Copied to
clipboard

G = C22⋊C4⋊4C8  order 128 = 27

3rd semidirect product of C22⋊C4 and C8 acting via C8/C4=C2

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

Aliases: C22⋊C4⋊4C8, C2.13(C8×D4), (C2×C8).327D4, C23.7(C2×C8), C2.3(C8⋊6D4), C2.7(C8⋊9D4), C24.60(C2×C4), C22.101(C4×D4), (C2×C4).37M4(2), C4.191(C4⋊D4), C4.84(C4.4D4), C22.32(C8○D4), C22.42(C22×C8), (C23×C4).22C22, (C22×C8).46C22, C4.48(C42⋊2C2), C2.C42.11C4, (C2×C42).998C22, C23.271(C22×C4), C22.7C42⋊8C2, C22.53(C2×M4(2)), (C22×C4).1633C23, C2.10(C42.12C4), C22.59(C42⋊C2), C4.140(C22.D4), C2.5(C42.7C22), C2.4(C24.C22), (C2×C4×C8)⋊9C2, (C2×C4⋊C8)⋊14C2, (C2×C4).21(C2×C8), (C2×C4).1535(C2×D4), (C4×C22⋊C4).15C2, (C2×C22⋊C4).30C4, (C2×C22⋊C8).24C2, (C2×C4).939(C4○D4), (C22×C4).123(C2×C4), SmallGroup(128,655)

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

Derived series
C1 — C22 — C22⋊C4⋊4C8
Chief series
C1 — C2 — C4 — C2×C4 — C22×C4 — C2×C42 — C2×C4×C8 — C22⋊C4⋊4C8
Lower central
C1 — C22 — C22⋊C4⋊4C8
Upper central
C1 — C22×C4 — C22⋊C4⋊4C8
Jennings
C1 — C2 — C2 — C22×C4 — C22⋊C4⋊4C8

Generators and relations for C22⋊C4⋊4C8
 G = < a,b,c,d | a2=b2=c4=d8=1, cac-1=ab=ba, dad-1=ac2, bc=cb, bd=db, cd=dc >

Subgroups: 252 in 146 conjugacy classes, 68 normal (52 characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C2×C4, C2×C4, C2×C4, C23, C23, C23, C42, C22⋊C4, C22⋊C4, C2×C8, C2×C8, C22×C4, C22×C4, C24, C2.C42, C4×C8, C22⋊C8, C4⋊C8, C2×C42, C2×C22⋊C4, C22×C8, C23×C4, C22.7C42, C4×C22⋊C4, C2×C4×C8, C2×C22⋊C8, C2×C4⋊C8, C22⋊C4⋊4C8
Quotients: C1, C2, C4, C22, C8, C2×C4, D4, C23, C2×C8, M4(2), C22×C4, C2×D4, C4○D4, C42⋊C2, C4×D4, C4⋊D4, C22.D4, C4.4D4, C42⋊2C2, C22×C8, C2×M4(2), C8○D4, C24.C22, C42.12C4, C42.7C22, C8×D4, C8⋊9D4, C8⋊6D4, C22⋊C4⋊4C8

Smallest permutation representation of C22⋊C4⋊4C8
►On 64 points
Generators in S64
(1 52)(2 6)(3 54)(4 8)(5 56)(7 50)(9 28)(10 19)(11 30)(12 21)(13 32)(14 23)(15 26)(16 17)(18 42)(20 44)(22 46)(24 48)(25 47)(27 41)(29 43)(31 45)(33 37)(34 61)(35 39)(36 63)(38 57)(40 59)(49 53)(51 55)(58 62)(60 64)

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

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

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

56 conjugacy classes

class 1 2A···2G2H2I4A···4H4I···4P4Q···4V8A···8P8Q···8X
order12···2224···44···44···48···88···8
size11···1441···12···24···42···24···4

56 irreducible representations

dim1111111112222
type+++++++
imageC1C2C2C2C2C2C4C4C8D4M4(2)C4○D4C8○D4
kernelC22⋊C4⋊4C8C22.7C42C4×C22⋊C4C2×C4×C8C2×C22⋊C8C2×C4⋊C8C2.C42C2×C22⋊C4C22⋊C4C2×C8C2×C4C2×C4C22
# reps12112144164488

Matrix representation of C22⋊C4⋊4C8 ►in GL5(𝔽17)

10000
016000
00100
00010
000016
,
10000
016000
001600
000160
000016
,
160000
001600
016000
000016
00010
,
20000
02000
00200
00008
00090

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

C22⋊C4⋊4C8 in GAP, Magma, Sage, TeX 

C_2^2\rtimes C_4\rtimes_4C_8
 
% in TeX
 
G:=Group("C2^2:C4:4C8");
 
// GroupNames label
 
G:=SmallGroup(128,655);
 
// by ID
 
G=gap.SmallGroup(128,655);
 
# by ID
 
G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,288,422,58,124]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^2=b^2=c^4=d^8=1,c*a*c^-1=a*b=b*a,d*a*d^-1=a*c^2,b*c=c*b,b*d=d*b,c*d=d*c>;
 
// generators/relations
 

׿
×
⋊
ℤ
𝔽
○
≀
ℚ
◁