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G = C2×D4○C16order 128 = 27

Direct product of C2 and D4○C16

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

Aliases: C2×D4○C16, C8.24C24, C16.14C23, M5(2)⋊16C22, D4(C2×C16), C16(C2×D4), Q8(C2×C16), C16(C2×Q8), C8(D4○C16), C4(D4○C16), C4○D4.6C8, C8○D4.7C4, D4.8(C2×C8), Q8.9(C2×C8), C162(C4○D4), C16(D4○C16), C162(C8○D4), (C2×D4).13C8, (C2×Q8).12C8, (C22×C16)⋊15C2, (C2×C16)⋊22C22, (C2×C16)2M5(2), C162(C2×M5(2)), C162(C2×M4(2)), M4(2)2(C2×C16), C4.63(C23×C4), C8.50(C22×C4), C2.11(C23×C8), C4.22(C22×C8), C23.24(C2×C8), (C2×M5(2))⋊22C2, (C2×C8).617C23, C8○D4.19C22, C22.4(C22×C8), (C2×M4(2)).38C4, M4(2).35(C2×C4), (C22×C8).587C22, C4○D4(C2×C16), C16(C2×C4○D4), C16(C2×C8○D4), (C2×Q8)(C2×C16), (C2×C16)(C8○D4), (C2×C4).57(C2×C8), (C2×C8).198(C2×C4), C4○D4.38(C2×C4), (C2×C4○D4).35C4, (C2×C8○D4).24C2, (C2×C16)(C2×M5(2)), (C2×C16)(C2×M4(2)), (C22×C4).421(C2×C4), (C2×C4).476(C22×C4), (C2×C16)(C2×C8○D4), (C2×C16)(C2×C4○D4), SmallGroup(128,2138)

Series: Derived Chief Lower central Upper central Jennings

C1C2 — C2×D4○C16
C1C2C4C8C2×C8C22×C8C2×C8○D4 — C2×D4○C16
C1C2 — C2×D4○C16
C1C2×C16 — C2×D4○C16
C1C2C2C2C2C4C4C8 — C2×D4○C16

Generators and relations for C2×D4○C16
 G = < a,b,c,d | a2=b4=c2=1, d8=b2, ab=ba, ac=ca, ad=da, cbc=b-1, bd=db, cd=dc >

Subgroups: 196 in 184 conjugacy classes, 172 normal (14 characteristic)
C1, C2, C2 [×2], C2 [×6], C4 [×2], C4 [×6], C22, C22 [×6], C22 [×6], C8 [×2], C8 [×6], C2×C4, C2×C4 [×15], D4 [×12], Q8 [×4], C23 [×3], C16 [×8], C2×C8, C2×C8 [×15], M4(2) [×12], C22×C4 [×3], C2×D4 [×3], C2×Q8, C4○D4 [×8], C2×C16, C2×C16 [×15], M5(2) [×12], C22×C8 [×3], C2×M4(2) [×3], C8○D4 [×8], C2×C4○D4, C22×C16 [×3], C2×M5(2) [×3], D4○C16 [×8], C2×C8○D4, C2×D4○C16
Quotients: C1, C2 [×15], C4 [×8], C22 [×35], C8 [×8], C2×C4 [×28], C23 [×15], C2×C8 [×28], C22×C4 [×14], C24, C22×C8 [×14], C23×C4, D4○C16 [×2], C23×C8, C2×D4○C16

Smallest permutation representation of C2×D4○C16
On 64 points
Generators in S64
(1 35)(2 36)(3 37)(4 38)(5 39)(6 40)(7 41)(8 42)(9 43)(10 44)(11 45)(12 46)(13 47)(14 48)(15 33)(16 34)(17 58)(18 59)(19 60)(20 61)(21 62)(22 63)(23 64)(24 49)(25 50)(26 51)(27 52)(28 53)(29 54)(30 55)(31 56)(32 57)
(1 51 9 59)(2 52 10 60)(3 53 11 61)(4 54 12 62)(5 55 13 63)(6 56 14 64)(7 57 15 49)(8 58 16 50)(17 34 25 42)(18 35 26 43)(19 36 27 44)(20 37 28 45)(21 38 29 46)(22 39 30 47)(23 40 31 48)(24 41 32 33)
(1 59)(2 60)(3 61)(4 62)(5 63)(6 64)(7 49)(8 50)(9 51)(10 52)(11 53)(12 54)(13 55)(14 56)(15 57)(16 58)(17 34)(18 35)(19 36)(20 37)(21 38)(22 39)(23 40)(24 41)(25 42)(26 43)(27 44)(28 45)(29 46)(30 47)(31 48)(32 33)
(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)

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

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

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

80 conjugacy classes

class 1 2A2B2C2D···2I4A4B4C4D4E···4J8A···8H8I···8T16A···16P16Q···16AN
order12222···244444···48···88···816···1616···16
size11112···211112···21···12···21···12···2

80 irreducible representations

dim111111111112
type+++++
imageC1C2C2C2C2C4C4C4C8C8C8D4○C16
kernelC2×D4○C16C22×C16C2×M5(2)D4○C16C2×C8○D4C2×M4(2)C8○D4C2×C4○D4C2×D4C2×Q8C4○D4C2
# reps133816821241616

Matrix representation of C2×D4○C16 in GL3(𝔽17) generated by

1600
010
001
,
100
0016
010
,
1600
0016
0160
,
1600
0140
0014
G:=sub<GL(3,GF(17))| [16,0,0,0,1,0,0,0,1],[1,0,0,0,0,1,0,16,0],[16,0,0,0,0,16,0,16,0],[16,0,0,0,14,0,0,0,14] >;

C2×D4○C16 in GAP, Magma, Sage, TeX

C_2\times D_4\circ C_{16}
% in TeX

G:=Group("C2xD4oC16");
// GroupNames label

G:=SmallGroup(128,2138);
// by ID

G=gap.SmallGroup(128,2138);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,-2,-2,112,723,102,124]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^4=c^2=1,d^8=b^2,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c=b^-1,b*d=d*b,c*d=d*c>;
// generators/relations

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