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G = C2xC4oD16  order 128 = 27

Direct product of C2 and C4oD16

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

Aliases: C2xC4oD16, D16:7C22, C8.12C24, Q32:6C22, D8.2C23, C23.29D8, C16.13C23, SD32:6C22, Q16.2C23, (C2xC4)oD16, C4o(C2xD16), C4o(C2xQ32), (C2xC4)oQ32, C4o(C2xSD32), (C2xC4)oSD32, C4o(C4oD16), (C2xC4).97D8, C4.95(C2xD8), C8.57(C2xD4), (C2xD16):14C2, (C2xQ32):14C2, (C2xC8).272D4, C4oD8:5C22, C22.3(C2xD8), (C22xC16):10C2, (C2xC16):19C22, (C2xSD32):18C2, C2.27(C22xD8), C4.18(C22xD4), (C2xC8).585C23, (C22xC4).624D4, (C2xD8).151C22, (C22xC8).560C22, (C2xQ16).147C22, (C2xC4)o(C2xD16), (C2xC4)o(C2xQ32), (C2xC4)o(C2xSD32), (C2xC4oD8):15C2, (C2xC4).875(C2xD4), SmallGroup(128,2143)

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

Derived series
C1 — C8 — C2xC4oD16
Chief series
C1 — C2 — C4 — C8 — C2xC8 — C22xC8 — C2xC4oD8 — C2xC4oD16
Lower central
C1 — C2 — C4 — C8 — C2xC4oD16
Upper central
C1 — C2xC4 — C22xC4 — C22xC8 — C2xC4oD16
Jennings
C1 — C2 — C2 — C2 — C2 — C4 — C4 — C8 — C2xC4oD16

Generators and relations for C2xC4oD16
 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, C2xC4, C2xC4, C2xC4, D4, Q8, C23, C23, C16, C2xC8, C2xC8, D8, D8, SD16, Q16, Q16, C22xC4, C22xC4, C2xD4, C2xQ8, C4oD4, C2xC16, C2xC16, D16, SD32, Q32, C22xC8, C2xD8, C2xSD16, C2xQ16, C4oD8, C4oD8, C2xC4oD4, C22xC16, C2xD16, C2xSD32, C2xQ32, C4oD16, C2xC4oD8, C2xC4oD16
Quotients: C1, C2, C22, D4, C23, D8, C2xD4, C24, C2xD8, C22xD4, C4oD16, C22xD8, C2xC4oD16

Smallest permutation representation of C2xC4oD16
►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+++++++++++
imageC1C2C2C2C2C2C2D4D4D8D8C4oD16
kernelC2xC4oD16C22xC16C2xD16C2xSD32C2xQ32C4oD16C2xC4oD8C2xC8C22xC4C2xC4C23C2
# reps1112182316216

Matrix representation of C2xC4oD16 ►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] >;
 

C2xC4oD16 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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