Copied to
clipboard

G = C4×M4(2)  order 64 = 26

Direct product of C4 and M4(2)

direct product, p-group, metabelian, nilpotent (class 2), monomial

Aliases: C4×M4(2), C4.4C42, C42.9C4, C22.4C42, C42.59C22, C86(C2×C4), (C4×C8)⋊13C2, C42(C8⋊C4), C8⋊C412C2, C2.6(C2×C42), C42(C8⋊C4), (C2×C8).98C22, C23.27(C2×C4), C4.33(C22×C4), (C22×C4).10C4, (C2×C42).12C2, C2.2(C2×M4(2)), C42(C2×M4(2)), (C2×C4).142C23, (C2×M4(2)).16C2, C22.18(C22×C4), (C22×C4).105C22, (C2×C4).84(C2×C4), SmallGroup(64,85)

Series: Derived Chief Lower central Upper central Jennings

C1C2 — C4×M4(2)
C1C2C22C2×C4C22×C4C2×C42 — C4×M4(2)
C1C2 — C4×M4(2)
C1C42 — C4×M4(2)
C1C2C2C2×C4 — C4×M4(2)

Generators and relations for C4×M4(2)
 G = < a,b,c | a4=b8=c2=1, ab=ba, ac=ca, cbc=b5 >

Subgroups: 81 in 71 conjugacy classes, 61 normal (11 characteristic)
C1, C2, C2 [×2], C2 [×2], C4 [×8], C4 [×2], C22, C22 [×2], C22 [×2], C8 [×8], C2×C4 [×2], C2×C4 [×8], C2×C4 [×4], C23, C42 [×2], C42 [×2], C2×C8 [×4], M4(2) [×8], C22×C4, C22×C4 [×2], C4×C8 [×2], C8⋊C4 [×2], C2×C42, C2×M4(2) [×2], C4×M4(2)
Quotients: C1, C2 [×7], C4 [×12], C22 [×7], C2×C4 [×18], C23, C42 [×4], M4(2) [×4], C22×C4 [×3], C2×C42, C2×M4(2) [×2], C4×M4(2)

Smallest permutation representation of C4×M4(2)
On 32 points
Generators in S32
(1 19 32 16)(2 20 25 9)(3 21 26 10)(4 22 27 11)(5 23 28 12)(6 24 29 13)(7 17 30 14)(8 18 31 15)
(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 32)(2 29)(3 26)(4 31)(5 28)(6 25)(7 30)(8 27)(9 24)(10 21)(11 18)(12 23)(13 20)(14 17)(15 22)(16 19)

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

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

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

C4×M4(2) is a maximal subgroup of
M4(2)⋊C8  C42.3Q8  C42.6Q8  C42.26D4  C42.388D4  C42.9Q8  C89M4(2)  C8215C2  C86M4(2)  C42.47D4  C42.400D4  C42.401D4  D44M4(2)  D45M4(2)  Q85M4(2)  C42.66D4  C42.405D4  C42.406D4  C42.407D4  C42.408D4  C42.376D4  M4(2)⋊1C8  C81M4(2)  C8.6C42  C8⋊C417C4  C42.427D4  C42.430D4  M4(2)⋊12D4  C42.114D4  C42.115D4  M4(2)⋊13D4  M4(2)⋊7Q8  M4(2)⋊8Q8  C42.128D4  C8.5M4(2)  M4(2)○2M4(2)  C42.677C23  C42.259C23  C42.260C23  C42.261C23  M4(2)⋊23D4  M4(2).51D4  M4(2)⋊9Q8  C42.290C23  C42.292C23  C42.294C23  D46M4(2)  Q86M4(2)  C42.240D4  C42.241D4  C42.242D4  C42.243D4  C42.244D4  M4(2)⋊7D4  M4(2)⋊8D4  M4(2)⋊9D4  M4(2)⋊5Q8  M4(2)⋊6Q8  C42.255D4  C42.256D4  C42.259D4  C42.260D4  C42.261D4  C42.262D4
 D2p.C42: D4.C42  D4.5C42  Dic35M4(2)  D10.6C42  Dic7.C42 ...
C4×M4(2) is a maximal quotient of
C89M4(2)  C8215C2  C86M4(2)  C23.28C42  C43.C2  C23.17C42  C4⋊C814C4
 C42.D2p: C42.378D4  C43.7C2  Dic35M4(2)  D10.6C42  Dic7.C42 ...

40 conjugacy classes

class 1 2A2B2C2D2E4A···4L4M···4R8A···8P
order1222224···44···48···8
size1111221···12···22···2

40 irreducible representations

dim111111112
type+++++
imageC1C2C2C2C2C4C4C4M4(2)
kernelC4×M4(2)C4×C8C8⋊C4C2×C42C2×M4(2)C42M4(2)C22×C4C4
# reps1221241648

Matrix representation of C4×M4(2) in GL3(𝔽17) generated by

1300
010
001
,
400
0415
0613
,
100
010
0416
G:=sub<GL(3,GF(17))| [13,0,0,0,1,0,0,0,1],[4,0,0,0,4,6,0,15,13],[1,0,0,0,1,4,0,0,16] >;

C4×M4(2) in GAP, Magma, Sage, TeX

C_4\times M_4(2)
% in TeX

G:=Group("C4xM4(2)");
// GroupNames label

G:=SmallGroup(64,85);
// by ID

G=gap.SmallGroup(64,85);
# by ID

G:=PCGroup([6,-2,2,2,-2,2,-2,48,103,650,117]);
// Polycyclic

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

׿
×
𝔽