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G = C2×C4○D16order 128 = 27

Direct product of C2 and C4○D16

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

Aliases: C2×C4○D16, D167C22, C8.12C24, Q326C22, D8.2C23, C23.29D8, C16.13C23, SD326C22, 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○D85C22, 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

C1C8 — C2×C4○D16
C1C2C4C8C2×C8C22×C8C2×C4○D8 — C2×C4○D16
C1C2C4C8 — C2×C4○D16
C1C2×C4C22×C4C22×C8 — C2×C4○D16
C1C2C2C2C2C4C4C8 — 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 [×2], C2 [×6], C4 [×2], C4 [×2], C4 [×4], C22, C22 [×2], C22 [×10], C8 [×2], C8 [×2], C2×C4 [×2], C2×C4 [×4], C2×C4 [×10], D4 [×14], Q8 [×6], C23, C23 [×2], C16 [×4], C2×C8 [×2], C2×C8 [×4], D8 [×4], D8 [×2], SD16 [×8], Q16 [×4], Q16 [×2], C22×C4, C22×C4 [×2], C2×D4 [×4], C2×Q8 [×2], C4○D4 [×12], C2×C16 [×2], C2×C16 [×4], D16 [×4], SD32 [×8], Q32 [×4], C22×C8, C2×D8 [×2], C2×SD16 [×2], C2×Q16 [×2], C4○D8 [×8], C4○D8 [×4], C2×C4○D4 [×2], C22×C16, C2×D16, C2×SD32 [×2], C2×Q32, C4○D16 [×8], C2×C4○D8 [×2], C2×C4○D16
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], D8 [×4], C2×D4 [×6], C24, C2×D8 [×6], C22×D4, C4○D16 [×2], C22×D8, C2×C4○D16

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