Copied to
clipboard

G = C4xC8.C4order 128 = 27

Direct product of C4 and C8.C4

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

Aliases: C4xC8.C4, C8.10C42, C42.460D4, (C4xC8).32C4, C4.173(C4xD4), C22.3(C4xQ8), C4.26(C2xC42), (C22xC4).76Q8, C23.77(C2xQ8), C42.318(C2xC4), C4o2(C4.C42), M4(2).21(C2xC4), (C4xM4(2)).21C2, C4.50(C42:C2), C4.C42.12C2, C42o(C4.C42), (C22xC8).548C22, (C2xC42).1053C22, (C22xC4).1318C23, (C2xM4(2)).312C22, (C2xC4xC8).43C2, C2.16(C4xC4:C4), (C2xC8).207(C2xC4), C2.5(C2xC8.C4), C22.61(C2xC4:C4), C42o(C2xC8.C4), (C2xC4).166(C4:C4), (C2xC4).1509(C2xD4), (C2xC8.C4).27C2, (C2xC4).549(C4oD4), (C2xC4).525(C22xC4), SmallGroup(128,509)

Series: Derived Chief Lower central Upper central Jennings

C1C4 — C4xC8.C4
C1C2C4C2xC4C22xC4C2xC42C4xM4(2) — C4xC8.C4
C1C2C4 — C4xC8.C4
C1C42C2xC42 — C4xC8.C4
C1C2C2C22xC4 — C4xC8.C4

Generators and relations for C4xC8.C4
 G = < a,b,c | a4=b8=1, c4=b4, ab=ba, ac=ca, cbc-1=b-1 >

Subgroups: 148 in 114 conjugacy classes, 80 normal (18 characteristic)
C1, C2, C2, C2, C4, C4, C4, C22, C22, C22, C8, C8, C2xC4, C2xC4, C2xC4, C23, C42, C42, C2xC8, C2xC8, C2xC8, M4(2), M4(2), C22xC4, C22xC4, C4xC8, C4xC8, C8:C4, C8.C4, C2xC42, C22xC8, C2xM4(2), C4.C42, C2xC4xC8, C4xM4(2), C2xC8.C4, C4xC8.C4
Quotients: C1, C2, C4, C22, C2xC4, D4, Q8, C23, C42, C4:C4, C22xC4, C2xD4, C2xQ8, C4oD4, C8.C4, C2xC42, C2xC4:C4, C42:C2, C4xD4, C4xQ8, C4xC4:C4, C2xC8.C4, C4xC8.C4

Smallest permutation representation of C4xC8.C4
On 64 points
Generators in S64
(1 14 21 49)(2 15 22 50)(3 16 23 51)(4 9 24 52)(5 10 17 53)(6 11 18 54)(7 12 19 55)(8 13 20 56)(25 44 64 40)(26 45 57 33)(27 46 58 34)(28 47 59 35)(29 48 60 36)(30 41 61 37)(31 42 62 38)(32 43 63 39)
(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 32 19 57 5 28 23 61)(2 31 20 64 6 27 24 60)(3 30 21 63 7 26 17 59)(4 29 22 62 8 25 18 58)(9 48 50 38 13 44 54 34)(10 47 51 37 14 43 55 33)(11 46 52 36 15 42 56 40)(12 45 53 35 16 41 49 39)

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

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

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

56 conjugacy classes

class 1 2A2B2C2D2E4A···4L4M···4R8A···8P8Q···8AF
order1222224···44···48···88···8
size1111221···12···22···24···4

56 irreducible representations

dim11111112222
type++++++-
imageC1C2C2C2C2C4C4D4Q8C4oD4C8.C4
kernelC4xC8.C4C4.C42C2xC4xC8C4xM4(2)C2xC8.C4C4xC8C8.C4C42C22xC4C2xC4C4
# reps1212281622416

Matrix representation of C4xC8.C4 in GL3(F17) generated by

1300
0160
0016
,
1600
090
002
,
100
001
040
G:=sub<GL(3,GF(17))| [13,0,0,0,16,0,0,0,16],[16,0,0,0,9,0,0,0,2],[1,0,0,0,0,4,0,1,0] >;

C4xC8.C4 in GAP, Magma, Sage, TeX

C_4\times C_8.C_4
% in TeX

G:=Group("C4xC8.C4");
// GroupNames label

G:=SmallGroup(128,509);
// by ID

G=gap.SmallGroup(128,509);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,112,141,232,142,1018,248,1411,172,124]);
// Polycyclic

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

׿
x
:
Z
F
o
wr
Q
<