Copied to
clipboard

G = C42.57D4order 128 = 27

39th non-split extension by C42 of D4 acting via D4/C2=C22

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

Aliases: C42.57D4, (C4×D4)⋊4C4, (C4×Q8)⋊4C4, C4.34C4≀C2, C4.4D48C4, C42.C24C4, C42.74(C2×C4), (C22×C4).662D4, C23.499(C2×D4), C42.6C430C2, C22.SD16.5C2, C23.31D419C2, C4⋊D4.135C22, C22⋊C8.131C22, C22.15(C8⋊C22), (C22×C4).631C23, (C2×C42).177C22, C22⋊Q8.140C22, C2.9(C23.36D4), C22.11(C8.C22), C23.36C23.6C2, C2.C42.507C22, C2.17(C23.C23), (C4×C4⋊C4)⋊2C2, C4⋊C4.9(C2×C4), C2.26(C2×C4≀C2), (C2×D4).8(C2×C4), (C2×Q8).8(C2×C4), (C2×C4).1155(C2×D4), (C2×C4).121(C22×C4), (C2×C4).173(C22⋊C4), C22.185(C2×C22⋊C4), SmallGroup(128,241)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C42.57D4
C1C2C22C23C22×C4C2×C42C23.36C23 — C42.57D4
C1C22C2×C4 — C42.57D4
C1C22C2×C42 — C42.57D4
C1C2C22C22×C4 — C42.57D4

Generators and relations for C42.57D4
 G = < a,b,c,d | a4=b4=c4=1, d2=b-1, ab=ba, ac=ca, dad-1=a-1b2, cbc-1=a2b-1, bd=db, dcd-1=b-1c-1 >

Subgroups: 252 in 121 conjugacy classes, 46 normal (36 characteristic)
C1, C2 [×3], C2 [×3], C4 [×2], C4 [×12], C22 [×3], C22 [×5], C8 [×2], C2×C4 [×6], C2×C4 [×18], D4 [×3], Q8, C23, C23, C42 [×4], C42 [×3], C22⋊C4 [×5], C4⋊C4 [×2], C4⋊C4 [×8], C2×C8 [×2], C22×C4 [×3], C22×C4 [×3], C2×D4, C2×D4, C2×Q8, C2.C42 [×2], C8⋊C4, C22⋊C8 [×2], C4⋊C8, C2×C42, C2×C42, C2×C4⋊C4, C42⋊C2, C4×D4, C4×D4, C4×Q8, C4⋊D4, C22⋊Q8, C22.D4, C4.4D4, C42.C2, C422C2, C22.SD16 [×2], C23.31D4 [×2], C4×C4⋊C4, C42.6C4, C23.36C23, C42.57D4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×4], C23, C22⋊C4 [×4], C22×C4, C2×D4 [×2], C4≀C2 [×2], C2×C22⋊C4, C8⋊C22, C8.C22, C23.C23, C23.36D4, C2×C4≀C2, C42.57D4

Smallest permutation representation of C42.57D4
On 32 points
Generators in S32
(1 10 31 21)(2 18 32 15)(3 12 25 23)(4 20 26 9)(5 14 27 17)(6 22 28 11)(7 16 29 19)(8 24 30 13)
(1 7 5 3)(2 8 6 4)(9 15 13 11)(10 16 14 12)(17 23 21 19)(18 24 22 20)(25 31 29 27)(26 32 30 28)
(1 17 31 14)(2 24)(3 12 25 23)(4 11)(5 21 27 10)(6 20)(7 16 29 19)(8 15)(9 28)(13 32)(18 30)(22 26)
(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)

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

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

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

32 conjugacy classes

class 1 2A2B2C2D2E2F4A···4H4I···4R4S4T4U8A8B8C8D
order12222224···44···44448888
size11112282···24···48888888

32 irreducible representations

dim1111111111222444
type+++++++++-
imageC1C2C2C2C2C2C4C4C4C4D4D4C4≀C2C8⋊C22C8.C22C23.C23
kernelC42.57D4C22.SD16C23.31D4C4×C4⋊C4C42.6C4C23.36C23C4×D4C4×Q8C4.4D4C42.C2C42C22×C4C4C22C22C2
# reps1221112222228112

Matrix representation of C42.57D4 in GL6(𝔽17)

1300000
0130000
004000
000400
0000130
0000013
,
1300000
0130000
0041600
00151300
0000131
000024
,
1300000
0160000
0013100
000400
0000160
000091
,
0160000
1300000
0000160
000091
0013100
000400

G:=sub<GL(6,GF(17))| [13,0,0,0,0,0,0,13,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,13,0,0,0,0,0,0,13],[13,0,0,0,0,0,0,13,0,0,0,0,0,0,4,15,0,0,0,0,16,13,0,0,0,0,0,0,13,2,0,0,0,0,1,4],[13,0,0,0,0,0,0,16,0,0,0,0,0,0,13,0,0,0,0,0,1,4,0,0,0,0,0,0,16,9,0,0,0,0,0,1],[0,13,0,0,0,0,16,0,0,0,0,0,0,0,0,0,13,0,0,0,0,0,1,4,0,0,16,9,0,0,0,0,0,1,0,0] >;

C42.57D4 in GAP, Magma, Sage, TeX

C_4^2._{57}D_4
% in TeX

G:=Group("C4^2.57D4");
// GroupNames label

G:=SmallGroup(128,241);
// by ID

G=gap.SmallGroup(128,241);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,2,112,141,387,520,1123,1018,248,1971]);
// Polycyclic

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

׿
×
𝔽