Copied to
clipboard

G = C42.63D4order 128 = 27

45th non-split extension by C42 of D4 acting via D4/C2=C22

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

Aliases: C42.63D4, (C4×D4)⋊8C4, (C4×Q8)⋊8C4, C42.C26C4, C425C42C2, C4.4D410C4, C42.80(C2×C4), C23.511(C2×D4), (C22×C4).222D4, C22.19(C4○D8), C23.31D43C2, C22.SD16.1C2, C42.12C418C2, C4⋊D4.143C22, C22⋊C8.169C22, (C2×C42).183C22, (C22×C4).643C23, C22⋊Q8.148C22, C2.26(C42⋊C22), C2.11(C23.24D4), C2.C42.10C22, C23.36C23.8C2, C2.23(C23.C23), C4⋊C4.21(C2×C4), (C2×D4).16(C2×C4), (C2×Q8).16(C2×C4), (C2×C4).1167(C2×D4), (C2×C4).133(C22×C4), (C2×C4).319(C22⋊C4), C22.197(C2×C22⋊C4), SmallGroup(128,253)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C42.63D4
C1C2C22C23C22×C4C2×C42C23.36C23 — C42.63D4
C1C22C2×C4 — C42.63D4
C1C22C2×C42 — C42.63D4
C1C2C22C22×C4 — C42.63D4

Generators and relations for C42.63D4
 G = < a,b,c,d | a4=b4=c4=1, d2=b-1, ab=ba, cac-1=ab2, dad-1=a-1, cbc-1=a2b-1, bd=db, dcd-1=b-1c-1 >

Subgroups: 236 in 109 conjugacy classes, 44 normal (34 characteristic)
C1, C2 [×3], C2 [×3], C4 [×11], C22, C22 [×2], C22 [×5], C8 [×2], C2×C4 [×6], C2×C4 [×15], D4 [×3], Q8, C23, C23, C42 [×4], C42, C22⋊C4 [×5], C4⋊C4 [×2], C4⋊C4 [×4], C2×C8 [×2], C22×C4 [×3], C22×C4 [×3], C2×D4, C2×D4, C2×Q8, C2.C42 [×2], C2.C42 [×2], C4×C8, C22⋊C8 [×2], C4⋊C8, C2×C42, 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], C425C4, C42.12C4, C23.36C23, C42.63D4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×4], C23, C22⋊C4 [×4], C22×C4, C2×D4 [×2], C2×C22⋊C4, C4○D8 [×2], C23.C23, C23.24D4, C42⋊C22, C42.63D4

Smallest permutation representation of C42.63D4
On 32 points
Generators in S32
(1 16 29 22)(2 23 30 9)(3 10 31 24)(4 17 32 11)(5 12 25 18)(6 19 26 13)(7 14 27 20)(8 21 28 15)
(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)
(2 28 30 8)(3 27)(4 6 32 26)(7 31)(9 17 23 11)(10 24)(12 16)(13 21 19 15)(14 20)(18 22)
(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)| (1,16,29,22)(2,23,30,9)(3,10,31,24)(4,17,32,11)(5,12,25,18)(6,19,26,13)(7,14,27,20)(8,21,28,15), (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), (2,28,30,8)(3,27)(4,6,32,26)(7,31)(9,17,23,11)(10,24)(12,16)(13,21,19,15)(14,20)(18,22), (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( (1,16,29,22)(2,23,30,9)(3,10,31,24)(4,17,32,11)(5,12,25,18)(6,19,26,13)(7,14,27,20)(8,21,28,15), (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), (2,28,30,8)(3,27)(4,6,32,26)(7,31)(9,17,23,11)(10,24)(12,16)(13,21,19,15)(14,20)(18,22), (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([(1,16,29,22),(2,23,30,9),(3,10,31,24),(4,17,32,11),(5,12,25,18),(6,19,26,13),(7,14,27,20),(8,21,28,15)], [(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)], [(2,28,30,8),(3,27),(4,6,32,26),(7,31),(9,17,23,11),(10,24),(12,16),(13,21,19,15),(14,20),(18,22)], [(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···4H4I4J4K···4Q8A···8H
order12222224···4444···48···8
size11112282···2448···84···4

32 irreducible representations

dim111111111122244
type++++++++
imageC1C2C2C2C2C2C4C4C4C4D4D4C4○D8C23.C23C42⋊C22
kernelC42.63D4C22.SD16C23.31D4C425C4C42.12C4C23.36C23C4×D4C4×Q8C4.4D4C42.C2C42C22×C4C22C2C2
# reps122111222222822

Matrix representation of C42.63D4 in GL6(𝔽17)

040000
1300000
000100
0016000
000001
0000160
,
0160000
100000
004000
000400
000040
000004
,
400000
0130000
001000
0001600
0000130
000004
,
5120000
550000
0000130
000004
001000
0001600

G:=sub<GL(6,GF(17))| [0,13,0,0,0,0,4,0,0,0,0,0,0,0,0,16,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,1,0],[0,1,0,0,0,0,16,0,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[4,0,0,0,0,0,0,13,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,13,0,0,0,0,0,0,4],[5,5,0,0,0,0,12,5,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,13,0,0,0,0,0,0,4,0,0] >;

C42.63D4 in GAP, Magma, Sage, TeX

C_4^2._{63}D_4
% in TeX

G:=Group("C4^2.63D4");
// GroupNames label

G:=SmallGroup(128,253);
// by ID

G=gap.SmallGroup(128,253);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,2,112,141,1430,387,520,1123,1018,248,1971]);
// Polycyclic

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

׿
×
𝔽