Copied to
clipboard

G = C8.C22⋊C4order 128 = 27

2nd semidirect product of C8.C22 and C4 acting via C4/C2=C2

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

Aliases: C4○D4.5D4, C4.64(C4×D4), C4⋊C4.314D4, C8.C222C4, C426C42C2, (C2×D4).275D4, M4(2)⋊4(C2×C4), (C2×Q8).216D4, C22.36(C4×D4), (C22×C4).23D4, D4.4(C22⋊C4), C23.558(C2×D4), Q8.4(C22⋊C4), C4.130(C4⋊D4), C22.C423C2, C2.4(D4.9D4), C22.97C22≀C2, C2.4(D4.10D4), C23.38D421C2, C22.43(C4⋊D4), (C22×C4).680C23, (C2×C42).281C22, (C2×M4(2)).7C22, (C22×Q8).14C22, C42⋊C2.17C22, C23.67C233C2, C4.67(C22.D4), C2.21(C23.23D4), C23.33C23.3C2, (C2×Q8)⋊7(C2×C4), (C2×C4≀C2).9C2, C4○D4.8(C2×C4), C4.16(C2×C22⋊C4), (C2×C4).1001(C2×D4), (C2×C4⋊C4).58C22, (C2×C4).10(C22×C4), (C2×C8.C22).1C2, (C2×C4).319(C4○D4), (C2×C4○D4).17C22, SmallGroup(128,614)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C8.C22⋊C4
C1C2C4C2×C4C22×C4C2×C4○D4C23.33C23 — C8.C22⋊C4
C1C2C2×C4 — C8.C22⋊C4
C1C22C22×C4 — C8.C22⋊C4
C1C2C2C22×C4 — C8.C22⋊C4

Generators and relations for C8.C22⋊C4
 G = < a,b,c,d | a8=b2=c2=d4=1, bab=a3, cac=a5, dad-1=a-1c, cbc=a4b, bd=db, dcd-1=a4c >

Subgroups: 332 in 167 conjugacy classes, 56 normal (38 characteristic)
C1, C2 [×3], C2 [×4], C4 [×4], C4 [×10], C22 [×3], C22 [×6], C8 [×3], C2×C4 [×6], C2×C4 [×21], D4 [×2], D4 [×5], Q8 [×2], Q8 [×7], C23, C23, C42 [×5], C22⋊C4 [×3], C4⋊C4 [×2], C4⋊C4 [×4], C2×C8 [×2], M4(2) [×2], M4(2) [×3], SD16 [×4], Q16 [×4], C22×C4, C22×C4 [×2], C22×C4 [×5], C2×D4, C2×D4, C2×Q8, C2×Q8 [×2], C2×Q8 [×5], C4○D4 [×4], C4○D4 [×2], C2.C42 [×2], Q8⋊C4 [×2], C4≀C2 [×2], C2×C42, C2×C4⋊C4, C2×C4⋊C4, C42⋊C2, C42⋊C2, C4×D4 [×3], C4×Q8, C2×M4(2) [×2], C2×SD16, C2×Q16, C8.C22 [×4], C8.C22 [×2], C22×Q8, C2×C4○D4, C426C4, C22.C42, C23.67C23, C23.38D4, C2×C4≀C2, C23.33C23, C2×C8.C22, C8.C22⋊C4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×8], C23, C22⋊C4 [×4], C22×C4, C2×D4 [×4], C4○D4 [×2], C2×C22⋊C4, C4×D4 [×2], C22≀C2, C4⋊D4 [×2], C22.D4, C23.23D4, D4.9D4, D4.10D4, C8.C22⋊C4

Smallest permutation representation of C8.C22⋊C4
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 20)(2 23)(3 18)(4 21)(5 24)(6 19)(7 22)(8 17)(9 28)(10 31)(11 26)(12 29)(13 32)(14 27)(15 30)(16 25)
(1 29)(2 26)(3 31)(4 28)(5 25)(6 30)(7 27)(8 32)(9 17)(10 22)(11 19)(12 24)(13 21)(14 18)(15 23)(16 20)
(1 28 25 4)(2 7 26 31)(3 30 27 6)(5 32 29 8)(9 16 21 20)(10 23 22 11)(12 17 24 13)(14 19 18 15)

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

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

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

32 conjugacy classes

class 1 2A2B2C2D2E2F2G4A4B4C4D4E···4R4S4T8A8B8C8D
order1222222244444···4448888
size1111224422224···4888888

32 irreducible representations

dim11111111122222244
type+++++++++++++-
imageC1C2C2C2C2C2C2C2C4D4D4D4D4D4C4○D4D4.9D4D4.10D4
kernelC8.C22⋊C4C426C4C22.C42C23.67C23C23.38D4C2×C4≀C2C23.33C23C2×C8.C22C8.C22C4⋊C4C22×C4C2×D4C2×Q8C4○D4C2×C4C2C2
# reps11111111822112422

Matrix representation of C8.C22⋊C4 in GL6(𝔽17)

1620000
1610000
0002013
000080
000900
001040
,
1620000
010000
0000130
000409
004000
0004013
,
100000
010000
001000
000100
0000160
0001016
,
1380000
040000
0001504
000090
0001500
0040130

G:=sub<GL(6,GF(17))| [16,16,0,0,0,0,2,1,0,0,0,0,0,0,0,0,0,1,0,0,2,0,9,0,0,0,0,8,0,4,0,0,13,0,0,0],[16,0,0,0,0,0,2,1,0,0,0,0,0,0,0,0,4,0,0,0,0,4,0,4,0,0,13,0,0,0,0,0,0,9,0,13],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,1,0,0,0,0,16,0,0,0,0,0,0,16],[13,0,0,0,0,0,8,4,0,0,0,0,0,0,0,0,0,4,0,0,15,0,15,0,0,0,0,9,0,13,0,0,4,0,0,0] >;

C8.C22⋊C4 in GAP, Magma, Sage, TeX

C_8.C_2^2\rtimes C_4
% in TeX

G:=Group("C8.C2^2:C4");
// GroupNames label

G:=SmallGroup(128,614);
// by ID

G=gap.SmallGroup(128,614);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,288,422,1018,248,2804,718,172,2028,124]);
// Polycyclic

G:=Group<a,b,c,d|a^8=b^2=c^2=d^4=1,b*a*b=a^3,c*a*c=a^5,d*a*d^-1=a^-1*c,c*b*c=a^4*b,b*d=d*b,d*c*d^-1=a^4*c>;
// generators/relations

׿
×
𝔽