C24 — C42.352C23
C1C2C2×C4 — C42.352C23
C1C22C42⋊C2 — C42.352C23
C1C2C2C2×C4 — C42.352C23

Generators and relations for C42.352C23
 G = < a,b,c,d,e | a4=b4=c2=d2=e2=1, ab=ba, ac=ca, dad=ab2, ae=ea, cbc=ebe=b-1, bd=db, dcd=a2c, ece=bc, de=ed >

Subgroups: 436 in 203 conjugacy classes, 88 normal (84 characteristic)
C1, C2 [×3], C2 [×6], C4 [×2], C4 [×10], C22, C22 [×18], C8 [×4], C2×C4 [×6], C2×C4 [×12], D4 [×4], D4 [×9], Q8 [×3], C23, C23 [×9], C42 [×2], C42 [×2], C22⋊C4 [×2], C22⋊C4 [×10], C4⋊C4 [×6], C4⋊C4 [×3], C2×C8 [×4], D8 [×4], SD16 [×4], C22×C4, C22×C4 [×5], C2×D4 [×3], C2×D4 [×6], C2×Q8, C2×Q8, C24, C4×C8, C8⋊C4, C22⋊C8 [×2], D4⋊C4 [×6], Q8⋊C4 [×2], C4⋊C8 [×2], C4.Q8 [×2], C2.D8 [×2], C2×C22⋊C4 [×2], C42⋊C2 [×2], C4×D4 [×5], C4×D4 [×2], C4×Q8, C4⋊D4, C22⋊Q8, C22⋊Q8, C22.D4, C4.4D4, C4.4D4, C422C2, C4⋊Q8, C2×D8 [×2], C2×SD16 [×2], C22×D4, C42.7C22, C4×D8, C4×SD16, SD16⋊C4, D8⋊C4, C22⋊D8, C22⋊SD16, D4.2D4 [×2], D4⋊Q8, D42Q8, C23.19D4, C23.20D4, C22.11C24, C22.36C24, C42.352C23
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], C2×D4 [×6], C4○D4 [×4], C24, C22×D4, C2×C4○D4 [×2], C22.19C24, D4○D8, D4○SD16, C42.352C23

Smallest permutation representation of C42.352C23
On 32 points
Generators in S32
(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 23 27 19)(2 24 28 20)(3 21 25 17)(4 22 26 18)(5 32 15 9)(6 29 16 10)(7 30 13 11)(8 31 14 12)
(1 10)(2 11)(3 12)(4 9)(5 18)(6 19)(7 20)(8 17)(13 24)(14 21)(15 22)(16 23)(25 31)(26 32)(27 29)(28 30)
(2 28)(4 26)(5 13)(6 8)(7 15)(9 30)(10 12)(11 32)(14 16)(18 22)(20 24)(29 31)
(1 3)(2 4)(5 30)(6 31)(7 32)(8 29)(9 13)(10 14)(11 15)(12 16)(17 23)(18 24)(19 21)(20 22)(25 27)(26 28)

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

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

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

32 conjugacy classes

class 1 2A2B2C2D···2H2I4A···4F4G···4M4N4O4P8A8B8C8D8E8F
order12222···224···44···4444888888
size11114···482···24···4888444488

32 irreducible representations

dim11111111111111122244
type++++++++++++++++++
imageC1C2C2C2C2C2C2C2C2C2C2C2C2C2C2D4D4C4○D4D4○D8D4○SD16
kernelC42.352C23C42.7C22C4×D8C4×SD16SD16⋊C4D8⋊C4C22⋊D8C22⋊SD16D4.2D4D4⋊Q8D42Q8C23.19D4C23.20D4C22.11C24C22.36C24C22⋊C4C4⋊C4D4C2C2
# reps11111111211111122822

Matrix representation of C42.352C23 in GL6(𝔽17)

400000
040000
000010
00161115
001000
0000016
,
100000
010000
000100
0016000
00161115
00160116
,
6150000
9110000
0031400
00141400
0031406
0014030
,
1600000
1110000
001000
000100
0000160
00161016
,
100000
010000
001000
0001600
000010
00160116

G:=sub<GL(6,GF(17))| [4,0,0,0,0,0,0,4,0,0,0,0,0,0,0,16,1,0,0,0,0,1,0,0,0,0,1,1,0,0,0,0,0,15,0,16],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,16,16,16,0,0,1,0,1,0,0,0,0,0,1,1,0,0,0,0,15,16],[6,9,0,0,0,0,15,11,0,0,0,0,0,0,3,14,3,14,0,0,14,14,14,0,0,0,0,0,0,3,0,0,0,0,6,0],[16,11,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,16,0,0,0,1,0,1,0,0,0,0,16,0,0,0,0,0,0,16],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,16,0,0,0,16,0,0,0,0,0,0,1,1,0,0,0,0,0,16] >;

C42.352C23 in GAP, Magma, Sage, TeX

C_4^2._{352}C_2^3
% in TeX

G:=Group("C4^2.352C2^3");
// GroupNames label

G:=SmallGroup(128,1850);
// by ID

G=gap.SmallGroup(128,1850);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,758,1018,80,4037,1027,124]);
// Polycyclic

G:=Group<a,b,c,d,e|a^4=b^4=c^2=d^2=e^2=1,a*b=b*a,a*c=c*a,d*a*d=a*b^2,a*e=e*a,c*b*c=e*b*e=b^-1,b*d=d*b,d*c*d=a^2*c,e*c*e=b*c,d*e=e*d>;
// generators/relations

׿
×
𝔽