Copied to
clipboard

G = C4×C8.C4order 128 = 27

Direct product of C4 and C8.C4

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

Aliases: C4×C8.C4, C8.10C42, C42.460D4, (C4×C8).32C4, C4.173(C4×D4), C22.3(C4×Q8), C4.26(C2×C42), (C22×C4).76Q8, C23.77(C2×Q8), C42.318(C2×C4), C42(C4.C42), M4(2).21(C2×C4), (C4×M4(2)).21C2, C4.50(C42⋊C2), C4.C42.12C2, C42(C4.C42), (C22×C8).548C22, (C2×C42).1053C22, (C22×C4).1318C23, (C2×M4(2)).312C22, (C2×C4×C8).43C2, C2.16(C4×C4⋊C4), (C2×C8).207(C2×C4), C2.5(C2×C8.C4), C22.61(C2×C4⋊C4), C42(C2×C8.C4), (C2×C4).166(C4⋊C4), (C2×C4).1509(C2×D4), (C2×C8.C4).27C2, (C2×C4).549(C4○D4), (C2×C4).525(C22×C4), SmallGroup(128,509)

Series: Derived Chief Lower central Upper central Jennings

C1C4 — C4×C8.C4
C1C2C4C2×C4C22×C4C2×C42C4×M4(2) — C4×C8.C4
C1C2C4 — C4×C8.C4
C1C42C2×C42 — C4×C8.C4
C1C2C2C22×C4 — C4×C8.C4

Generators and relations for C4×C8.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 [×2], C2 [×2], C4 [×2], C4 [×6], C4 [×2], C22, C22 [×2], C22 [×2], C8 [×4], C8 [×10], C2×C4 [×2], C2×C4 [×8], C2×C4 [×4], C23, C42 [×2], C42 [×2], C2×C8 [×2], C2×C8 [×6], C2×C8 [×6], M4(2) [×8], M4(2) [×4], C22×C4, C22×C4 [×2], C4×C8 [×4], C4×C8 [×2], C8⋊C4 [×2], C8.C4 [×8], C2×C42, C22×C8 [×2], C2×M4(2) [×4], C4.C42 [×2], C2×C4×C8, C4×M4(2) [×2], C2×C8.C4 [×2], C4×C8.C4
Quotients: C1, C2 [×7], C4 [×12], C22 [×7], C2×C4 [×18], D4 [×2], Q8 [×2], C23, C42 [×4], C4⋊C4 [×4], C22×C4 [×3], C2×D4, C2×Q8, C4○D4 [×2], C8.C4 [×4], C2×C42, C2×C4⋊C4, C42⋊C2, C4×D4 [×2], C4×Q8 [×2], C4×C4⋊C4, C2×C8.C4 [×2], C4×C8.C4

Smallest permutation representation of C4×C8.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 33)(26 45 57 34)(27 46 58 35)(28 47 59 36)(29 48 60 37)(30 41 61 38)(31 42 62 39)(32 43 63 40)
(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 39 13 44 54 35)(10 47 51 38 14 43 55 34)(11 46 52 37 15 42 56 33)(12 45 53 36 16 41 49 40)

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)(25,44,64,33)(26,45,57,34)(27,46,58,35)(28,47,59,36)(29,48,60,37)(30,41,61,38)(31,42,62,39)(32,43,63,40), (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)(9,48,50,39,13,44,54,35)(10,47,51,38,14,43,55,34)(11,46,52,37,15,42,56,33)(12,45,53,36,16,41,49,40)>;

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)(25,44,64,33)(26,45,57,34)(27,46,58,35)(28,47,59,36)(29,48,60,37)(30,41,61,38)(31,42,62,39)(32,43,63,40), (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)(9,48,50,39,13,44,54,35)(10,47,51,38,14,43,55,34)(11,46,52,37,15,42,56,33)(12,45,53,36,16,41,49,40) );

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),(25,44,64,33),(26,45,57,34),(27,46,58,35),(28,47,59,36),(29,48,60,37),(30,41,61,38),(31,42,62,39),(32,43,63,40)], [(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),(9,48,50,39,13,44,54,35),(10,47,51,38,14,43,55,34),(11,46,52,37,15,42,56,33),(12,45,53,36,16,41,49,40)])

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++++++-
imageC1C2C2C2C2C4C4D4Q8C4○D4C8.C4
kernelC4×C8.C4C4.C42C2×C4×C8C4×M4(2)C2×C8.C4C4×C8C8.C4C42C22×C4C2×C4C4
# reps1212281622416

Matrix representation of C4×C8.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] >;

C4×C8.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

׿
×
𝔽