Copied to
clipboard

G = C4×C8⋊C22order 128 = 27

Direct product of C4 and C8⋊C22

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

Aliases: C4×C8⋊C22, C42.441D4, C42.273C23, D84(C2×C4), (C4×D8)⋊27C2, C81(C22×C4), (C4×C8)⋊20C22, SD161(C2×C4), C4.133(C4×D4), D43(C22×C4), C42(D8⋊C4), D8⋊C433C2, Q83(C22×C4), (C4×SD16)⋊11C2, (C4×D4)⋊80C22, M4(2)⋊7(C2×C4), (C4×M4(2))⋊1C2, C4.21(C23×C4), (C4×Q8)⋊76C22, C22.46(C4×D4), C4.Q845C22, C2.D865C22, C8⋊C436C22, C4⋊C4.361C23, (C2×C8).412C23, (C2×C4).201C24, C23.643(C2×D4), (C22×C4).710D4, C42(SD16⋊C4), SD16⋊C454C2, D4⋊C488C22, Q8⋊C491C22, (C2×D4).370C23, (C2×D8).157C22, (C2×Q8).343C23, C42(M4(2)⋊C4), M4(2)⋊C446C2, C42(C23.37D4), C42(C23.36D4), C2.4(D8⋊C22), C23.37D439C2, C23.36D448C2, (C22×C4).922C23, (C2×C42).766C22, C22.145(C22×D4), (C2×SD16).107C22, (C22×D4).559C22, C42⋊C2.297C22, (C2×M4(2)).350C22, (C2×C4×D4)⋊58C2, C2.61(C2×C4×D4), (C4×C4○D4)⋊5C2, C4○D49(C2×C4), C4.9(C2×C4○D4), (C2×D4)⋊34(C2×C4), C2.6(C2×C8⋊C22), (C2×C4)(D8⋊C4), (C2×C4).691(C2×D4), (C2×C8⋊C22).13C2, (C2×C4).68(C22×C4), (C2×C4)(SD16⋊C4), (C2×C4).262(C4○D4), (C2×C4⋊C4).912C22, (C2×C4○D4).291C22, (C2×C4)(C23.37D4), (C2×C4)(C2×C8⋊C22), SmallGroup(128,1676)

Series: Derived Chief Lower central Upper central Jennings

C1C4 — C4×C8⋊C22
C1C2C22C2×C4C22×C4C2×C42C2×C4×D4 — C4×C8⋊C22
C1C2C4 — C4×C8⋊C22
C1C2×C4C2×C42 — C4×C8⋊C22
C1C2C2C2×C4 — C4×C8⋊C22

Generators and relations for C4×C8⋊C22
 G = < a,b,c,d | a4=b8=c2=d2=1, ab=ba, ac=ca, ad=da, cbc=b3, dbd=b5, cd=dc >

Subgroups: 500 in 274 conjugacy classes, 142 normal (44 characteristic)
C1, C2 [×3], C2 [×8], C4 [×2], C4 [×4], C4 [×9], C22, C22 [×2], C22 [×22], C8 [×4], C8 [×2], C2×C4 [×6], C2×C4 [×4], C2×C4 [×26], D4 [×6], D4 [×11], Q8 [×2], Q8, C23, C23 [×11], C42 [×4], C42 [×3], C22⋊C4 [×7], C4⋊C4 [×2], C4⋊C4 [×2], C4⋊C4 [×3], C2×C8 [×4], M4(2) [×4], M4(2) [×2], D8 [×8], SD16 [×8], C22×C4 [×3], C22×C4 [×12], C2×D4, C2×D4 [×6], C2×D4 [×4], C2×Q8, C4○D4 [×4], C4○D4 [×2], C24, C4×C8 [×2], C8⋊C4 [×2], D4⋊C4 [×6], Q8⋊C4 [×2], C4.Q8 [×2], C2.D8 [×2], C2×C42, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C42⋊C2, C42⋊C2, C4×D4 [×2], C4×D4 [×4], C4×D4 [×4], C4×Q8 [×2], C2×M4(2) [×2], C2×D8 [×2], C2×SD16 [×2], C8⋊C22 [×8], C23×C4, C22×D4, C2×C4○D4, C4×M4(2), C23.36D4, C23.37D4, M4(2)⋊C4, C4×D8 [×2], C4×SD16 [×2], SD16⋊C4 [×2], D8⋊C4 [×2], C2×C4×D4, C4×C4○D4, C2×C8⋊C22, C4×C8⋊C22
Quotients: C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], D4 [×4], C23 [×15], C22×C4 [×14], C2×D4 [×6], C4○D4 [×2], C24, C4×D4 [×4], C8⋊C22 [×2], C23×C4, C22×D4, C2×C4○D4, C2×C4×D4, C2×C8⋊C22, D8⋊C22, C4×C8⋊C22

Smallest permutation representation of C4×C8⋊C22
On 32 points
Generators in S32
(1 19 26 13)(2 20 27 14)(3 21 28 15)(4 22 29 16)(5 23 30 9)(6 24 31 10)(7 17 32 11)(8 18 25 12)
(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 4)(3 7)(6 8)(10 12)(11 15)(14 16)(17 21)(18 24)(20 22)(25 31)(27 29)(28 32)
(2 6)(4 8)(10 14)(12 16)(18 22)(20 24)(25 29)(27 31)

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

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

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

44 conjugacy classes

class 1 2A2B2C2D2E2F···2K4A4B4C4D4E···4N4O···4X8A···8H
order1222222···244444···44···48···8
size1111224···411112···24···44···4

44 irreducible representations

dim111111111111122244
type+++++++++++++++
imageC1C2C2C2C2C2C2C2C2C2C2C2C4D4D4C4○D4C8⋊C22D8⋊C22
kernelC4×C8⋊C22C4×M4(2)C23.36D4C23.37D4M4(2)⋊C4C4×D8C4×SD16SD16⋊C4D8⋊C4C2×C4×D4C4×C4○D4C2×C8⋊C22C8⋊C22C42C22×C4C2×C4C4C2
# reps1111122221111622422

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

400000
040000
004000
000400
000040
000004
,
1680000
410000
0010130
0010013
0013400
0090016
,
100000
13160000
0010413
0001640
000010
0000216
,
100000
010000
0010013
0001013
0000160
0000016

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

C4×C8⋊C22 in GAP, Magma, Sage, TeX

C_4\times C_8\rtimes C_2^2
% in TeX

G:=Group("C4xC8:C2^2");
// GroupNames label

G:=SmallGroup(128,1676);
// by ID

G=gap.SmallGroup(128,1676);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,224,253,184,2019,2804,1411,172]);
// Polycyclic

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

׿
×
𝔽