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G = C2×C4○D16  order 128 = 27

Direct product of C2 and C4○D16

direct product, p-group, metabelian, nilpotent (class 4), monomial

Aliases: C2×C4○D16, D16⋊7C22, C8.12C24, Q32⋊6C22, D8.2C23, C23.29D8, C16.13C23, SD32⋊6C22, Q16.2C23, (C2×C4)○D16, C4○(C2×D16), C4○(C2×Q32), (C2×C4)○Q32, C4○(C2×SD32), (C2×C4)○SD32, C4○(C4○D16), (C2×C4).97D8, C4.95(C2×D8), C8.57(C2×D4), (C2×D16)⋊14C2, (C2×Q32)⋊14C2, (C2×C8).272D4, C4○D8⋊5C22, C22.3(C2×D8), (C22×C16)⋊10C2, (C2×C16)⋊19C22, (C2×SD32)⋊18C2, C2.27(C22×D8), C4.18(C22×D4), (C2×C8).585C23, (C22×C4).624D4, (C2×D8).151C22, (C22×C8).560C22, (C2×Q16).147C22, (C2×C4)○(C2×D16), (C2×C4)○(C2×Q32), (C2×C4)○(C2×SD32), (C2×C4○D8)⋊15C2, (C2×C4).875(C2×D4), SmallGroup(128,2143)

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

Derived series
C1 — C8 — C2×C4○D16
Chief series
C1 — C2 — C4 — C8 — C2×C8 — C22×C8 — C2×C4○D8 — C2×C4○D16
Lower central
C1 — C2 — C4 — C8 — C2×C4○D16
Upper central
C1 — C2×C4 — C22×C4 — C22×C8 — C2×C4○D16
Jennings
C1 — C2 — C2 — C2 — C2 — C4 — C4 — C8 — C2×C4○D16

Generators and relations for C2×C4○D16
 G = < a,b,c,d | a2=b4=d2=1, c8=b2, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=b2c7 >

Subgroups: 404 in 184 conjugacy classes, 92 normal (20 characteristic)
C1, C2, C2, C2, C4, C4, C4, C22, C22, C22, C8, C8, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, C16, C2×C8, C2×C8, D8, D8, SD16, Q16, Q16, C22×C4, C22×C4, C2×D4, C2×Q8, C4○D4, C2×C16, C2×C16, D16, SD32, Q32, C22×C8, C2×D8, C2×SD16, C2×Q16, C4○D8, C4○D8, C2×C4○D4, C22×C16, C2×D16, C2×SD32, C2×Q32, C4○D16, C2×C4○D8, C2×C4○D16
Quotients: C1, C2, C22, D4, C23, D8, C2×D4, C24, C2×D8, C22×D4, C4○D16, C22×D8, C2×C4○D16

Smallest permutation representation of C2×C4○D16
►On 64 points
Generators in S64
(1 58)(2 59)(3 60)(4 61)(5 62)(6 63)(7 64)(8 49)(9 50)(10 51)(11 52)(12 53)(13 54)(14 55)(15 56)(16 57)(17 40)(18 41)(19 42)(20 43)(21 44)(22 45)(23 46)(24 47)(25 48)(26 33)(27 34)(28 35)(29 36)(30 37)(31 38)(32 39)

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

(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)

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

44 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I4A4B4C4D4E4F4G4H4I4J8A···8H16A···16P
order122222222244444444448···816···16
size111122888811112288882···22···2

44 irreducible representations

dim111111122222
type+++++++++++
imageC1C2C2C2C2C2C2D4D4D8D8C4○D16
kernelC2×C4○D16C22×C16C2×D16C2×SD32C2×Q32C4○D16C2×C4○D8C2×C8C22×C4C2×C4C23C2
# reps1112182316216

Matrix representation of C2×C4○D16 ►in GL3(𝔽17) generated by

1600
0160
0016
,
1600
040
004
,
1600
0712
0112
,
1600
010
0116
G:=sub<GL(3,GF(17))| [16,0,0,0,16,0,0,0,16],[16,0,0,0,4,0,0,0,4],[16,0,0,0,7,11,0,12,2],[16,0,0,0,1,1,0,0,16] >;
 

C2×C4○D16 in GAP, Magma, Sage, TeX 

C_2\times C_4\circ D_{16}
 
% in TeX
 
G:=Group("C2xC4oD16");
 
// GroupNames label
 
G:=SmallGroup(128,2143);
 
// by ID
 
G=gap.SmallGroup(128,2143);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,-2,-2,253,352,1684,851,242,4037,2028,124]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^2=b^4=d^2=1,c^8=b^2,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=b^2*c^7>;
 
// generators/relations
 

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