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G = C2×C4○D8order 64 = 26

Direct product of C2 and C4○D8

direct product, p-group, metabelian, nilpotent (class 3), monomial

Aliases: C2×C4○D8, D86C22, C4.4C24, C8.13C23, Q166C22, D4.2C23, C23.29D4, Q8.2C23, SD165C22, (C2×C4)D8, C4(C2×D8), (C2×C4)Q16, C4(C2×Q16), C4(C4○D8), (C2×C4)SD16, C4(C2×SD16), (C2×D8)⋊13C2, (C22×C8)⋊8C2, (C2×C4).90D4, C4.83(C2×D4), (C2×C8)⋊13C22, (C2×Q16)⋊13C2, C4○D43C22, (C2×SD16)⋊16C2, C2.26(C22×D4), C22.67(C2×D4), (C2×C4).138C23, (C2×D4).73C22, (C2×Q8).69C22, (C22×C4).132C22, (C2×C4)(C2×D8), (C2×C4)(C2×Q16), (C2×C4)(C2×SD16), (C2×C4○D4)⋊10C2, SmallGroup(64,253)

Series: Derived Chief Lower central Upper central Jennings

C1C4 — C2×C4○D8
C1C2C4C2×C4C22×C4C2×C4○D4 — C2×C4○D8
C1C2C4 — C2×C4○D8
C1C2×C4C22×C4 — C2×C4○D8
C1C2C2C4 — C2×C4○D8

Generators and relations for C2×C4○D8
 G = < a,b,c,d | a2=b4=d2=1, c4=b2, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=b2c3 >

Subgroups: 201 in 133 conjugacy classes, 81 normal (15 characteristic)
C1, C2, C2 [×2], C2 [×6], C4 [×2], C4 [×2], C4 [×4], C22, C22 [×2], C22 [×10], C8 [×4], C2×C4 [×2], C2×C4 [×4], C2×C4 [×10], D4 [×4], D4 [×10], Q8 [×4], Q8 [×2], C23, C23 [×2], C2×C8 [×2], C2×C8 [×4], D8 [×4], SD16 [×8], Q16 [×4], C22×C4, C22×C4 [×2], C2×D4 [×2], C2×D4 [×2], C2×Q8 [×2], C4○D4 [×8], C4○D4 [×4], C22×C8, C2×D8, C2×SD16 [×2], C2×Q16, C4○D8 [×8], C2×C4○D4 [×2], C2×C4○D8
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], C2×D4 [×6], C24, C4○D8 [×2], C22×D4, C2×C4○D8

Character table of C2×C4○D8

 class 12A2B2C2D2E2F2G2H2I4A4B4C4D4E4F4G4H4I4J8A8B8C8D8E8F8G8H
 size 1111224444111122444422222222
ρ11111111111111111111111111111    trivial
ρ21111-1-1-1-1-1-1-1-1-1-1111111-11-1-111-11    linear of order 2
ρ3111111-1-1-1-1111111-1-1-1-111111111    linear of order 2
ρ41-11-1-11-11-11-1-111-111-11-1-1-11-11-111    linear of order 2
ρ51-11-11-1-11-1111-1-1-11-11-111-1-111-1-11    linear of order 2
ρ61111-1-11111-1-1-1-111-1-1-1-1-11-1-111-11    linear of order 2
ρ71-11-11-11-11-111-1-1-111-11-11-1-111-1-11    linear of order 2
ρ81-11-1-111-11-1-1-111-11-11-11-1-11-11-111    linear of order 2
ρ91111-1-1-1-111-1-1-1-1111-1-111-111-1-11-1    linear of order 2
ρ10111111-1-111111111-111-1-1-1-1-1-1-1-1-1    linear of order 2
ρ111-11-1-11-111-1-1-111-1111-1-111-11-11-1-1    linear of order 2
ρ121-11-11-1-111-111-1-1-11-1-111-111-1-111-1    linear of order 2
ρ1311111111-1-11111111-1-11-1-1-1-1-1-1-1-1    linear of order 2
ρ141111-1-111-1-1-1-1-1-111-111-11-111-1-11-1    linear of order 2
ρ151-11-11-11-1-1111-1-1-1111-1-1-111-1-111-1    linear of order 2
ρ161-11-1-111-1-11-1-111-11-1-11111-11-11-1-1    linear of order 2
ρ172222-2-200002222-2-2000000000000    orthogonal lifted from D4
ρ182-22-2-22000022-2-22-2000000000000    orthogonal lifted from D4
ρ192222220000-2-2-2-2-2-2000000000000    orthogonal lifted from D4
ρ202-22-22-20000-2-2222-2000000000000    orthogonal lifted from D4
ρ2122-2-2000000-2i2i2i-2i000000-22-2--22-2--2-2    complex lifted from C4○D8
ρ2222-2-20000002i-2i-2i2i000000--22--2-22-2-2-2    complex lifted from C4○D8
ρ232-2-22000000-2i2i-2i2i000000--22-2-2-2-2--22    complex lifted from C4○D8
ρ242-2-220000002i-2i2i-2i000000-22--2--2-2-2-22    complex lifted from C4○D8
ρ2522-2-2000000-2i2i2i-2i000000--2-2--2-2-22-22    complex lifted from C4○D8
ρ2622-2-20000002i-2i-2i2i000000-2-2-2--2-22--22    complex lifted from C4○D8
ρ272-2-22000000-2i2i-2i2i000000-2-2--2--222-2-2    complex lifted from C4○D8
ρ282-2-220000002i-2i2i-2i000000--2-2-2-222--2-2    complex lifted from C4○D8

Smallest permutation representation of C2×C4○D8
On 32 points
Generators in S32
(1 31)(2 32)(3 25)(4 26)(5 27)(6 28)(7 29)(8 30)(9 19)(10 20)(11 21)(12 22)(13 23)(14 24)(15 17)(16 18)
(1 16 5 12)(2 9 6 13)(3 10 7 14)(4 11 8 15)(17 26 21 30)(18 27 22 31)(19 28 23 32)(20 29 24 25)
(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)
(2 8)(3 7)(4 6)(9 15)(10 14)(11 13)(17 19)(20 24)(21 23)(25 29)(26 28)(30 32)

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

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

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

C2×C4○D8 is a maximal subgroup of
M4(2).43D4  C42.326D4  C42.116D4  M4(2).30D4  C23.24D8  C23.39D8  C23.20SD16  C23.13D8  C23.21SD16  D8.10D4  C42.383D4  C42.280C23  C42.281C23  M4(2)○D8  (C2×D4)⋊21D4  (C2×Q8)⋊17D4  C42.18C23  C42.19C23  M4(2).20D4  D810D4  SD168D4  SD1610D4  D813D4  D16⋊C22  C8.C24
 C4○(C8pD4): C24.144D4  C24.110D4  C42.360D4  C42.247D4 ...
 C8pD4⋊C2: D88D4  (C2×C8)⋊13D4  (C2×C8)⋊14D4  M4(2)⋊10D4  M4(2)⋊11D4  SD167D4  Q1610D4  D812D4 ...
 C4○(D4.pD4): C24.103D4  C42.443D4  M4(2).10C23 ...
C2×C4○D8 is a maximal quotient of
C2×C4×D8  C2×C4×SD16  C2×C4×Q16  C24.103D4  C42.443D4  C24.144D4  C42.447D4  C24.115D4  C42.384D4  C42.225D4  C42.450D4  C42.451D4  C42.226D4  C42.355D4  C42.360D4  C42.364D4  C42.365D4  C42.308D4  C42.366D4  C42.367D4  C24.121D4  C24.123D4  C24.124D4  C42.265D4  C42.268D4  C42.269D4  C42.270D4  C42.280D4  C42.283D4  C42.284D4  C42.285D4  C42.295D4  C42.296D4  C42.298D4  D812D4  SD1610D4  D813D4  SD1611D4  Q1612D4  Q1613D4  C42.461C23  C42.462C23  C42.465C23  C42.466C23  C42.467C23  C42.468C23  C42.469C23  C42.470C23  C42.485C23  C42.486C23  C42.488C23  C42.489C23  C42.490C23  C42.491C23  C42.501C23  C42.502C23  C42.505C23  C42.506C23  D86Q8  SD164Q8  Q166Q8  C42.527C23  C42.528C23  C42.530C23

Matrix representation of C2×C4○D8 in GL3(𝔽17) generated by

1600
0160
0016
,
100
0130
0013
,
1600
0314
033
,
1600
010
0016
G:=sub<GL(3,GF(17))| [16,0,0,0,16,0,0,0,16],[1,0,0,0,13,0,0,0,13],[16,0,0,0,3,3,0,14,3],[16,0,0,0,1,0,0,0,16] >;

C2×C4○D8 in GAP, Magma, Sage, TeX

C_2\times C_4\circ D_8
% in TeX

G:=Group("C2xC4oD8");
// GroupNames label

G:=SmallGroup(64,253);
// by ID

G=gap.SmallGroup(64,253);
# by ID

G:=PCGroup([6,-2,2,2,2,-2,-2,217,158,1444,730,88]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^4=d^2=1,c^4=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^3>;
// generators/relations

Export

Character table of C2×C4○D8 in TeX

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