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G = C42.2C22order 64 = 26

2nd non-split extension by C42 of C22 acting faithfully

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

Aliases: C42.2C22, C2.7C4≀C2, C4⋊C4.1C4, (C2×C4).97D4, C8⋊C4.3C2, C42.C2.1C2, C2.3(C4.10D4), C22.38(C22⋊C4), (C2×C4).11(C2×C4), SmallGroup(64,11)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C42.2C22
C1C2C22C2×C4C42C42.C2 — C42.2C22
C1C22C2×C4 — C42.2C22
C1C22C42 — C42.2C22
C1C22C22C42 — C42.2C22

Generators and relations for C42.2C22
 G = < a,b,c,d | a4=b4=1, c2=b, d2=b2, ab=ba, cac-1=ab2, dad-1=a-1, bc=cb, dbd-1=a2b-1, dcd-1=a-1b2c >

2C4
2C4
2C4
4C4
4C4
2C8
2C2×C4
2C8
2C8
2C8
2C2×C4
2C2×C8
2C4⋊C4
2C2×C8
2C4⋊C4

Character table of C42.2C22

 class 12A2B2C4A4B4C4D4E4F4G8A8B8C8D8E8F8G8H
 size 1111222248844444444
ρ11111111111111111111    trivial
ρ211111111111-1-1-1-1-1-1-1-1    linear of order 2
ρ3111111111-1-11-1-111-1-11    linear of order 2
ρ4111111111-1-1-111-1-111-1    linear of order 2
ρ51111-1-1-1-11-11-i-iii-ii-ii    linear of order 4
ρ61111-1-1-1-11-11ii-i-ii-ii-i    linear of order 4
ρ71111-1-1-1-111-1-ii-ii-i-iii    linear of order 4
ρ81111-1-1-1-111-1i-ii-iii-i-i    linear of order 4
ρ92222-222-2-20000000000    orthogonal lifted from D4
ρ1022222-2-22-20000000000    orthogonal lifted from D4
ρ112-22-22i00-2i0000-1+i1+i00-1-i1-i0    complex lifted from C4≀C2
ρ122-22-2-2i002i00001+i-1+i001-i-1-i0    complex lifted from C4≀C2
ρ132-2-2202i-2i0000-1-i00-1+i1+i001-i    complex lifted from C4≀C2
ρ142-2-220-2i2i0000-1+i00-1-i1-i001+i    complex lifted from C4≀C2
ρ152-22-2-2i002i0000-1-i1-i00-1+i1+i0    complex lifted from C4≀C2
ρ162-2-220-2i2i00001-i001+i-1+i00-1-i    complex lifted from C4≀C2
ρ172-2-2202i-2i00001+i001-i-1-i00-1+i    complex lifted from C4≀C2
ρ182-22-22i00-2i00001-i-1-i001+i-1+i0    complex lifted from C4≀C2
ρ1944-4-4000000000000000    symplectic lifted from C4.10D4, Schur index 2

Smallest permutation representation of C42.2C22
Regular action on 64 points
Generators in S64
(1 31 55 47)(2 28 56 44)(3 25 49 41)(4 30 50 46)(5 27 51 43)(6 32 52 48)(7 29 53 45)(8 26 54 42)(9 57 40 17)(10 62 33 22)(11 59 34 19)(12 64 35 24)(13 61 36 21)(14 58 37 18)(15 63 38 23)(16 60 39 20)
(1 3 5 7)(2 4 6 8)(9 11 13 15)(10 12 14 16)(17 19 21 23)(18 20 22 24)(25 27 29 31)(26 28 30 32)(33 35 37 39)(34 36 38 40)(41 43 45 47)(42 44 46 48)(49 51 53 55)(50 52 54 56)(57 59 61 63)(58 60 62 64)
(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)(33 34 35 36 37 38 39 40)(41 42 43 44 45 46 47 48)(49 50 51 52 53 54 55 56)(57 58 59 60 61 62 63 64)
(1 63 5 59)(2 12 6 16)(3 21 7 17)(4 33 8 37)(9 41 13 45)(10 54 14 50)(11 31 15 27)(18 46 22 42)(19 55 23 51)(20 28 24 32)(25 36 29 40)(26 58 30 62)(34 47 38 43)(35 52 39 56)(44 64 48 60)(49 61 53 57)

G:=sub<Sym(64)| (1,31,55,47)(2,28,56,44)(3,25,49,41)(4,30,50,46)(5,27,51,43)(6,32,52,48)(7,29,53,45)(8,26,54,42)(9,57,40,17)(10,62,33,22)(11,59,34,19)(12,64,35,24)(13,61,36,21)(14,58,37,18)(15,63,38,23)(16,60,39,20), (1,3,5,7)(2,4,6,8)(9,11,13,15)(10,12,14,16)(17,19,21,23)(18,20,22,24)(25,27,29,31)(26,28,30,32)(33,35,37,39)(34,36,38,40)(41,43,45,47)(42,44,46,48)(49,51,53,55)(50,52,54,56)(57,59,61,63)(58,60,62,64), (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)(33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56)(57,58,59,60,61,62,63,64), (1,63,5,59)(2,12,6,16)(3,21,7,17)(4,33,8,37)(9,41,13,45)(10,54,14,50)(11,31,15,27)(18,46,22,42)(19,55,23,51)(20,28,24,32)(25,36,29,40)(26,58,30,62)(34,47,38,43)(35,52,39,56)(44,64,48,60)(49,61,53,57)>;

G:=Group( (1,31,55,47)(2,28,56,44)(3,25,49,41)(4,30,50,46)(5,27,51,43)(6,32,52,48)(7,29,53,45)(8,26,54,42)(9,57,40,17)(10,62,33,22)(11,59,34,19)(12,64,35,24)(13,61,36,21)(14,58,37,18)(15,63,38,23)(16,60,39,20), (1,3,5,7)(2,4,6,8)(9,11,13,15)(10,12,14,16)(17,19,21,23)(18,20,22,24)(25,27,29,31)(26,28,30,32)(33,35,37,39)(34,36,38,40)(41,43,45,47)(42,44,46,48)(49,51,53,55)(50,52,54,56)(57,59,61,63)(58,60,62,64), (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)(33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56)(57,58,59,60,61,62,63,64), (1,63,5,59)(2,12,6,16)(3,21,7,17)(4,33,8,37)(9,41,13,45)(10,54,14,50)(11,31,15,27)(18,46,22,42)(19,55,23,51)(20,28,24,32)(25,36,29,40)(26,58,30,62)(34,47,38,43)(35,52,39,56)(44,64,48,60)(49,61,53,57) );

G=PermutationGroup([(1,31,55,47),(2,28,56,44),(3,25,49,41),(4,30,50,46),(5,27,51,43),(6,32,52,48),(7,29,53,45),(8,26,54,42),(9,57,40,17),(10,62,33,22),(11,59,34,19),(12,64,35,24),(13,61,36,21),(14,58,37,18),(15,63,38,23),(16,60,39,20)], [(1,3,5,7),(2,4,6,8),(9,11,13,15),(10,12,14,16),(17,19,21,23),(18,20,22,24),(25,27,29,31),(26,28,30,32),(33,35,37,39),(34,36,38,40),(41,43,45,47),(42,44,46,48),(49,51,53,55),(50,52,54,56),(57,59,61,63),(58,60,62,64)], [(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),(33,34,35,36,37,38,39,40),(41,42,43,44,45,46,47,48),(49,50,51,52,53,54,55,56),(57,58,59,60,61,62,63,64)], [(1,63,5,59),(2,12,6,16),(3,21,7,17),(4,33,8,37),(9,41,13,45),(10,54,14,50),(11,31,15,27),(18,46,22,42),(19,55,23,51),(20,28,24,32),(25,36,29,40),(26,58,30,62),(34,47,38,43),(35,52,39,56),(44,64,48,60),(49,61,53,57)])

C42.2C22 is a maximal subgroup of
C42.2C23  C42.3C23  C42.4C23  C42.6C23  C42.7C23  C42.8C23  C42.9C23  C42.10C23  C10.C4≀C2
 C42.D2p: C42.66D4  C42.406D4  C42.408D4  C42.376D4  C42.68D4  C42.69D4  C42.71D4  C42.72D4 ...
C42.2C22 is a maximal quotient of
C4⋊C4⋊C8  C10.C4≀C2
 C42.D2p: C42.7Q8  C42.2D6  C42.8D6  C42.2D10  C42.8D10  C42.2D14  C42.8D14 ...

Matrix representation of C42.2C22 in GL4(𝔽17) generated by

01300
13000
0001
00160
,
0100
1000
0040
0004
,
61000
10600
0002
0020
,
41000
71300
00512
001212
G:=sub<GL(4,GF(17))| [0,13,0,0,13,0,0,0,0,0,0,16,0,0,1,0],[0,1,0,0,1,0,0,0,0,0,4,0,0,0,0,4],[6,10,0,0,10,6,0,0,0,0,0,2,0,0,2,0],[4,7,0,0,10,13,0,0,0,0,5,12,0,0,12,12] >;

C42.2C22 in GAP, Magma, Sage, TeX

C_4^2._2C_2^2
% in TeX

G:=Group("C4^2.2C2^2");
// GroupNames label

G:=SmallGroup(64,11);
// by ID

G=gap.SmallGroup(64,11);
# by ID

G:=PCGroup([6,-2,2,-2,2,-2,2,48,73,103,362,332,158,681,69]);
// Polycyclic

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

Export

Subgroup lattice of C42.2C22 in TeX
Character table of C42.2C22 in TeX

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