Copied to
clipboard

G = C42.304D4order 128 = 27

286th non-split extension by C42 of D4 acting via D4/C2=C22

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

Aliases: C42.304D4, C42.438C23, C4.122- 1+4, C4.312+ 1+4, C8⋊D422C2, C82D4.6C2, Q8.Q829C2, D4.Q829C2, C8.D416C2, C4⋊C8.84C22, (C2×C8).80C23, D4.2D430C2, C4⋊C4.195C23, (C2×C4).454C24, Q8.D430C2, (C2×D8).76C22, (C22×C4).531D4, C23.311(C2×D4), C4⋊M4(2)⋊14C2, C4.Q8.49C22, (C2×D4).195C23, (C4×D4).133C22, (C4×Q8).130C22, (C2×Q8).183C23, (C2×Q16).76C22, C2.D8.114C22, D4⋊C4.60C22, C4⋊D4.215C22, (C2×C42).911C22, Q8⋊C4.58C22, (C2×SD16).43C22, C22.714(C22×D4), C22⋊Q8.220C22, C2.74(D8⋊C22), (C22×C4).1109C23, C4.4D4.168C22, (C2×M4(2)).92C22, C42.C2.145C22, C23.36C2316C2, C2.73(C22.31C24), (C2×C4).578(C2×D4), SmallGroup(128,1988)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C42.304D4
C1C2C4C2×C4C42C4×D4C23.36C23 — C42.304D4
C1C2C2×C4 — C42.304D4
C1C22C2×C42 — C42.304D4
C1C2C2C2×C4 — C42.304D4

Generators and relations for C42.304D4
 G = < a,b,c,d | a4=b4=d2=1, c4=a2, ab=ba, cac-1=a-1, dad=a-1b2, cbc-1=dbd=a2b-1, dcd=a2c3 >

Subgroups: 340 in 177 conjugacy classes, 84 normal (24 characteristic)
C1, C2, C2 [×2], C2 [×3], C4 [×2], C4 [×11], C22, C22 [×9], C8 [×4], C2×C4 [×2], C2×C4 [×4], C2×C4 [×14], D4 [×8], Q8 [×4], C23, C23 [×2], C42 [×2], C42 [×2], C42 [×2], C22⋊C4 [×10], C4⋊C4 [×4], C4⋊C4 [×8], C2×C8 [×2], C2×C8 [×2], M4(2) [×2], D8, SD16 [×2], Q16, C22×C4, C22×C4 [×2], C22×C4 [×2], C2×D4 [×2], C2×D4 [×2], C2×Q8 [×2], D4⋊C4 [×4], Q8⋊C4 [×4], C4⋊C8 [×4], C4.Q8 [×2], C2.D8 [×2], C2×C42, C42⋊C2 [×2], C4×D4 [×2], C4×D4 [×2], C4×Q8 [×2], C4⋊D4 [×2], C22⋊Q8 [×2], C22.D4 [×2], C4.4D4 [×2], C42.C2 [×2], C422C2 [×2], C2×M4(2) [×2], C2×D8, C2×SD16 [×2], C2×Q16, C4⋊M4(2), D4.2D4 [×2], Q8.D4 [×2], C8⋊D4 [×2], C82D4, C8.D4, D4.Q8 [×2], Q8.Q8 [×2], C23.36C23 [×2], C42.304D4
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], C2×D4 [×6], C24, C22×D4, 2+ 1+4, 2- 1+4, C22.31C24, D8⋊C22 [×2], C42.304D4

Character table of C42.304D4

 class 12A2B2C2D2E2F4A4B4C4D4E4F4G4H4I4J4K4L4M4N4O8A8B8C8D
 size 11114882222224448888888888
ρ111111111111111111111111111    trivial
ρ21111-1-11-111-1111-1-11-1-1-11111-1-1    linear of order 2
ρ31111-1111-1111-1-11-1-11-1-1-11-111-1    linear of order 2
ρ411111-11-1-11-11-1-1-11-1-111-11-11-11    linear of order 2
ρ51111-11-11-1111-1-11-11-11-1-111-1-11    linear of order 2
ρ611111-1-1-1-11-11-1-1-1111-11-111-11-1    linear of order 2
ρ7111111-1111111111-1-1-1111-1-1-1-1    linear of order 2
ρ81111-1-1-1-111-1111-1-1-111-111-1-111    linear of order 2
ρ91111-1-1-11-1111-1-11-11-1111-1-111-1    linear of order 2
ρ10111111-1-1-11-11-1-1-1111-1-11-1-11-11    linear of order 2
ρ1111111-1-1111111111-1-1-1-1-1-11111    linear of order 2
ρ121111-11-1-111-1111-1-1-1111-1-111-1-1    linear of order 2
ρ1311111-11111111111111-1-1-1-1-1-1-1    linear of order 2
ρ141111-111-111-1111-1-11-1-11-1-1-1-111    linear of order 2
ρ151111-1-111-1111-1-11-1-11-111-11-1-11    linear of order 2
ρ161111111-1-11-11-1-1-11-1-11-11-11-11-1    linear of order 2
ρ172222-200-2-2-2-2-2-22220000000000    orthogonal lifted from D4
ρ182222200-22-2-2-22-22-20000000000    orthogonal lifted from D4
ρ1922222002-2-22-2-22-2-20000000000    orthogonal lifted from D4
ρ202222-20022-22-22-2-220000000000    orthogonal lifted from D4
ρ214-44-400000-40400000000000000    orthogonal lifted from 2+ 1+4
ρ224-44-40000040-400000000000000    symplectic lifted from 2- 1+4, Schur index 2
ρ234-4-440000-4i0004i0000000000000    complex lifted from D8⋊C22
ρ2444-4-4000-4i004i000000000000000    complex lifted from D8⋊C22
ρ2544-4-40004i00-4i000000000000000    complex lifted from D8⋊C22
ρ264-4-4400004i000-4i0000000000000    complex lifted from D8⋊C22

Smallest permutation representation of C42.304D4
On 64 points
Generators in S64
(1 63 5 59)(2 60 6 64)(3 57 7 61)(4 62 8 58)(9 40 13 36)(10 37 14 33)(11 34 15 38)(12 39 16 35)(17 26 21 30)(18 31 22 27)(19 28 23 32)(20 25 24 29)(41 50 45 54)(42 55 46 51)(43 52 47 56)(44 49 48 53)
(1 31 51 16)(2 13 52 28)(3 25 53 10)(4 15 54 30)(5 27 55 12)(6 9 56 32)(7 29 49 14)(8 11 50 26)(17 62 38 41)(18 46 39 59)(19 64 40 43)(20 48 33 61)(21 58 34 45)(22 42 35 63)(23 60 36 47)(24 44 37 57)
(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 8)(2 7)(3 6)(4 5)(9 14)(10 13)(11 12)(15 16)(17 39)(18 38)(19 37)(20 36)(21 35)(22 34)(23 33)(24 40)(25 28)(26 27)(29 32)(30 31)(41 63)(42 62)(43 61)(44 60)(45 59)(46 58)(47 57)(48 64)(49 52)(50 51)(53 56)(54 55)

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

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

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

Matrix representation of C42.304D4 in GL8(𝔽17)

001300000
000130000
130000000
013000000
00000010
00000001
000016000
000001600
,
00100000
00010000
10000000
01000000
00004000
00000400
00000040
00000004
,
005120000
00550000
125000000
1212000000
000016133
00001616143
000033116
000014311
,
005120000
0012120000
125000000
55000000
000016133
000011314
00001414161
000014311

G:=sub<GL(8,GF(17))| [0,0,13,0,0,0,0,0,0,0,0,13,0,0,0,0,13,0,0,0,0,0,0,0,0,13,0,0,0,0,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0],[0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4],[0,0,12,12,0,0,0,0,0,0,5,12,0,0,0,0,5,5,0,0,0,0,0,0,12,5,0,0,0,0,0,0,0,0,0,0,16,16,3,14,0,0,0,0,1,16,3,3,0,0,0,0,3,14,1,1,0,0,0,0,3,3,16,1],[0,0,12,5,0,0,0,0,0,0,5,5,0,0,0,0,5,12,0,0,0,0,0,0,12,12,0,0,0,0,0,0,0,0,0,0,16,1,14,14,0,0,0,0,1,1,14,3,0,0,0,0,3,3,16,1,0,0,0,0,3,14,1,1] >;

C42.304D4 in GAP, Magma, Sage, TeX

C_4^2._{304}D_4
% in TeX

G:=Group("C4^2.304D4");
// GroupNames label

G:=SmallGroup(128,1988);
// by ID

G=gap.SmallGroup(128,1988);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,456,758,891,675,304,4037,1027,124]);
// Polycyclic

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

Export

Character table of C42.304D4 in TeX

׿
×
𝔽