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G = C2×C8○D4order 64 = 26

Direct product of C2 and C8○D4

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

Aliases: C2×C8○D4, C8.14C23, C4.16C24, M4(2)⋊11C22, D4(C2×C8), C8(C2×D4), C8(C2×Q8), Q8(C2×C8), C4(C8○D4), C8(C8○D4), C82(C4○D4), C4○D4.4C4, D4.8(C2×C4), Q8.9(C2×C4), (C2×C8)⋊16C22, (C22×C8)⋊13C2, (C2×D4).12C4, C82(C2×M4(2)), (C2×C8)2M4(2), (C2×Q8).10C4, C2.11(C23×C4), C23.19(C2×C4), C4.22(C22×C4), (C2×M4(2))⋊17C2, (C2×C4).162C23, C4○D4.14C22, C22.4(C22×C4), (C22×C4).128C22, C8(C2×C4○D4), (C2×C8)(C2×Q8), (C2×C8)(C4○D4), (C2×C4).50(C2×C4), (C2×C8)(C2×M4(2)), (C2×C4○D4).14C2, (C2×C8)(C2×C4○D4), SmallGroup(64,248)

Series: Derived Chief Lower central Upper central Jennings

C1C2 — C2×C8○D4
C1C2C4C2×C4C22×C4C2×C4○D4 — C2×C8○D4
C1C2 — C2×C8○D4
C1C2×C8 — C2×C8○D4
C1C2C2C4 — C2×C8○D4

Generators and relations for C2×C8○D4
 G = < a,b,c,d | a2=b8=d2=1, c2=b4, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=b4c >

Subgroups: 145 in 133 conjugacy classes, 121 normal (10 characteristic)
C1, C2, C2 [×2], C2 [×6], C4 [×2], C4 [×6], C22, C22 [×6], C22 [×6], C8 [×8], C2×C4, C2×C4 [×15], D4 [×12], Q8 [×4], C23 [×3], C2×C8, C2×C8 [×15], M4(2) [×12], C22×C4 [×3], C2×D4 [×3], C2×Q8, C4○D4 [×8], C22×C8 [×3], C2×M4(2) [×3], C8○D4 [×8], C2×C4○D4, C2×C8○D4
Quotients: C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], C23 [×15], C22×C4 [×14], C24, C8○D4 [×2], C23×C4, C2×C8○D4

Smallest permutation representation of C2×C8○D4
On 32 points
Generators in S32
(1 27)(2 28)(3 29)(4 30)(5 31)(6 32)(7 25)(8 26)(9 18)(10 19)(11 20)(12 21)(13 22)(14 23)(15 24)(16 17)
(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 14 5 10)(2 15 6 11)(3 16 7 12)(4 9 8 13)(17 25 21 29)(18 26 22 30)(19 27 23 31)(20 28 24 32)
(1 10)(2 11)(3 12)(4 13)(5 14)(6 15)(7 16)(8 9)(17 25)(18 26)(19 27)(20 28)(21 29)(22 30)(23 31)(24 32)

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

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

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

C2×C8○D4 is a maximal subgroup of
C23.5C42  Q8.C42  D4.3C42  (C2×D4).24Q8  (C2×C8).103D4  C8○D4⋊C4  C4○D4.4Q8  C4○D4.5Q8  M4(2).42D4  M4(2).43D4  M4(2).48D4  M4(2).49D4  (C2×D4).5C8  M5(2).19C22  M5(2)⋊12C22  D4.5C42  D4○(C22⋊C8)  2+ 1+45C4  2- 1+44C4  C42.674C23  C4○D4.7Q8  C4○D4.8Q8  M4(2).29C23  C42.264C23  C42.265C23  C42.681C23  C42.266C23  M4(2)⋊22D4  M4(2)⋊23D4  C42.283C23  (C2×C8)⋊11D4  (C2×C8)⋊12D4  C8.D4⋊C2  (C2×C8)⋊13D4  (C2×C8)⋊14D4  M4(2)⋊16D4  M4(2)⋊17D4  M4(2).10C23  Q8○M5(2)  C4.22C25  C8.C24  Dic5.21C24
C2×C8○D4 is a maximal quotient of
D4○(C22⋊C8)  C42.674C23  C42.260C23  C42.262C23  C42.678C23  D4×C2×C8  C42.264C23  C42.681C23  M4(2)⋊22D4  M4(2)⋊23D4  Q8×C2×C8  C42.286C23  M4(2)⋊9Q8  C8×C4○D4  C42.290C23  C42.291C23  C42.293C23  C42.294C23  D47M4(2)  C42.297C23  C42.298C23  C42.694C23  C42.301C23  Q8.4M4(2)  C42.696C23  C42.304C23  D48M4(2)  Q87M4(2)  C42.307C23  C42.308C23  C42.309C23  Dic5.21C24

40 conjugacy classes

class 1 2A2B2C2D···2I4A4B4C4D4E···4J8A···8H8I···8T
order12222···244444···48···88···8
size11112···211112···21···12···2

40 irreducible representations

dim111111112
type+++++
imageC1C2C2C2C2C4C4C4C8○D4
kernelC2×C8○D4C22×C8C2×M4(2)C8○D4C2×C4○D4C2×D4C2×Q8C4○D4C2
# reps133816288

Matrix representation of C2×C8○D4 in GL3(𝔽17) generated by

1600
0160
0016
,
400
080
008
,
1600
01615
011
,
100
01615
001
G:=sub<GL(3,GF(17))| [16,0,0,0,16,0,0,0,16],[4,0,0,0,8,0,0,0,8],[16,0,0,0,16,1,0,15,1],[1,0,0,0,16,0,0,15,1] >;

C2×C8○D4 in GAP, Magma, Sage, TeX

C_2\times C_8\circ D_4
% in TeX

G:=Group("C2xC8oD4");
// GroupNames label

G:=SmallGroup(64,248);
// by ID

G=gap.SmallGroup(64,248);
# by ID

G:=PCGroup([6,-2,2,2,2,-2,-2,96,332,88]);
// Polycyclic

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

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