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G = C4235D4order 128 = 27

29th semidirect product of C42 and D4 acting via D4/C2=C22

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

Aliases: C4235D4, C24.105C23, C23.723C24, C22.3792- 1+4, C22.4962+ 1+4, C429C439C2, C23.Q894C2, (C2×C42).735C22, (C22×C4).234C23, C22.455(C22×D4), C23.10D4.73C2, (C22×D4).298C22, C23.81C23135C2, C2.46(C22.54C24), C2.C42.426C22, C2.61(C22.31C24), C2.62(C22.57C24), (C2×C4).440(C2×D4), (C2×C422C2)⋊28C2, (C2×C4⋊C4).532C22, (C2×C22⋊C4).341C22, SmallGroup(128,1555)

Series: Derived Chief Lower central Upper central Jennings

C1C23 — C4235D4
C1C2C22C23C22×C4C2×C22⋊C4C23.10D4 — C4235D4
C1C23 — C4235D4
C1C23 — C4235D4
C1C23 — C4235D4

Generators and relations for C4235D4
 G = < a,b,c,d | a4=b4=c4=d2=1, ab=ba, cac-1=a-1, dad=a-1b2, cbc-1=b-1, dbd=a2b, dcd=c-1 >

Subgroups: 468 in 228 conjugacy classes, 92 normal (9 characteristic)
C1, C2, C2 [×6], C2 [×2], C4 [×16], C22, C22 [×6], C22 [×14], C2×C4 [×6], C2×C4 [×36], D4 [×4], C23, C23 [×14], C42 [×4], C22⋊C4 [×18], C4⋊C4 [×21], C22×C4, C22×C4 [×12], C2×D4 [×3], C24 [×2], C2.C42 [×6], C2×C42, C2×C22⋊C4 [×12], C2×C4⋊C4 [×15], C422C2 [×8], C22×D4, C429C4, C23.10D4 [×3], C23.Q8 [×6], C23.81C23 [×3], C2×C422C2 [×2], C4235D4
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], C2×D4 [×6], C24, C22×D4, 2+ 1+4 [×3], 2- 1+4 [×3], C22.31C24 [×3], C22.54C24, C22.57C24 [×3], C4235D4

Character table of C4235D4

 class 12A2B2C2D2E2F2G2H2I4A4B4C4D4E4F4G4H4I4J4K4L4M4N4O4P
 size 11111111884444448888888888
ρ111111111111111111111111111    trivial
ρ211111111-11-1-111-1-1-1-11-1-1-11111    linear of order 2
ρ311111111-11111111-1-11-111-1-1-1-1    linear of order 2
ρ41111111111-1-111-1-11111-1-1-1-1-1-1    linear of order 2
ρ511111111-1-1-1-111-1-111-1-11111-1-1    linear of order 2
ρ6111111111-1111111-1-1-11-1-111-1-1    linear of order 2
ρ7111111111-1-1-111-1-1-1-1-1111-1-111    linear of order 2
ρ811111111-1-111111111-1-1-1-1-1-111    linear of order 2
ρ911111111-1111-1-1-1-1-11-11-111-11-1    linear of order 2
ρ101111111111-1-1-1-1111-1-1-11-11-11-1    linear of order 2
ρ11111111111111-1-1-1-11-1-1-1-11-11-11    linear of order 2
ρ1211111111-11-1-1-1-111-11-111-1-11-11    linear of order 2
ρ13111111111-1-1-1-1-111-111-1-111-1-11    linear of order 2
ρ1411111111-1-111-1-1-1-11-1111-11-1-11    linear of order 2
ρ1511111111-1-1-1-1-1-1111-111-11-111-1    linear of order 2
ρ16111111111-111-1-1-1-1-111-11-1-111-1    linear of order 2
ρ172-22-22-22-200-222-2-220000000000    orthogonal lifted from D4
ρ182-22-22-22-200-22-222-20000000000    orthogonal lifted from D4
ρ192-22-22-22-2002-2-22-220000000000    orthogonal lifted from D4
ρ202-22-22-22-2002-22-22-20000000000    orthogonal lifted from D4
ρ214-4-4-444-44000000000000000000    orthogonal lifted from 2+ 1+4
ρ2244-4-4-4-444000000000000000000    orthogonal lifted from 2+ 1+4
ρ234-444-4-4-44000000000000000000    orthogonal lifted from 2+ 1+4
ρ24444-4-44-4-4000000000000000000    symplectic lifted from 2- 1+4, Schur index 2
ρ254-4-44-444-4000000000000000000    symplectic lifted from 2- 1+4, Schur index 2
ρ2644-444-4-4-4000000000000000000    symplectic lifted from 2- 1+4, Schur index 2

Smallest permutation representation of C4235D4
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 42 53 13)(2 43 54 14)(3 44 55 15)(4 41 56 16)(5 18 40 47)(6 19 37 48)(7 20 38 45)(8 17 39 46)(9 57 22 32)(10 58 23 29)(11 59 24 30)(12 60 21 31)(25 64 51 36)(26 61 52 33)(27 62 49 34)(28 63 50 35)
(1 33 11 19)(2 36 12 18)(3 35 9 17)(4 34 10 20)(5 43 51 60)(6 42 52 59)(7 41 49 58)(8 44 50 57)(13 26 30 37)(14 25 31 40)(15 28 32 39)(16 27 29 38)(21 47 54 64)(22 46 55 63)(23 45 56 62)(24 48 53 61)
(2 56)(4 54)(5 25)(6 50)(7 27)(8 52)(10 21)(12 23)(13 15)(14 43)(16 41)(17 35)(18 62)(19 33)(20 64)(26 39)(28 37)(29 58)(30 32)(31 60)(34 47)(36 45)(38 49)(40 51)(42 44)(46 63)(48 61)(57 59)

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

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

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

Matrix representation of C4235D4 in GL10(𝔽5)

4000000000
0400000000
0030000000
0003000000
0000200000
0022020000
0000004230
0000002403
0000002313
0000003231
,
4000000000
0400000000
0004000000
0010000000
0011120000
0040440000
0000003000
0000000200
0000000430
0000001002
,
2300000000
0300000000
0000400000
0044430000
0010000000
0001110000
0000000100
0000001000
0000000001
0000000010
,
1000000000
2400000000
0010000000
0004000000
0000400000
0001110000
0000001000
0000000100
0000004240
0000002404

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

C4235D4 in GAP, Magma, Sage, TeX

C_4^2\rtimes_{35}D_4
% in TeX

G:=Group("C4^2:35D4");
// GroupNames label

G:=SmallGroup(128,1555);
// by ID

G=gap.SmallGroup(128,1555);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,336,253,758,723,794,185,136]);
// 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^-1*b^2,c*b*c^-1=b^-1,d*b*d=a^2*b,d*c*d=c^-1>;
// generators/relations

Export

Character table of C4235D4 in TeX

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