Copied to
clipboard

G = (C2×D8)⋊10C4  order 128 = 27

6th semidirect product of C2×D8 and C4 acting via C4/C2=C2

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

Aliases: (C2×D8)⋊10C4, C4.21(C4×D4), C8⋊2(C22⋊C4), (C2×C8).207D4, C2.4(C8⋊3D4), C2.5(C8⋊2D4), C4.85(C4⋊D4), C4.5(C4.4D4), (C22×D8).11C2, C23.805(C2×D4), C22.185(C4×D4), (C22×C4).135D4, C2.12(D8⋊C4), C22.41(C4⋊1D4), C22.96(C8⋊C22), C24.3C22⋊5C2, (C22×C8).407C22, (C2×C42).323C22, (C22×D4).50C22, C22.147(C4⋊D4), (C22×C4).1412C23, C2.24(C24.3C22), (C2×C4.Q8)⋊3C2, (C2×C8⋊C4)⋊1C2, (C2×C8).72(C2×C4), C4.40(C2×C22⋊C4), (C2×D4⋊C4)⋊47C2, (C2×D4).111(C2×C4), (C2×C4).1357(C2×D4), (C2×C4⋊C4).93C22, (C2×C4).603(C4○D4), (C2×C4).426(C22×C4), SmallGroup(128,704)

Series: Derived ►Chief ►Lower central ►Upper central ►Jennings

Derived series
C1 — C2×C4 — (C2×D8)⋊10C4
Chief series
C1 — C2 — C22 — C2×C4 — C22×C4 — C22×C8 — C2×C8⋊C4 — (C2×D8)⋊10C4
Lower central
C1 — C2 — C2×C4 — (C2×D8)⋊10C4
Upper central
C1 — C23 — C2×C42 — (C2×D8)⋊10C4
Jennings
C1 — C2 — C2 — C22×C4 — (C2×D8)⋊10C4

Generators and relations for (C2×D8)⋊10C4
 G = < a,b,c,d | a2=b8=c2=d4=1, ab=ba, ac=ca, ad=da, cbc=b-1, dbd-1=b3, dcd-1=ab2c >

Subgroups: 484 in 188 conjugacy classes, 64 normal (24 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C8, C8, C2×C4, C2×C4, D4, C23, C23, C42, C22⋊C4, C4⋊C4, C2×C8, C2×C8, D8, C22×C4, C22×C4, C22×C4, C2×D4, C2×D4, C24, C8⋊C4, D4⋊C4, C4.Q8, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C22×C8, C2×D8, C2×D8, C22×D4, C24.3C22, C2×C8⋊C4, C2×D4⋊C4, C2×C4.Q8, C22×D8, (C2×D8)⋊10C4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22⋊C4, C22×C4, C2×D4, C4○D4, C2×C22⋊C4, C4×D4, C4⋊D4, C4.4D4, C4⋊1D4, C8⋊C22, C24.3C22, D8⋊C4, C8⋊2D4, C8⋊3D4, (C2×D8)⋊10C4

Smallest permutation representation of (C2×D8)⋊10C4
►On 64 points
Generators in S64
(1 21)(2 22)(3 23)(4 24)(5 17)(6 18)(7 19)(8 20)(9 47)(10 48)(11 41)(12 42)(13 43)(14 44)(15 45)(16 46)(25 63)(26 64)(27 57)(28 58)(29 59)(30 60)(31 61)(32 62)(33 49)(34 50)(35 51)(36 52)(37 53)(38 54)(39 55)(40 56)

(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)(33 34 35 36 37 38 39 40)(41 42 43 44 45 46 47 48)(49 50 51 52 53 54 55 56)(57 58 59 60 61 62 63 64)

(1 20)(2 19)(3 18)(4 17)(5 24)(6 23)(7 22)(8 21)(9 48)(10 47)(11 46)(12 45)(13 44)(14 43)(15 42)(16 41)(25 30)(26 29)(27 28)(31 32)(33 36)(34 35)(37 40)(38 39)(49 52)(50 51)(53 56)(54 55)(57 58)(59 64)(60 63)(61 62)

(1 35 48 58)(2 38 41 61)(3 33 42 64)(4 36 43 59)(5 39 44 62)(6 34 45 57)(7 37 46 60)(8 40 47 63)(9 25 20 56)(10 28 21 51)(11 31 22 54)(12 26 23 49)(13 29 24 52)(14 32 17 55)(15 27 18 50)(16 30 19 53)
 
G:=sub<Sym(64)| (1,21)(2,22)(3,23)(4,24)(5,17)(6,18)(7,19)(8,20)(9,47)(10,48)(11,41)(12,42)(13,43)(14,44)(15,45)(16,46)(25,63)(26,64)(27,57)(28,58)(29,59)(30,60)(31,61)(32,62)(33,49)(34,50)(35,51)(36,52)(37,53)(38,54)(39,55)(40,56), (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)(33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56)(57,58,59,60,61,62,63,64), (1,20)(2,19)(3,18)(4,17)(5,24)(6,23)(7,22)(8,21)(9,48)(10,47)(11,46)(12,45)(13,44)(14,43)(15,42)(16,41)(25,30)(26,29)(27,28)(31,32)(33,36)(34,35)(37,40)(38,39)(49,52)(50,51)(53,56)(54,55)(57,58)(59,64)(60,63)(61,62), (1,35,48,58)(2,38,41,61)(3,33,42,64)(4,36,43,59)(5,39,44,62)(6,34,45,57)(7,37,46,60)(8,40,47,63)(9,25,20,56)(10,28,21,51)(11,31,22,54)(12,26,23,49)(13,29,24,52)(14,32,17,55)(15,27,18,50)(16,30,19,53)>;
 
G:=Group( (1,21)(2,22)(3,23)(4,24)(5,17)(6,18)(7,19)(8,20)(9,47)(10,48)(11,41)(12,42)(13,43)(14,44)(15,45)(16,46)(25,63)(26,64)(27,57)(28,58)(29,59)(30,60)(31,61)(32,62)(33,49)(34,50)(35,51)(36,52)(37,53)(38,54)(39,55)(40,56), (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)(33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56)(57,58,59,60,61,62,63,64), (1,20)(2,19)(3,18)(4,17)(5,24)(6,23)(7,22)(8,21)(9,48)(10,47)(11,46)(12,45)(13,44)(14,43)(15,42)(16,41)(25,30)(26,29)(27,28)(31,32)(33,36)(34,35)(37,40)(38,39)(49,52)(50,51)(53,56)(54,55)(57,58)(59,64)(60,63)(61,62), (1,35,48,58)(2,38,41,61)(3,33,42,64)(4,36,43,59)(5,39,44,62)(6,34,45,57)(7,37,46,60)(8,40,47,63)(9,25,20,56)(10,28,21,51)(11,31,22,54)(12,26,23,49)(13,29,24,52)(14,32,17,55)(15,27,18,50)(16,30,19,53) );
 
G=PermutationGroup([[(1,21),(2,22),(3,23),(4,24),(5,17),(6,18),(7,19),(8,20),(9,47),(10,48),(11,41),(12,42),(13,43),(14,44),(15,45),(16,46),(25,63),(26,64),(27,57),(28,58),(29,59),(30,60),(31,61),(32,62),(33,49),(34,50),(35,51),(36,52),(37,53),(38,54),(39,55),(40,56)], [(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),(33,34,35,36,37,38,39,40),(41,42,43,44,45,46,47,48),(49,50,51,52,53,54,55,56),(57,58,59,60,61,62,63,64)], [(1,20),(2,19),(3,18),(4,17),(5,24),(6,23),(7,22),(8,21),(9,48),(10,47),(11,46),(12,45),(13,44),(14,43),(15,42),(16,41),(25,30),(26,29),(27,28),(31,32),(33,36),(34,35),(37,40),(38,39),(49,52),(50,51),(53,56),(54,55),(57,58),(59,64),(60,63),(61,62)], [(1,35,48,58),(2,38,41,61),(3,33,42,64),(4,36,43,59),(5,39,44,62),(6,34,45,57),(7,37,46,60),(8,40,47,63),(9,25,20,56),(10,28,21,51),(11,31,22,54),(12,26,23,49),(13,29,24,52),(14,32,17,55),(15,27,18,50),(16,30,19,53)]])
 

32 conjugacy classes

class 1 2A···2G2H2I2J2K4A4B4C4D4E4F4G4H4I4J4K4L8A···8H
order12···222224444444444448···8
size11···188882222444488884···4

32 irreducible representations

dim11111112224
type+++++++++
imageC1C2C2C2C2C2C4D4D4C4○D4C8⋊C22
kernel(C2×D8)⋊10C4C24.3C22C2×C8⋊C4C2×D4⋊C4C2×C4.Q8C22×D8C2×D8C2×C8C22×C4C2×C4C22
# reps12121186244

Matrix representation of (C2×D8)⋊10C4 ►in GL8(𝔽17)

160000000
016000000
00100000
00010000
000016000
000001600
000000160
000000016
,
01000000
160000000
00010000
001600000
00001002
00001011
0000160016
0000161016
,
016000000
160000000
00010000
00100000
0000160015
000000116
00000101
00000001
,
016000000
160000000
00400000
000130000
00005899
000013409
000012984
00004040

G:=sub<GL(8,GF(17))| [16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16],[0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,1,16,16,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,2,1,16,16],[0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,15,16,1,1],[0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,13,0,0,0,0,0,0,0,0,5,13,12,4,0,0,0,0,8,4,9,0,0,0,0,0,9,0,8,4,0,0,0,0,9,9,4,0] >;
 

(C2×D8)⋊10C4 in GAP, Magma, Sage, TeX 

(C_2\times D_8)\rtimes_{10}C_4
 
% in TeX
 
G:=Group("(C2xD8):10C4");
 
// GroupNames label
 
G:=SmallGroup(128,704);
 
// by ID
 
G=gap.SmallGroup(128,704);
 
# by ID
 
G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,736,422,723,100,2019,248,2804,172]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^2=b^8=c^2=d^4=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c=b^-1,d*b*d^-1=b^3,d*c*d^-1=a*b^2*c>;
 
// generators/relations
 

׿
×
⋊
ℤ
𝔽
○
≀
ℚ
◁