Copied to
clipboard

G = C42.32D4order 128 = 27

14th non-split extension by C42 of D4 acting via D4/C2=C22

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

Aliases: C42.32D4, C42.13Q8, C23.11C42, (C2×M4(2)).4C4, C4.12(C4.D4), C4.12(C4.10D4), C4⋊M4(2).10C2, (C2×C42).141C22, C2.11(C22.C42), C22.59(C2.C42), (C2×C4).27(C4⋊C4), (C22×C4).163(C2×C4), (C2×C4).341(C22⋊C4), SmallGroup(128,41)

Series: Derived Chief Lower central Upper central Jennings

C1C23 — C42.32D4
C1C2C22C2×C4C42C2×C42C4⋊M4(2) — C42.32D4
C1C22C23 — C42.32D4
C1C22C2×C42 — C42.32D4
C1C22C22C2×C42 — C42.32D4

Generators and relations for C42.32D4
 G = < a,b,c,d | a4=b4=1, c4=a2b2, d2=a, ab=ba, cac-1=a-1, ad=da, cbc-1=dbd-1=b-1, dcd-1=a-1b-1c3 >

Subgroups: 136 in 73 conjugacy classes, 36 normal (6 characteristic)
C1, C2 [×3], C2, C4 [×6], C4 [×3], C22, C22 [×3], C8 [×6], C2×C4 [×6], C2×C4 [×6], C23, C42, C42 [×3], C2×C8 [×6], M4(2) [×6], C22×C4 [×3], C4⋊C8 [×6], C2×C42, C2×M4(2) [×6], C4⋊M4(2) [×3], C42.32D4
Quotients: C1, C2 [×3], C4 [×6], C22, C2×C4 [×3], D4 [×3], Q8, C42, C22⋊C4 [×3], C4⋊C4 [×3], C2.C42, C4.D4 [×3], C4.10D4 [×3], C22.C42 [×3], C42.32D4

Character table of C42.32D4

 class 12A2B2C2D4A4B4C4D4E4F4G4H4I8A8B8C8D8E8F8G8H8I8J8K8L
 size 11114222222444888888888888
ρ111111111111111111111111111    trivial
ρ2111111111111111-1-1-1-1-1-11-11-11    linear of order 2
ρ311111111111111-111-1-1-11-1-1-11-1    linear of order 2
ρ411111111111111-1-1-1111-1-11-1-1-1    linear of order 2
ρ511111-11-11-1-1-1-111-i-ii-i-ii-1i-1i1    linear of order 4
ρ6111111-11-1-1-1-11-1-ii-i-1-11-i-i1iii    linear of order 4
ρ711111-11-11-1-1-1-11-1-i-i-iiii1-i1i-1    linear of order 4
ρ811111-1-1-1-1111-1-1-i-11i-ii-1i-i-i1i    linear of order 4
ρ9111111-11-1-1-1-11-1-i-ii11-1i-i-1i-ii    linear of order 4
ρ1011111-1-1-1-1111-1-1i-11-ii-i-1-iii1-i    linear of order 4
ρ1111111-1-1-1-1111-1-1i1-1i-ii1-i-ii-1-i    linear of order 4
ρ1211111-1-1-1-1111-1-1-i1-1-ii-i1ii-i-1i    linear of order 4
ρ13111111-11-1-1-1-11-1ii-i11-1-ii-1-ii-i    linear of order 4
ρ1411111-11-11-1-1-1-11-1iii-i-i-i1i1-i-1    linear of order 4
ρ15111111-11-1-1-1-11-1i-ii-1-11ii1-i-i-i    linear of order 4
ρ1611111-11-11-1-1-1-111ii-iii-i-1-i-1-i1    linear of order 4
ρ172222-22-22-222-2-22000000000000    orthogonal lifted from D4
ρ182222-22222-2-22-2-2000000000000    orthogonal lifted from D4
ρ192222-2-22-2222-22-2000000000000    orthogonal lifted from D4
ρ202222-2-2-2-2-2-2-2222000000000000    symplectic lifted from Q8, Schur index 2
ρ214-4-44040-4000000000000000000    orthogonal lifted from C4.D4
ρ224-44-400000-44000000000000000    orthogonal lifted from C4.D4
ρ2344-4-40040-400000000000000000    orthogonal lifted from C4.D4
ρ2444-4-400-40400000000000000000    symplectic lifted from C4.10D4, Schur index 2
ρ254-44-4000004-4000000000000000    symplectic lifted from C4.10D4, Schur index 2
ρ264-4-440-404000000000000000000    symplectic lifted from C4.10D4, Schur index 2

Smallest permutation representation of C42.32D4
On 64 points
Generators in S64
(1 20 58 10)(2 11 59 21)(3 22 60 12)(4 13 61 23)(5 24 62 14)(6 15 63 17)(7 18 64 16)(8 9 57 19)(25 56 44 40)(26 33 45 49)(27 50 46 34)(28 35 47 51)(29 52 48 36)(30 37 41 53)(31 54 42 38)(32 39 43 55)
(1 18 62 12)(2 13 63 19)(3 20 64 14)(4 15 57 21)(5 22 58 16)(6 9 59 23)(7 24 60 10)(8 11 61 17)(25 38 48 50)(26 51 41 39)(27 40 42 52)(28 53 43 33)(29 34 44 54)(30 55 45 35)(31 36 46 56)(32 49 47 37)
(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 27 20 50 58 46 10 34)(2 43 11 55 59 32 21 39)(3 25 22 56 60 44 12 40)(4 41 13 53 61 30 23 37)(5 31 24 54 62 42 14 38)(6 47 15 51 63 28 17 35)(7 29 18 52 64 48 16 36)(8 45 9 49 57 26 19 33)

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

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

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

Matrix representation of C42.32D4 in GL8(𝔽17)

016000000
10000000
06010000
1101600000
00001000
00000100
00000010
00000001
,
01000000
160000000
100010000
071600000
00000100
000016000
00000001
000000160
,
125000000
55000000
1445120000
4512120000
00000001
00000010
00001000
000001600
,
55770000
3147100000
15612140000
671230000
00000100
00001000
00000001
00000010

G:=sub<GL(8,GF(17))| [0,1,0,11,0,0,0,0,16,0,6,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[0,16,10,0,0,0,0,0,1,0,0,7,0,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0],[12,5,14,4,0,0,0,0,5,5,4,5,0,0,0,0,0,0,5,12,0,0,0,0,0,0,12,12,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0],[5,3,15,6,0,0,0,0,5,14,6,7,0,0,0,0,7,7,12,12,0,0,0,0,7,10,14,3,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0] >;

C42.32D4 in GAP, Magma, Sage, TeX

C_4^2._{32}D_4
% in TeX

G:=Group("C4^2.32D4");
// GroupNames label

G:=SmallGroup(128,41);
// by ID

G=gap.SmallGroup(128,41);
# by ID

G:=PCGroup([7,-2,2,-2,2,2,-2,2,56,85,120,758,723,570,136,2804,172]);
// Polycyclic

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

Export

Character table of C42.32D4 in TeX

׿
×
𝔽