Copied to
clipboard

G = C4234D4order 128 = 27

28th semidirect product of C42 and D4 acting via D4/C2=C22

p-group, metabelian, nilpotent (class 2), monomial, rational

Aliases: C4234D4, C23.719C24, C24.102C23, C22.4922+ 1+4, C22.3752- 1+4, C428C472C2, C232D449C2, C23⋊Q862C2, (C2×C42).731C22, (C22×C4).230C23, C22.451(C22×D4), C23.10D4110C2, (C22×D4).296C22, (C22×Q8).234C22, C2.71(C22.29C24), C2.16(C24⋊C22), C2.C42.422C22, C2.51(C22.56C24), C2.58(C22.31C24), (C2×C4).436(C2×D4), (C2×C4.4D4)⋊33C2, (C2×C4⋊C4).528C22, (C2×C22⋊C4).338C22, SmallGroup(128,1551)

Series: Derived Chief Lower central Upper central Jennings

C1C23 — C4234D4
C1C2C22C23C22×C4C22×D4C232D4 — C4234D4
C1C23 — C4234D4
C1C23 — C4234D4
C1C23 — C4234D4

Generators and relations for C4234D4
 G = < a,b,c,d | a4=b4=c4=d2=1, ab=ba, cac-1=a-1, dad=ab2, cbc-1=a2b-1, dbd=a2b, dcd=c-1 >

Subgroups: 708 in 290 conjugacy classes, 92 normal (10 characteristic)
C1, C2, C2 [×6], C2 [×4], C4 [×14], C22 [×3], C22 [×4], C22 [×28], C2×C4 [×6], C2×C4 [×30], D4 [×24], Q8 [×8], C23, C23 [×28], C42 [×4], C22⋊C4 [×24], C4⋊C4 [×2], C22×C4, C22×C4 [×10], C2×D4 [×20], C2×Q8 [×8], C24 [×4], C2.C42 [×8], C2×C42, C2×C22⋊C4 [×16], C2×C4⋊C4 [×2], C4.4D4 [×8], C22×D4 [×6], C22×Q8 [×2], C428C4, C232D4 [×4], C23⋊Q8 [×4], C23.10D4 [×4], C2×C4.4D4 [×2], C4234D4
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], C2×D4 [×6], C24, C22×D4, 2+ 1+4 [×5], 2- 1+4, C22.29C24 [×2], C22.31C24, C24⋊C22 [×2], C22.56C24 [×2], C4234D4

Character table of C4234D4

 class 12A2B2C2D2E2F2G2H2I2J2K4A4B4C4D4E4F4G4H4I4J4K4L4M4N
 size 11111111888844444488888888
ρ111111111111111111111111111    trivial
ρ211111111-11-11-1-11-1-111-11-11-1-11    linear of order 2
ρ311111111-1-1-111-1-11-1-1-1-111-1111    linear of order 2
ρ4111111111-111-11-1-11-1-111-1-1-1-11    linear of order 2
ρ511111111-111-1-1-11-1-11-11-11-1-111    linear of order 2
ρ61111111111-1-1111111-1-1-1-1-11-11    linear of order 2
ρ7111111111-1-1-1-11-1-11-11-1-111-111    linear of order 2
ρ811111111-1-11-11-1-11-1-111-1-111-11    linear of order 2
ρ911111111-1-1-1-1111111111-1-1-11-1    linear of order 2
ρ10111111111-11-1-1-11-1-111-111-11-1-1    linear of order 2
ρ1111111111111-11-1-11-1-1-1-11-11-11-1    linear of order 2
ρ1211111111-11-1-1-11-1-11-1-111111-1-1    linear of order 2
ρ13111111111-1-11-1-11-1-11-11-1-1111-1    linear of order 2
ρ1411111111-1-111111111-1-1-111-1-1-1    linear of order 2
ρ1511111111-1111-11-1-11-11-1-1-1-111-1    linear of order 2
ρ161111111111-111-1-11-1-111-11-1-1-1-1    linear of order 2
ρ172-2-22-22-2200002-22-22-200000000    orthogonal lifted from D4
ρ182-2-22-22-22000022-2-2-2200000000    orthogonal lifted from D4
ρ192-2-22-22-220000-2222-2-200000000    orthogonal lifted from D4
ρ202-2-22-22-220000-2-2-222200000000    orthogonal lifted from D4
ρ2144-4-444-4-4000000000000000000    orthogonal lifted from 2+ 1+4
ρ224-44-4-444-4000000000000000000    orthogonal lifted from 2+ 1+4
ρ2344-44-4-44-4000000000000000000    orthogonal lifted from 2+ 1+4
ρ24444-4-4-4-44000000000000000000    orthogonal lifted from 2+ 1+4
ρ254-4444-4-4-4000000000000000000    orthogonal lifted from 2+ 1+4
ρ264-4-4-44-444000000000000000000    symplectic lifted from 2- 1+4, Schur index 2

Smallest permutation representation of C4234D4
On 64 points
Generators in S64
(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 15 23 29)(2 16 24 30)(3 13 21 31)(4 14 22 32)(5 9 62 40)(6 10 63 37)(7 11 64 38)(8 12 61 39)(17 45 27 43)(18 46 28 44)(19 47 25 41)(20 48 26 42)(33 53 51 59)(34 54 52 60)(35 55 49 57)(36 56 50 58)
(1 5 49 26)(2 8 50 25)(3 7 51 28)(4 6 52 27)(9 53 42 31)(10 56 43 30)(11 55 44 29)(12 54 41 32)(13 40 59 48)(14 39 60 47)(15 38 57 46)(16 37 58 45)(17 22 63 34)(18 21 64 33)(19 24 61 36)(20 23 62 35)
(1 14)(2 29)(3 16)(4 31)(5 47)(6 42)(7 45)(8 44)(9 27)(10 18)(11 25)(12 20)(13 22)(15 24)(17 40)(19 38)(21 30)(23 32)(26 39)(28 37)(33 56)(34 59)(35 54)(36 57)(41 62)(43 64)(46 61)(48 63)(49 60)(50 55)(51 58)(52 53)

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

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

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

Matrix representation of C4234D4 in GL10(𝔽5)

1000000000
0100000000
0003040000
0030400000
0003020000
0030200000
0000001342
0000002210
0000004401
0000003042
,
4000000000
0400000000
0004000000
0010000000
0004010000
0010400000
0000000100
0000001000
0000004142
0000004101
,
0400000000
1000000000
0020000000
0003000000
0020300000
0003020000
0000001342
0000003340
0000002241
0000004042
,
4000000000
0100000000
0030400000
0002010000
0030200000
0002030000
0000002210
0000004213
0000004214
0000000410

G:=sub<GL(10,GF(5))| [1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,3,0,3,0,0,0,0,0,0,3,0,3,0,0,0,0,0,0,0,0,4,0,2,0,0,0,0,0,0,4,0,2,0,0,0,0,0,0,0,0,0,0,0,1,2,4,3,0,0,0,0,0,0,3,2,4,0,0,0,0,0,0,0,4,1,0,4,0,0,0,0,0,0,2,0,1,2],[4,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0,1,0,1,0,0,0,0,0,0,4,0,4,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,4,4,0,0,0,0,0,0,1,0,1,1,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,2,1],[0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0,2,0,2,0,0,0,0,0,0,0,0,3,0,3,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,1,3,2,4,0,0,0,0,0,0,3,3,2,0,0,0,0,0,0,0,4,4,4,4,0,0,0,0,0,0,2,0,1,2],[4,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,3,0,3,0,0,0,0,0,0,0,0,2,0,2,0,0,0,0,0,0,4,0,2,0,0,0,0,0,0,0,0,1,0,3,0,0,0,0,0,0,0,0,0,0,2,4,4,0,0,0,0,0,0,0,2,2,2,4,0,0,0,0,0,0,1,1,1,1,0,0,0,0,0,0,0,3,4,0] >;

C4234D4 in GAP, Magma, Sage, TeX

C_4^2\rtimes_{34}D_4
% in TeX

G:=Group("C4^2:34D4");
// GroupNames label

G:=SmallGroup(128,1551);
// by ID

G=gap.SmallGroup(128,1551);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,336,253,758,723,794,185,80]);
// Polycyclic

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

Export

Character table of C4234D4 in TeX

׿
×
𝔽