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G = C4xC8.C22  order 128 = 27

Direct product of C4 and C8.C22

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

Aliases: C4xC8.C22, C42.442D4, C42.274C23, Q16:4(C2xC4), (C4xQ16):27C2, SD16:2(C2xC4), C4.134(C4xD4), C8.1(C22xC4), (C4xSD16):12C2, (C4xM4(2)):2C2, M4(2):8(C2xC4), C4.22(C23xC4), D4.5(C22xC4), C22.47(C4xD4), C4o2(Q16:C4), Q16:C4:33C2, Q8.5(C22xC4), C4:C4.362C23, (C4xC8).173C22, (C2xC8).413C23, (C2xC4).202C24, (C22xC4).711D4, C23.644(C2xD4), C4o3(SD16:C4), SD16:C4:55C2, (C2xD4).371C23, (C4xD4).292C22, (C2xQ8).344C23, (C4xQ8).275C22, C4o3(M4(2):C4), M4(2):C4:47C2, C2.D8.212C22, C4.Q8.126C22, C8:C4.112C22, C4o2(C23.38D4), C4o3(C23.36D4), C23.38D4:39C2, C2.5(D8:C22), (C2xC42).767C22, (C22xC4).923C23, (C2xQ16).152C22, C22.146(C22xD4), D4:C4.196C22, Q8:C4.196C22, (C2xSD16).108C22, C23.36D4.15C2, (C22xQ8).463C22, C42:C2.298C22, (C2xM4(2)).351C22, (C2xC4xQ8):33C2, C2.62(C2xC4xD4), (C2xQ8):29(C2xC4), C4.10(C2xC4oD4), C4oD4.20(C2xC4), (C4xC4oD4).14C2, (C2xC4).692(C2xD4), C2.6(C2xC8.C22), (C2xC4).69(C22xC4), (C2xC4).263(C4oD4), (C2xC4:C4).913C22, (C2xC8.C22).13C2, (C2xC4)o(M4(2):C4), (C2xC4oD4).292C22, SmallGroup(128,1677)

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

Derived series
C1 — C4 — C4xC8.C22
Chief series
C1 — C2 — C22 — C2xC4 — C22xC4 — C2xC42 — C2xC4xQ8 — C4xC8.C22
Lower central
C1 — C2 — C4 — C4xC8.C22
Upper central
C1 — C2xC4 — C2xC42 — C4xC8.C22
Jennings
C1 — C2 — C2 — C2xC4 — C4xC8.C22

Generators and relations for C4xC8.C22
 G = < a,b,c,d | a4=b8=c2=d2=1, ab=ba, ac=ca, ad=da, cbc=b3, dbd=b5, dcd=b4c >

Subgroups: 372 in 242 conjugacy classes, 142 normal (44 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C22, C8, C8, C2xC4, C2xC4, C2xC4, D4, D4, Q8, Q8, C23, C23, C42, C42, C22:C4, C4:C4, C4:C4, C4:C4, C2xC8, M4(2), M4(2), SD16, Q16, C22xC4, C22xC4, C2xD4, C2xD4, C2xQ8, C2xQ8, C2xQ8, C4oD4, C4oD4, C4xC8, C8:C4, D4:C4, Q8:C4, C4.Q8, C2.D8, C2xC42, C2xC42, C2xC4:C4, C2xC4:C4, C42:C2, C42:C2, C4xD4, C4xD4, C4xQ8, C4xQ8, C4xQ8, C2xM4(2), C2xSD16, C2xQ16, C8.C22, C22xQ8, C2xC4oD4, C4xM4(2), C23.36D4, C23.38D4, M4(2):C4, C4xSD16, C4xQ16, SD16:C4, Q16:C4, C2xC4xQ8, C4xC4oD4, C2xC8.C22, C4xC8.C22
Quotients: C1, C2, C4, C22, C2xC4, D4, C23, C22xC4, C2xD4, C4oD4, C24, C4xD4, C8.C22, C23xC4, C22xD4, C2xC4oD4, C2xC4xD4, C2xC8.C22, D8:C22, C4xC8.C22

Smallest permutation representation of C4xC8.C22
►On 64 points
Generators in S64
(1 61 53 21)(2 62 54 22)(3 63 55 23)(4 64 56 24)(5 57 49 17)(6 58 50 18)(7 59 51 19)(8 60 52 20)(9 46 34 27)(10 47 35 28)(11 48 36 29)(12 41 37 30)(13 42 38 31)(14 43 39 32)(15 44 40 25)(16 45 33 26)

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

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

44 conjugacy classes

class 1 2A2B2C2D2E2F2G4A4B4C4D4E···4N4O···4AB8A···8H
order1222222244444···44···48···8
size1111224411112···24···44···4

44 irreducible representations

dim111111111111122244
type++++++++++++++-
imageC1C2C2C2C2C2C2C2C2C2C2C2C4D4D4C4oD4C8.C22D8:C22
kernelC4xC8.C22C4xM4(2)C23.36D4C23.38D4M4(2):C4C4xSD16C4xQ16SD16:C4Q16:C4C2xC4xQ8C4xC4oD4C2xC8.C22C8.C22C42C22xC4C2xC4C4C2
# reps1111122221111622422

Matrix representation of C4xC8.C22 ►in GL6(F17)

400000
040000
001000
000100
000010
000001
,
1620000
1610000
007101212
0077512
0022107
001521010
,
1600000
1610000
0016000
000100
000010
0000016
,
1600000
0160000
000400
0013000
0010013
000140

G:=sub<GL(6,GF(17))| [4,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[16,16,0,0,0,0,2,1,0,0,0,0,0,0,7,7,2,15,0,0,10,7,2,2,0,0,12,5,10,10,0,0,12,12,7,10],[16,16,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,16],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,0,13,1,0,0,0,4,0,0,1,0,0,0,0,0,4,0,0,0,0,13,0] >;
 

C4xC8.C22 in GAP, Magma, Sage, TeX 

C_4\times C_8.C_2^2
 
% in TeX
 
G:=Group("C4xC8.C2^2");
 
// GroupNames label
 
G:=SmallGroup(128,1677);
 
// by ID
 
G=gap.SmallGroup(128,1677);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,-2,224,253,456,184,2019,2804,1411,172]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^4=b^8=c^2=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c=b^3,d*b*d=b^5,d*c*d=b^4*c>;
 
// generators/relations
 

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