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G = C2×C4.10D4order 64 = 26

Direct product of C2 and C4.10D4

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

Aliases: C2×C4.10D4, M4(2).7C22, C4.45(C2×D4), (C2×Q8).6C4, (C2×C4).121D4, (C22×C4).5C4, (C2×C4).2C23, C23.30(C2×C4), (C22×Q8).3C2, C4.11(C22⋊C4), C22.9(C22×C4), (C2×Q8).38C22, (C2×M4(2)).11C2, (C22×C4).32C22, C22.31(C22⋊C4), (C2×C4).6(C2×C4), C2.15(C2×C22⋊C4), SmallGroup(64,93)

Series: Derived Chief Lower central Upper central Jennings

C1C22 — C2×C4.10D4
C1C2C4C2×C4C22×C4C22×Q8 — C2×C4.10D4
C1C2C22 — C2×C4.10D4
C1C22C22×C4 — C2×C4.10D4
C1C2C2C2×C4 — C2×C4.10D4

Generators and relations for C2×C4.10D4
 G = < a,b,c,d | a2=b4=1, c4=b2, d2=cbc-1=b-1, ab=ba, ac=ca, ad=da, bd=db, dcd-1=b-1c3 >

Subgroups: 105 in 73 conjugacy classes, 41 normal (11 characteristic)
C1, C2, C2 [×2], C2 [×2], C4 [×4], C4 [×4], C22 [×3], C22 [×2], C8 [×4], C2×C4 [×2], C2×C4 [×8], C2×C4 [×4], Q8 [×8], C23, C2×C8 [×2], M4(2) [×4], M4(2) [×2], C22×C4, C22×C4 [×2], C2×Q8 [×4], C2×Q8 [×4], C4.10D4 [×4], C2×M4(2) [×2], C22×Q8, C2×C4.10D4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×4], C23, C22⋊C4 [×4], C22×C4, C2×D4 [×2], C4.10D4 [×2], C2×C22⋊C4, C2×C4.10D4

Character table of C2×C4.10D4

 class 12A2B2C2D2E4A4B4C4D4E4F4G4H8A8B8C8D8E8F8G8H
 size 1111222222444444444444
ρ11111111111111111111111    trivial
ρ21-11-1-111-1-11-11-11-111-11-1-11    linear of order 2
ρ31111111111-1-1-1-1-11-1-1111-1    linear of order 2
ρ41-11-1-111-1-111-11-111-111-1-1-1    linear of order 2
ρ51-11-1-111-1-111-11-1-1-11-1-1111    linear of order 2
ρ61111111111-1-1-1-11-111-1-1-11    linear of order 2
ρ71-11-1-111-1-11-11-111-1-11-111-1    linear of order 2
ρ811111111111111-1-1-1-1-1-1-1-1    linear of order 2
ρ91-11-1-11-111-1-1-111i-ii-ii-ii-i    linear of order 4
ρ101-11-1-11-111-111-1-1-i-i-iii-iii    linear of order 4
ρ11111111-1-1-1-11-1-11-i-iiiii-i-i    linear of order 4
ρ12111111-1-1-1-1-111-1i-i-i-iii-ii    linear of order 4
ρ131-11-1-11-111-1-1-111-ii-ii-ii-ii    linear of order 4
ρ141-11-1-11-111-111-1-1iii-i-ii-i-i    linear of order 4
ρ15111111-1-1-1-11-1-11ii-i-i-i-iii    linear of order 4
ρ16111111-1-1-1-1-111-1-iiii-i-ii-i    linear of order 4
ρ172-22-22-2-22-22000000000000    orthogonal lifted from D4
ρ182222-2-2-2-222000000000000    orthogonal lifted from D4
ρ192-22-22-22-22-2000000000000    orthogonal lifted from D4
ρ202222-2-222-2-2000000000000    orthogonal lifted from D4
ρ2144-4-4000000000000000000    symplectic lifted from C4.10D4, Schur index 2
ρ224-4-44000000000000000000    symplectic lifted from C4.10D4, Schur index 2

Smallest permutation representation of C2×C4.10D4
On 32 points
Generators in S32
(1 15)(2 16)(3 9)(4 10)(5 11)(6 12)(7 13)(8 14)(17 25)(18 26)(19 27)(20 28)(21 29)(22 30)(23 31)(24 32)
(1 13 5 9)(2 10 6 14)(3 15 7 11)(4 12 8 16)(17 31 21 27)(18 28 22 32)(19 25 23 29)(20 30 24 26)
(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 9 26 5 24 13 30)(2 25 14 19 6 29 10 23)(3 18 11 32 7 22 15 28)(4 31 16 17 8 27 12 21)

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

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

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

C2×C4.10D4 is a maximal subgroup of
(C2×Q8).Q8  C4.C22≀C2  (C23×C4).C4  2- 1+42C4  C42.97D4  (C2×Q8).211D4  C4.10D42C4  C4⋊Q815C4  C4.4D413C4  M4(2).45D4  M4(2).46D4  M4(2).49D4  C42.7D4  M4(2).50D4  C4.10D43C4  C42.114D4  C42.115D4  M4(2).31D4  M4(2).33D4  C42.129D4  C42.130D4  M4(2).5D4  M4(2).6D4  M4(2).7D4  C4⋊C4.97D4  C4⋊C4.98D4  M4(2).9D4  M4(2).11D4  M4(2).13D4  M4(2).15D4  (C2×C8).6D4  C4⋊Q8.C4  M4(2).25C23  M4(2).C23  M4(2).38D4  (C2×Q8).7F5
C2×C4.10D4 is a maximal quotient of
C42.394D4  (C2×C4)⋊M4(2)  C42.44D4  C42.396D4  C24.45(C2×C4)  C42.406D4  C42.408D4  C42.68D4  C42.71D4  C42.74D4  C42.412D4  C42.414D4  C42.416D4  C42.81D4  C42.83D4  C42.88D4  C4.C22≀C2  C42.97D4  (C22×C4).275D4  M4(2).45D4  C42.114D4  M4(2)⋊8Q8  (C2×Q8).7F5

Matrix representation of C2×C4.10D4 in GL6(𝔽17)

1600000
0160000
001000
000100
000010
000001
,
1600000
0160000
0016200
0016100
0016101
0001160
,
1600000
010000
0016020
0000116
0001610
0016010
,
0160000
100000
0080315
0070119
007101016
008161016

G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,16,16,0,0,0,2,1,1,1,0,0,0,0,0,16,0,0,0,0,1,0],[16,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,16,0,0,0,0,16,0,0,0,2,1,1,1,0,0,0,16,0,0],[0,1,0,0,0,0,16,0,0,0,0,0,0,0,8,7,7,8,0,0,0,0,10,16,0,0,3,11,10,10,0,0,15,9,16,16] >;

C2×C4.10D4 in GAP, Magma, Sage, TeX

C_2\times C_4._{10}D_4
% in TeX

G:=Group("C2xC4.10D4");
// GroupNames label

G:=SmallGroup(64,93);
// by ID

G=gap.SmallGroup(64,93);
# by ID

G:=PCGroup([6,-2,2,2,-2,2,-2,96,121,199,963,730,88]);
// Polycyclic

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

Export

Character table of C2×C4.10D4 in TeX

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