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G = C8○D16order 128 = 27

Central product of C8 and D16

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

Aliases: C8D16, C8Q32, C8SD32, D166C4, Q326C4, C8.23D8, SD325C4, C42.332D4, C8○D81C2, (C4×C16)⋊10C2, C8(C4○D16), D8.4(C2×C4), C4.92(C2×D8), C4.31(C4×D4), C2.19(C4×D8), C16.14(C2×C4), C4○D16.5C2, (C2×C8).289D4, Q16.4(C2×C4), C8(C8.4Q8), C8.4Q88C2, C8(D8.C4), D8.C47C2, C8.58(C4○D4), C8.41(C22×C4), (C4×C8).422C22, (C2×C8).580C23, (C2×C16).95C22, C4○D8.14C22, C22.1(C4○D8), C8.C4.16C22, (C2×C4).774(C2×D4), SmallGroup(128,910)

Series: Derived Chief Lower central Upper central Jennings

C1C8 — C8○D16
C1C2C4C2×C4C2×C8C4×C8C8○D8 — C8○D16
C1C2C4C8 — C8○D16
C1C8C2×C8C4×C8 — C8○D16
C1C2C2C2C2C4C4C2×C8 — C8○D16

Generators and relations for C8○D16
 G = < a,b,c | a8=c2=1, b8=a4, ab=ba, ac=ca, cbc=a4b7 >

Subgroups: 148 in 73 conjugacy classes, 38 normal (24 characteristic)
C1, C2, C2 [×3], C4 [×2], C4 [×4], C22, C22 [×2], C8 [×4], C8 [×2], C2×C4, C2×C4 [×3], D4 [×4], Q8 [×2], C16 [×2], C16, C42, C2×C8 [×2], C2×C8 [×2], M4(2) [×4], D8 [×2], SD16 [×2], Q16 [×2], C4○D4 [×2], C4×C8, C4≀C2 [×2], C8.C4 [×2], C2×C16 [×2], D16, SD32 [×2], Q32, C8○D4 [×2], C4○D8 [×2], C4×C16, D8.C4 [×2], C8.4Q8, C8○D8 [×2], C4○D16, C8○D16
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×2], C23, D8 [×2], C22×C4, C2×D4, C4○D4, C4×D4, C2×D8, C4○D8, C4×D8, C8○D16

Smallest permutation representation of C8○D16
On 32 points
Generators in S32
(1 15 13 11 9 7 5 3)(2 16 14 12 10 8 6 4)(17 19 21 23 25 27 29 31)(18 20 22 24 26 28 30 32)
(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)
(1 20)(2 19)(3 18)(4 17)(5 32)(6 31)(7 30)(8 29)(9 28)(10 27)(11 26)(12 25)(13 24)(14 23)(15 22)(16 21)

G:=sub<Sym(32)| (1,15,13,11,9,7,5,3)(2,16,14,12,10,8,6,4)(17,19,21,23,25,27,29,31)(18,20,22,24,26,28,30,32), (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), (1,20)(2,19)(3,18)(4,17)(5,32)(6,31)(7,30)(8,29)(9,28)(10,27)(11,26)(12,25)(13,24)(14,23)(15,22)(16,21)>;

G:=Group( (1,15,13,11,9,7,5,3)(2,16,14,12,10,8,6,4)(17,19,21,23,25,27,29,31)(18,20,22,24,26,28,30,32), (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), (1,20)(2,19)(3,18)(4,17)(5,32)(6,31)(7,30)(8,29)(9,28)(10,27)(11,26)(12,25)(13,24)(14,23)(15,22)(16,21) );

G=PermutationGroup([(1,15,13,11,9,7,5,3),(2,16,14,12,10,8,6,4),(17,19,21,23,25,27,29,31),(18,20,22,24,26,28,30,32)], [(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)], [(1,20),(2,19),(3,18),(4,17),(5,32),(6,31),(7,30),(8,29),(9,28),(10,27),(11,26),(12,25),(13,24),(14,23),(15,22),(16,21)])

44 conjugacy classes

class 1 2A2B2C2D4A4B4C···4G4H4I8A8B8C8D8E···8J8K8L8M8N16A···16P
order12222444···44488888···8888816···16
size11288112···28811112···288882···2

44 irreducible representations

dim111111111222222
type+++++++++
imageC1C2C2C2C2C2C4C4C4D4D4D8C4○D4C4○D8C8○D16
kernelC8○D16C4×C16D8.C4C8.4Q8C8○D8C4○D16D16SD32Q32C42C2×C8C8C8C22C1
# reps1121212421142416

Matrix representation of C8○D16 in GL2(𝔽17) generated by

20
02
,
30
06
,
01
10
G:=sub<GL(2,GF(17))| [2,0,0,2],[3,0,0,6],[0,1,1,0] >;

C8○D16 in GAP, Magma, Sage, TeX

C_8\circ D_{16}
% in TeX

G:=Group("C8oD16");
// GroupNames label

G:=SmallGroup(128,910);
// by ID

G=gap.SmallGroup(128,910);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,-2,112,141,100,1123,570,360,172,4037,2028,124]);
// Polycyclic

G:=Group<a,b,c|a^8=c^2=1,b^8=a^4,a*b=b*a,a*c=c*a,c*b*c=a^4*b^7>;
// generators/relations

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