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G = C2xC4.10D4  order 64 = 26

Direct product of C2 and C4.10D4

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

Aliases: C2xC4.10D4, M4(2).7C22, C4.45(C2xD4), (C2xQ8).6C4, (C2xC4).121D4, (C22xC4).5C4, (C2xC4).2C23, C23.30(C2xC4), (C22xQ8).3C2, C4.11(C22:C4), C22.9(C22xC4), (C2xQ8).38C22, (C2xM4(2)).11C2, (C22xC4).32C22, C22.31(C22:C4), (C2xC4).6(C2xC4), C2.15(C2xC22:C4), SmallGroup(64,93)

Series: Derived ►Chief ►Lower central ►Upper central ►Jennings

Derived series
C1 — C22 — C2xC4.10D4
Chief series
C1 — C2 — C4 — C2xC4 — C22xC4 — C22xQ8 — C2xC4.10D4
Lower central
C1 — C2 — C22 — C2xC4.10D4
Upper central
C1 — C22 — C22xC4 — C2xC4.10D4
Jennings
C1 — C2 — C2 — C2xC4 — C2xC4.10D4

Generators and relations for C2xC4.10D4
 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, C2xC4, C2xC4, C2xC4, Q8, C23, C2xC8, M4(2), M4(2), C22xC4, C22xC4, C2xQ8, C2xQ8, C4.10D4, C2xM4(2), C22xQ8, C2xC4.10D4
Quotients: C1, C2, C4, C22, C2xC4, D4, C23, C22:C4, C22xC4, C2xD4, C4.10D4, C2xC22:C4, C2xC4.10D4

Character table of C2xC4.10D4

 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 C2xC4.10D4
►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)]])
 

C2xC4.10D4 is a maximal subgroup of
 (C2xQ8).Q8  C4.C22wrC2  (C23xC4).C4  2- 1+4:2C4  C42.97D4  (C2xQ8).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  (C2xC8).6D4  C4:Q8.C4  M4(2).25C23  M4(2).C23  M4(2).38D4  (C2xQ8).7F5
C2xC4.10D4 is a maximal quotient of
 C42.394D4  (C2xC4):M4(2)  C42.44D4  C42.396D4  C24.45(C2xC4)  C42.406D4  C42.408D4  C42.68D4  C42.71D4  C42.74D4  C42.412D4  C42.414D4  C42.416D4  C42.81D4  C42.83D4  C42.88D4  C4.C22wrC2  C42.97D4  (C22xC4).275D4  M4(2).45D4  C42.114D4  M4(2):8Q8  (C2xQ8).7F5

Matrix representation of C2xC4.10D4 ►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] >;
 

C2xC4.10D4 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 C2xC4.10D4 in TeX

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