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G = C2×C4.10D4  order 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

Derived series
C1 — C22 — C2×C4.10D4
Chief series
C1 — C2 — C4 — C2×C4 — C22×C4 — C22×Q8 — C2×C4.10D4
Lower central
C1 — C2 — C22 — C2×C4.10D4
Upper central
C1 — C22 — C22×C4 — C2×C4.10D4
Jennings
C1 — C2 — C2 — C2×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, C2, C4, C4, C22, C22, C8, C2×C4, C2×C4, C2×C4, Q8, C23, C2×C8, M4(2), M4(2), C22×C4, C22×C4, C2×Q8, C2×Q8, C4.10D4, C2×M4(2), C22×Q8, C2×C4.10D4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22⋊C4, C22×C4, C2×D4, C4.10D4, 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 29)(2 30)(3 31)(4 32)(5 25)(6 26)(7 27)(8 28)(9 18)(10 19)(11 20)(12 21)(13 22)(14 23)(15 24)(16 17)

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

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

C2×C4.10D4 is a maximal subgroup of
 (C2×Q8).Q8  C4.C22≀C2  (C23×C4).C4  2- 1+4⋊2C4  C42.97D4  (C2×Q8).211D4  C4.10D4⋊2C4  C4⋊Q8⋊15C4  C4.4D4⋊13C4  M4(2).45D4  M4(2).46D4  M4(2).49D4  C42.7D4  M4(2).50D4  C4.10D4⋊3C4  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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