Copied to
clipboard

G = C2×C22⋊C8order 64 = 26

Direct product of C2 and C22⋊C8

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

Aliases: C2×C22⋊C8, C232C8, C24.3C4, C22.9M4(2), C4(C22⋊C8), (C22×C8)⋊1C2, C222(C2×C8), C4.66(C2×D4), (C2×C8)⋊10C22, (C2×C4).144D4, C2.1(C22×C8), (C23×C4).5C2, (C22×C4).11C4, C23.28(C2×C4), C2.3(C2×M4(2)), C4.31(C22⋊C4), (C2×C4).144C23, (C22×C4).91C22, C22.20(C22×C4), C22.28(C22⋊C4), (C2×C4)(C22⋊C8), (C2×C4).55(C2×C4), C2.3(C2×C22⋊C4), SmallGroup(64,87)

Series: Derived Chief Lower central Upper central Jennings

C1C2 — C2×C22⋊C8
C1C2C4C2×C4C22×C4C23×C4 — C2×C22⋊C8
C1C2 — C2×C22⋊C8
C1C22×C4 — C2×C22⋊C8
C1C2C2C2×C4 — C2×C22⋊C8

Generators and relations for C2×C22⋊C8
 G = < a,b,c,d | a2=b2=c2=d8=1, ab=ba, ac=ca, ad=da, dbd-1=bc=cb, cd=dc >

Subgroups: 145 in 101 conjugacy classes, 57 normal (13 characteristic)
C1, C2 [×3], C2 [×4], C2 [×4], C4 [×4], C4 [×2], C22, C22 [×10], C22 [×12], C8 [×4], C2×C4 [×2], C2×C4 [×6], C2×C4 [×10], C23, C23 [×6], C23 [×4], C2×C8 [×4], C2×C8 [×4], C22×C4 [×2], C22×C4 [×4], C22×C4 [×4], C24, C22⋊C8 [×4], C22×C8 [×2], C23×C4, C2×C22⋊C8
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C8 [×4], C2×C4 [×6], D4 [×4], C23, C22⋊C4 [×4], C2×C8 [×6], M4(2) [×2], C22×C4, C2×D4 [×2], C22⋊C8 [×4], C2×C22⋊C4, C22×C8, C2×M4(2), C2×C22⋊C8

Smallest permutation representation of C2×C22⋊C8
On 32 points
Generators in S32
(1 19)(2 20)(3 21)(4 22)(5 23)(6 24)(7 17)(8 18)(9 28)(10 29)(11 30)(12 31)(13 32)(14 25)(15 26)(16 27)
(1 5)(2 32)(3 7)(4 26)(6 28)(8 30)(9 24)(10 14)(11 18)(12 16)(13 20)(15 22)(17 21)(19 23)(25 29)(27 31)
(1 27)(2 28)(3 29)(4 30)(5 31)(6 32)(7 25)(8 26)(9 20)(10 21)(11 22)(12 23)(13 24)(14 17)(15 18)(16 19)
(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)| (1,19)(2,20)(3,21)(4,22)(5,23)(6,24)(7,17)(8,18)(9,28)(10,29)(11,30)(12,31)(13,32)(14,25)(15,26)(16,27), (1,5)(2,32)(3,7)(4,26)(6,28)(8,30)(9,24)(10,14)(11,18)(12,16)(13,20)(15,22)(17,21)(19,23)(25,29)(27,31), (1,27)(2,28)(3,29)(4,30)(5,31)(6,32)(7,25)(8,26)(9,20)(10,21)(11,22)(12,23)(13,24)(14,17)(15,18)(16,19), (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( (1,19)(2,20)(3,21)(4,22)(5,23)(6,24)(7,17)(8,18)(9,28)(10,29)(11,30)(12,31)(13,32)(14,25)(15,26)(16,27), (1,5)(2,32)(3,7)(4,26)(6,28)(8,30)(9,24)(10,14)(11,18)(12,16)(13,20)(15,22)(17,21)(19,23)(25,29)(27,31), (1,27)(2,28)(3,29)(4,30)(5,31)(6,32)(7,25)(8,26)(9,20)(10,21)(11,22)(12,23)(13,24)(14,17)(15,18)(16,19), (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([(1,19),(2,20),(3,21),(4,22),(5,23),(6,24),(7,17),(8,18),(9,28),(10,29),(11,30),(12,31),(13,32),(14,25),(15,26),(16,27)], [(1,5),(2,32),(3,7),(4,26),(6,28),(8,30),(9,24),(10,14),(11,18),(12,16),(13,20),(15,22),(17,21),(19,23),(25,29),(27,31)], [(1,27),(2,28),(3,29),(4,30),(5,31),(6,32),(7,25),(8,26),(9,20),(10,21),(11,22),(12,23),(13,24),(14,17),(15,18),(16,19)], [(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)])

C2×C22⋊C8 is a maximal subgroup of
C23.19C42  C23.21C42  C23.8D8  C24.2Q8  C23.30D8  C24.3Q8  C23⋊C16  C23.8M4(2)  C23⋊C8⋊C2  C24.(C2×C4)  C24.45(C2×C4)  C24.53D4  C24.59D4  C24.60D4  C24.61D4  C42.378D4  C42.379D4  C8×C22⋊C4  C23.36C42  C23.17C42  C243C8  C24.51(C2×C4)  C23.35D8  C24.155D4  C24.65D4  C42.425D4  C42.95D4  C23.32M4(2)  C24.53(C2×C4)  C23.36D8  C24.157D4  C24.69D4  C23.21M4(2)  (C2×C8).195D4  C23.37D8  C24.159D4  C24.71D4  C24.10Q8  C23.22M4(2)  C232M4(2)  C24.160D4  C24.73D4  C23.38D8  C24.74D4  C22⋊C44C8  C23.9M4(2)  C42.325D4  C42.109D4  C232D8  C233SD16  C232Q16  C24.83D4  C24.84D4  C24.85D4  C24.86D4  C23.12D8  C24.88D4  C24.89D4  D4○(C22⋊C8)  C42.262C23  D4×C2×C8  M4(2)⋊22D4  C42.691C23  C233M4(2)  D47M4(2)  C42.297C23  C42.298C23  C24.103D4  C24.115D4  C233D8  C234SD16  C24.121D4  C233Q16  C24.123D4  C24.124D4  D10.11M4(2)
C2×C22⋊C8 is a maximal quotient of
C42.371D4  C23.8M4(2)  C42.393D4  C42.394D4  C42.455D4  C42.397D4  C42.398D4  C42.399D4  C243C8  C42.425D4  C23.32M4(2)  C23.22M4(2)  C42.325D4  C42.327D4  C24.5C8  (C2×D4).5C8  M5(2).19C22  M5(2)⋊12C22  D10.11M4(2)

40 conjugacy classes

class 1 2A···2G2H2I2J2K4A···4H4I4J4K4L8A···8P
order12···222224···444448···8
size11···122221···122222···2

40 irreducible representations

dim111111122
type+++++
imageC1C2C2C2C4C4C8D4M4(2)
kernelC2×C22⋊C8C22⋊C8C22×C8C23×C4C22×C4C24C23C2×C4C22
# reps1421621644

Matrix representation of C2×C22⋊C8 in GL4(𝔽17) generated by

1000
01600
00160
00016
,
1000
01600
0010
00816
,
1000
0100
00160
00016
,
9000
01600
0092
00108
G:=sub<GL(4,GF(17))| [1,0,0,0,0,16,0,0,0,0,16,0,0,0,0,16],[1,0,0,0,0,16,0,0,0,0,1,8,0,0,0,16],[1,0,0,0,0,1,0,0,0,0,16,0,0,0,0,16],[9,0,0,0,0,16,0,0,0,0,9,10,0,0,2,8] >;

C2×C22⋊C8 in GAP, Magma, Sage, TeX

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

G:=Group("C2xC2^2:C8");
// GroupNames label

G:=SmallGroup(64,87);
// by ID

G=gap.SmallGroup(64,87);
# by ID

G:=PCGroup([6,-2,2,2,-2,2,-2,96,121,88]);
// Polycyclic

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

׿
×
𝔽