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G = D4×C16order 128 = 27

Direct product of C16 and D4

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

Aliases: D4×C16, C16(C4⋊C8), (C4×C16)⋊5C2, C41(C2×C16), C162(C4⋊C4), C16(C4⋊C16), C4⋊C1620C2, C2.3(C8×D4), C4⋊C8.27C4, C4⋊C4.13C8, C16(C22⋊C8), C221(C2×C16), (C22×C16)⋊7C2, (C8×D4).19C2, (C4×D4).40C4, (C2×D4).12C8, C8.138(C2×D4), C4.175(C4×D4), C22⋊C4.8C8, C16(C22⋊C16), C162(C22⋊C4), C22⋊C1617C2, C2.2(D4○C16), C4.57(C8○D4), C22⋊C8.25C4, C23.22(C2×C8), C2.4(C22×C16), C8.101(C4○D4), (C4×C8).375C22, C42.272(C2×C4), (C2×C8).632C23, (C2×C16).111C22, C22.29(C22×C8), (C22×C8).501C22, C4⋊C8(C2×C16), C4⋊C4(C2×C16), (C2×C16)(C4×D4), (C2×C16)(C8×D4), (C2×D4)(C2×C16), C22⋊C8(C2×C16), C22⋊C4(C2×C16), (C2×C16)(C4⋊C16), (C2×C4).41(C2×C8), (C2×C8).162(C2×C4), (C2×C16)(C22⋊C16), (C2×C4).617(C22×C4), (C22×C4).378(C2×C4), SmallGroup(128,899)

Series: Derived Chief Lower central Upper central Jennings

C1C2 — D4×C16
C1C2C4C8C2×C8C2×C16C22×C16 — D4×C16
C1C2 — D4×C16
C1C2×C16 — D4×C16
C1C2C2C2C2C4C4C2×C8 — D4×C16

Generators and relations for D4×C16
 G = < a,b,c | a16=b4=c2=1, ab=ba, ac=ca, cbc=b-1 >

Subgroups: 116 in 87 conjugacy classes, 58 normal (26 characteristic)
C1, C2 [×3], C2 [×4], C4 [×2], C4 [×2], C4 [×3], C22, C22 [×4], C22 [×4], C8 [×2], C8 [×3], C2×C4 [×3], C2×C4 [×2], C2×C4 [×4], D4 [×4], C23 [×2], C16 [×2], C16 [×3], C42, C22⋊C4 [×2], C4⋊C4, C2×C8 [×2], C2×C8 [×2], C2×C8 [×4], C22×C4 [×2], C2×D4, C4×C8, C22⋊C8 [×2], C4⋊C8, C2×C16 [×2], C2×C16 [×2], C2×C16 [×4], C4×D4, C22×C8 [×2], C4×C16, C22⋊C16 [×2], C4⋊C16, C8×D4, C22×C16 [×2], D4×C16
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C8 [×4], C2×C4 [×6], D4 [×2], C23, C16 [×4], C2×C8 [×6], C22×C4, C2×D4, C4○D4, C2×C16 [×6], C4×D4, C22×C8, C8○D4, C8×D4, C22×C16, D4○C16, D4×C16

Smallest permutation representation of D4×C16
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 36 29 50)(2 37 30 51)(3 38 31 52)(4 39 32 53)(5 40 17 54)(6 41 18 55)(7 42 19 56)(8 43 20 57)(9 44 21 58)(10 45 22 59)(11 46 23 60)(12 47 24 61)(13 48 25 62)(14 33 26 63)(15 34 27 64)(16 35 28 49)
(1 9)(2 10)(3 11)(4 12)(5 13)(6 14)(7 15)(8 16)(17 25)(18 26)(19 27)(20 28)(21 29)(22 30)(23 31)(24 32)(33 55)(34 56)(35 57)(36 58)(37 59)(38 60)(39 61)(40 62)(41 63)(42 64)(43 49)(44 50)(45 51)(46 52)(47 53)(48 54)

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

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

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

80 conjugacy classes

class 1 2A2B2C2D2E2F2G4A4B4C4D4E···4L8A···8H8I···8T16A···16P16Q···16AN
order1222222244444···48···88···816···1616···16
size1111222211112···21···12···21···12···2

80 irreducible representations

dim11111111111112222
type+++++++
imageC1C2C2C2C2C2C4C4C4C8C8C8C16D4C4○D4C8○D4D4○C16
kernelD4×C16C4×C16C22⋊C16C4⋊C16C8×D4C22×C16C22⋊C8C4⋊C8C4×D4C22⋊C4C4⋊C4C2×D4D4C16C8C4C2
# reps112112422844322248

Matrix representation of D4×C16 in GL3(𝔽17) generated by

600
060
006
,
1600
0115
0116
,
100
0162
001
G:=sub<GL(3,GF(17))| [6,0,0,0,6,0,0,0,6],[16,0,0,0,1,1,0,15,16],[1,0,0,0,16,0,0,2,1] >;

D4×C16 in GAP, Magma, Sage, TeX

D_4\times C_{16}
% in TeX

G:=Group("D4xC16");
// GroupNames label

G:=SmallGroup(128,899);
// by ID

G=gap.SmallGroup(128,899);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,-2,112,141,100,102,124]);
// Polycyclic

G:=Group<a,b,c|a^16=b^4=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

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