Copied to
clipboard

G = C42.30C22order 64 = 26

30th non-split extension by C42 of C22 acting faithfully

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

Aliases: C42.30C22, C4⋊Q8.8C2, (C2×C4).43D4, C8⋊C4.6C2, C4.18(C4○D4), C4⋊C4.21C22, (C2×C8).56C22, Q8⋊C4.7C2, C42.C2.3C2, (C2×C4).116C23, C22.112(C2×D4), (C2×Q8).23C22, C2.14(C4.4D4), C2.21(C8.C22), SmallGroup(64,172)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C42.30C22
C1C2C4C2×C4C2×C8C8⋊C4 — C42.30C22
C1C2C2×C4 — C42.30C22
C1C22C42 — C42.30C22
C1C2C2C2×C4 — C42.30C22

Generators and relations for C42.30C22
 G = < a,b,c,d | a4=b4=1, c2=b2, d2=b, ab=ba, cac-1=a-1, dad-1=ab2, cbc-1=b-1, bd=db, dcd-1=a2b-1c >

2C4
2C4
4C4
4C4
4C4
4C4
2C2×C4
2Q8
2Q8
2Q8
2C2×C4
2Q8
2C8
2C2×C4
2C8
2C2×C4
2C4⋊C4
2C4⋊C4
2C4⋊C4
2C4⋊C4

Character table of C42.30C22

 class 12A2B2C4A4B4C4D4E4F4G4H8A8B8C8D
 size 1111224488884444
ρ11111111111111111    trivial
ρ211111111-11-11-1-1-1-1    linear of order 2
ρ3111111-1-111-1-11-11-1    linear of order 2
ρ4111111-1-1-111-1-11-11    linear of order 2
ρ5111111-1-1-1-1111-11-1    linear of order 2
ρ6111111-1-11-1-11-11-11    linear of order 2
ρ711111111-1-1-1-11111    linear of order 2
ρ8111111111-11-1-1-1-1-1    linear of order 2
ρ92222-2-2-2200000000    orthogonal lifted from D4
ρ102222-2-22-200000000    orthogonal lifted from D4
ρ112-22-2-22000000-2i02i0    complex lifted from C4○D4
ρ122-22-22-200000002i0-2i    complex lifted from C4○D4
ρ132-22-22-20000000-2i02i    complex lifted from C4○D4
ρ142-22-2-220000002i0-2i0    complex lifted from C4○D4
ρ154-4-44000000000000    symplectic lifted from C8.C22, Schur index 2
ρ1644-4-4000000000000    symplectic lifted from C8.C22, Schur index 2

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

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

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

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

C42.30C22 is a maximal subgroup of
C42.(C2×C4)  C8⋊C4.C4
 C42.D2p: C42.239D4  C42.241D4  C42.244D4  C42.256D4  C42.258D4  C42.276D4  C42.277D4  C42.288D4 ...
 C4⋊C4.D2p: C42.8C23  C42.9C23  C42.10C23  C42.367C23  C42.386C23  C42.389C23  C42.407C23  C42.409C23 ...
C42.30C22 is a maximal quotient of
C42.24Q8  C2.(C8⋊D4)  (C2×Q8).109D4  (C2×C4).28D8
 C42.D2p: C42.111D4  C42.124D4  C42.14D6  C42.71D6  C42.77D6  C42.14D10  C42.71D10  C42.77D10 ...
 C4⋊C4.D2p: C4⋊C4.85D4  C4⋊C4.95D4  (C2×Q8).36D6  C408C4.C2  C56⋊C4.C2 ...

Matrix representation of C42.30C22 in GL6(𝔽17)

0130000
1300000
0016000
0001600
000010
000001
,
100000
010000
0001600
001000
0000016
000010
,
100000
0160000
0071600
00161000
000017
0000716
,
010000
100000
000001
0000160
0016000
0001600

G:=sub<GL(6,GF(17))| [0,13,0,0,0,0,13,0,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,16,0,0,0,0,0,0,0,0,1,0,0,0,0,16,0],[1,0,0,0,0,0,0,16,0,0,0,0,0,0,7,16,0,0,0,0,16,10,0,0,0,0,0,0,1,7,0,0,0,0,7,16],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,16,0,0,0,0,1,0,0,0] >;

C42.30C22 in GAP, Magma, Sage, TeX

C_4^2._{30}C_2^2
% in TeX

G:=Group("C4^2.30C2^2");
// GroupNames label

G:=SmallGroup(64,172);
// by ID

G=gap.SmallGroup(64,172);
# by ID

G:=PCGroup([6,-2,2,2,-2,2,-2,192,121,295,362,332,50,963,117,1444,88]);
// Polycyclic

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

Export

Subgroup lattice of C42.30C22 in TeX
Character table of C42.30C22 in TeX

׿
×
𝔽