Copied to
clipboard

G = C2×C426C4order 128 = 27

Direct product of C2 and C426C4

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

Aliases: C2×C426C4, C24.151D4, (C2×C42)⋊17C4, C4242(C2×C4), C4(C426C4), C22.40C4≀C2, (C2×C4).64C42, C4.14(C2×C42), C42⋊C213C4, (C22×C4).85Q8, C23.59(C4⋊C4), M4(2)⋊18(C2×C4), (C2×M4(2))⋊12C4, C23.536(C2×D4), (C22×C4).751D4, (C22×C42).15C2, (C23×C4).666C22, C23.225(C22⋊C4), C4.15(C2.C42), (C22×C4).1298C23, (C2×C42).1044C22, (C22×M4(2)).10C2, C42⋊C2.256C22, (C2×M4(2)).295C22, C22.28(C2.C42), (C2×C4⋊C4)⋊20C4, C4⋊C433(C2×C4), C2.4(C2×C4≀C2), C4.72(C2×C4⋊C4), C22.8(C2×C4⋊C4), (C2×C4).177(C2×Q8), (C2×C4).121(C4⋊C4), (C2×C4)(C426C4), (C2×C4).1491(C2×D4), C4.102(C2×C22⋊C4), (C2×C42⋊C2).5C2, (C22×C4).400(C2×C4), (C2×C4).344(C22×C4), C2.9(C2×C2.C42), (C2×C4).392(C22⋊C4), C22.103(C2×C22⋊C4), SmallGroup(128,464)

Series: Derived Chief Lower central Upper central Jennings

C1C4 — C2×C426C4
C1C2C22C23C22×C4C23×C4C22×C42 — C2×C426C4
C1C2C4 — C2×C426C4
C1C22×C4C23×C4 — C2×C426C4
C1C2C2C22×C4 — C2×C426C4

Generators and relations for C2×C426C4
 G = < a,b,c,d | a2=b4=c4=d4=1, ab=ba, ac=ca, ad=da, dbd-1=bc=cb, dcd-1=c-1 >

Subgroups: 388 in 244 conjugacy classes, 116 normal (32 characteristic)
C1, C2, C2 [×6], C2 [×4], C4 [×4], C4 [×4], C4 [×12], C22 [×3], C22 [×8], C22 [×12], C8 [×4], C2×C4 [×8], C2×C4 [×20], C2×C4 [×36], C23 [×3], C23 [×4], C23 [×4], C42 [×4], C42 [×10], C22⋊C4 [×4], C4⋊C4 [×4], C4⋊C4 [×2], C2×C8 [×6], M4(2) [×4], M4(2) [×6], C22×C4 [×6], C22×C4 [×8], C22×C4 [×16], C24, C2×C42 [×6], C2×C42 [×4], C2×C22⋊C4, C2×C4⋊C4 [×2], C42⋊C2 [×4], C42⋊C2 [×2], C22×C8, C2×M4(2) [×6], C2×M4(2) [×3], C23×C4, C23×C4, C426C4 [×4], C22×C42, C2×C42⋊C2, C22×M4(2), C2×C426C4
Quotients: C1, C2 [×7], C4 [×12], C22 [×7], C2×C4 [×18], D4 [×6], Q8 [×2], C23, C42 [×4], C22⋊C4 [×12], C4⋊C4 [×12], C22×C4 [×3], C2×D4 [×3], C2×Q8, C2.C42 [×8], C4≀C2 [×4], C2×C42, C2×C22⋊C4 [×3], C2×C4⋊C4 [×3], C426C4 [×4], C2×C2.C42, C2×C4≀C2 [×2], C2×C426C4

Smallest permutation representation of C2×C426C4
On 32 points
Generators in S32
(1 14)(2 13)(3 12)(4 11)(5 16)(6 15)(7 10)(8 9)(17 21)(18 22)(19 23)(20 24)(25 32)(26 29)(27 30)(28 31)
(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 6 4 9)(2 5 3 10)(7 13 16 12)(8 14 15 11)(17 26 19 28)(18 27 20 25)(21 29 23 31)(22 30 24 32)
(1 30 2 21)(3 23 4 32)(5 31 6 22)(7 26 8 20)(9 24 10 29)(11 25 12 19)(13 17 14 27)(15 18 16 28)

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

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

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

56 conjugacy classes

class 1 2A···2G2H2I2J2K4A···4H4I···4AB4AC···4AJ8A···8H
order12···222224···44···44···48···8
size11···122221···12···24···44···4

56 irreducible representations

dim1111111112222
type++++++-+
imageC1C2C2C2C2C4C4C4C4D4Q8D4C4≀C2
kernelC2×C426C4C426C4C22×C42C2×C42⋊C2C22×M4(2)C2×C42C2×C4⋊C4C42⋊C2C2×M4(2)C22×C4C22×C4C24C22
# reps14111844852116

Matrix representation of C2×C426C4 in GL4(𝔽17) generated by

16000
01600
0010
0001
,
1000
01600
0010
0004
,
1000
0100
0040
00013
,
4000
01600
00016
00160
G:=sub<GL(4,GF(17))| [16,0,0,0,0,16,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,16,0,0,0,0,1,0,0,0,0,4],[1,0,0,0,0,1,0,0,0,0,4,0,0,0,0,13],[4,0,0,0,0,16,0,0,0,0,0,16,0,0,16,0] >;

C2×C426C4 in GAP, Magma, Sage, TeX

C_2\times C_4^2\rtimes_6C_4
% in TeX

G:=Group("C2xC4^2:6C4");
// GroupNames label

G:=SmallGroup(128,464);
// by ID

G=gap.SmallGroup(128,464);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,112,141,232,2019,248,4037]);
// Polycyclic

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

׿
×
𝔽