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G = C8×D8order 128 = 27

Direct product of C8 and D8

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

Aliases: C8×D8, C824C2, C42.632C23, C84(C2×C8), C8(D4⋊C8), D41(C2×C8), (C8×D4)⋊1C2, C8(C81C8), C8(C2.D8), C2.4(C8×D4), C2.1(C4×D8), D4⋊C838C2, C81C832C2, C4.85(C2×D8), C8(D4⋊C4), (C4×D8).21C2, (C2×D8).15C4, C4.7(C8○D4), C2.1(C8○D8), (C2×C8).217D4, C4.7(C22×C8), C2.D8.22C4, C22.77(C4×D4), D4⋊C4.13C4, C4.125(C4○D8), C4⋊C8.270C22, (C4×C8).366C22, (C4×D4).268C22, (C2×C8)(C4×D8), (C2×C8)(D4⋊C8), (C2×C8)(C81C8), C4⋊C4.126(C2×C4), (C2×C8).154(C2×C4), (C2×D4).148(C2×C4), (C2×C4).1468(C2×D4), (C2×C4).493(C4○D4), (C2×C4).324(C22×C4), 2-Sylow(CO(3,9)), SmallGroup(128,307)

Series: Derived Chief Lower central Upper central Jennings

C1C4 — C8×D8
C1C2C22C2×C4C42C4×C8C8×D4 — C8×D8
C1C2C4 — C8×D8
C1C2×C8C4×C8 — C8×D8
C1C22C22C42 — C8×D8

Generators and relations for C8×D8
 G = < a,b,c | a8=b8=c2=1, ab=ba, ac=ca, cbc=b-1 >

Subgroups: 184 in 98 conjugacy classes, 52 normal (26 characteristic)
C1, C2 [×3], C2 [×4], C4 [×4], C4 [×3], C22, C22 [×8], C8 [×4], C8 [×6], C2×C4 [×3], C2×C4 [×6], D4 [×4], D4 [×2], C23 [×2], C42, C22⋊C4 [×2], C4⋊C4 [×2], C2×C8 [×4], C2×C8 [×7], D8 [×4], C22×C4 [×2], C2×D4 [×2], C4×C8 [×3], C22⋊C8 [×2], D4⋊C4 [×2], C4⋊C8 [×2], C2.D8, C4×D4 [×2], C22×C8 [×2], C2×D8, C82, D4⋊C8 [×2], C81C8, C8×D4 [×2], C4×D8, C8×D8
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C8 [×4], C2×C4 [×6], D4 [×2], C23, C2×C8 [×6], D8 [×2], C22×C4, C2×D4, C4○D4, C4×D4, C22×C8, C8○D4, C2×D8, C4○D8, C8×D4, C4×D8, C8○D8, C8×D8

Smallest permutation representation of C8×D8
On 64 points
Generators in S64
(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 43 13 21 31 63 39 52)(2 44 14 22 32 64 40 53)(3 45 15 23 25 57 33 54)(4 46 16 24 26 58 34 55)(5 47 9 17 27 59 35 56)(6 48 10 18 28 60 36 49)(7 41 11 19 29 61 37 50)(8 42 12 20 30 62 38 51)
(1 5)(2 6)(3 7)(4 8)(9 39)(10 40)(11 33)(12 34)(13 35)(14 36)(15 37)(16 38)(17 63)(18 64)(19 57)(20 58)(21 59)(22 60)(23 61)(24 62)(25 29)(26 30)(27 31)(28 32)(41 54)(42 55)(43 56)(44 49)(45 50)(46 51)(47 52)(48 53)

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

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

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

56 conjugacy classes

class 1 2A2B2C2D2E2F2G4A4B4C4D4E4F4G4H4I4J4K4L8A···8H8I···8AB8AC···8AJ
order122222224444444444448···88···88···8
size111144441111222244441···12···24···4

56 irreducible representations

dim1111111111222222
type++++++++
imageC1C2C2C2C2C2C4C4C4C8D4D8C4○D4C8○D4C4○D8C8○D8
kernelC8×D8C82D4⋊C8C81C8C8×D4C4×D8D4⋊C4C2.D8C2×D8D8C2×C8C8C2×C4C4C4C2
# reps11212142216242448

Matrix representation of C8×D8 in GL4(𝔽17) generated by

15000
01500
00130
00013
,
91600
14800
0090
00112
,
16000
16100
0014
00016
G:=sub<GL(4,GF(17))| [15,0,0,0,0,15,0,0,0,0,13,0,0,0,0,13],[9,14,0,0,16,8,0,0,0,0,9,11,0,0,0,2],[16,16,0,0,0,1,0,0,0,0,1,0,0,0,4,16] >;

C8×D8 in GAP, Magma, Sage, TeX

C_8\times D_8
% in TeX

G:=Group("C8xD8");
// GroupNames label

G:=SmallGroup(128,307);
// by ID

G=gap.SmallGroup(128,307);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,2,112,141,100,1123,570,136,172]);
// Polycyclic

G:=Group<a,b,c|a^8=b^8=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

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