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G = C42.288D4order 128 = 27

270th non-split extension by C42 of D4 acting via D4/C2=C22

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

Aliases: C42.288D4, C42.418C23, C4.602- 1+4, C8⋊Q814C2, Q8.Q823C2, C4⋊C8.70C22, (C2×C8).70C23, C4⋊C4.175C23, (C2×C4).434C24, (C22×C4).516D4, C23.703(C2×D4), C4⋊Q8.317C22, C8⋊C4.27C22, C4.Q8.38C22, C22⋊C8.61C22, C42.6C4.2C2, (C4×Q8).113C22, (C2×Q8).166C23, C2.D8.104C22, (C2×C42).895C22, Q8⋊C4.48C22, C23.48D4.3C2, C23.47D4.2C2, C22.694(C22×D4), C22⋊Q8.206C22, C2.65(D8⋊C22), C42.30C225C2, (C22×C4).1099C23, C22.41(C8.C22), C42.C2.135C22, C23.37C23.41C2, C2.82(C23.38C23), (C2×C4).558(C2×D4), C2.62(C2×C8.C22), (C2×C4⋊C4).651C22, (C2×C42.C2).36C2, SmallGroup(128,1968)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C42.288D4
C1C2C4C2×C4C22×C4C2×C4⋊C4C2×C42.C2 — C42.288D4
C1C2C2×C4 — C42.288D4
C1C22C2×C42 — C42.288D4
C1C2C2C2×C4 — C42.288D4

Generators and relations for C42.288D4
 G = < a,b,c,d | a4=b4=1, c4=d2=b2, ab=ba, cac-1=ab2, dad-1=a-1, cbc-1=a2b-1, dbd-1=a2b, dcd-1=a2b2c3 >

Subgroups: 268 in 162 conjugacy classes, 86 normal (28 characteristic)
C1, C2 [×3], C2 [×2], C4 [×2], C4 [×13], C22, C22 [×2], C22 [×2], C8 [×4], C2×C4 [×6], C2×C4 [×17], Q8 [×6], C23, C42 [×4], C42 [×2], C22⋊C4 [×2], C4⋊C4 [×6], C4⋊C4 [×17], C2×C8 [×4], C22×C4 [×3], C22×C4 [×2], C2×Q8 [×2], C2×Q8, C8⋊C4 [×2], C22⋊C8 [×2], Q8⋊C4 [×8], C4⋊C8 [×2], C4.Q8 [×4], C2.D8 [×4], C2×C42, C2×C4⋊C4 [×2], C2×C4⋊C4 [×2], C42⋊C2, C4×Q8 [×2], C4×Q8, C22⋊Q8 [×2], C22⋊Q8, C42.C2 [×4], C42.C2 [×3], C4⋊Q8 [×2], C42.6C4, Q8.Q8 [×4], C23.47D4 [×2], C23.48D4 [×2], C42.30C22 [×2], C8⋊Q8 [×2], C2×C42.C2, C23.37C23, C42.288D4
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], C2×D4 [×6], C24, C8.C22 [×2], C22×D4, 2- 1+4 [×2], C23.38C23, C2×C8.C22, D8⋊C22, C42.288D4

Character table of C42.288D4

 class 12A2B2C2D2E4A4B4C4D4E4F4G4H4I4J4K4L4M4N4O4P8A8B8C8D
 size 11112222224444888888888888
ρ111111111111111111111111111    trivial
ρ21111111-11-11-1-1-111-1-11-1-11-1-111    linear of order 2
ρ311111111111111-1-1-1-1-1-1-1-11111    linear of order 2
ρ41111111-11-11-1-1-1-1-111-111-1-1-111    linear of order 2
ρ51111-1-11111-11-1-11-1-11-11-111-1-11    linear of order 2
ρ61111-1-11-11-1-1-1111-11-1-1-111-11-11    linear of order 2
ρ71111-1-11111-11-1-1-111-11-11-11-1-11    linear of order 2
ρ81111-1-11-11-1-1-111-11-1111-1-1-11-11    linear of order 2
ρ91111-1-11111-11-1-1-1-1-111-111-111-1    linear of order 2
ρ101111-1-11-11-1-1-111-1-11-111-111-11-1    linear of order 2
ρ111111-1-11111-11-1-1111-1-11-1-1-111-1    linear of order 2
ρ121111-1-11-11-1-1-11111-11-1-11-11-11-1    linear of order 2
ρ1311111111111111-1111-1-1-11-1-1-1-1    linear of order 2
ρ141111111-11-11-1-1-1-11-1-1-111111-1-1    linear of order 2
ρ15111111111111111-1-1-1111-1-1-1-1-1    linear of order 2
ρ161111111-11-11-1-1-11-1111-1-1-111-1-1    linear of order 2
ρ17222222-22-22-2-22-2000000000000    orthogonal lifted from D4
ρ18222222-2-2-2-2-22-22000000000000    orthogonal lifted from D4
ρ192222-2-2-2-2-2-2222-2000000000000    orthogonal lifted from D4
ρ202222-2-2-22-222-2-22000000000000    orthogonal lifted from D4
ρ214-44-40040-400000000000000000    symplectic lifted from 2- 1+4, Schur index 2
ρ224-4-444-400000000000000000000    symplectic lifted from C8.C22, Schur index 2
ρ234-44-400-40400000000000000000    symplectic lifted from 2- 1+4, Schur index 2
ρ244-4-44-4400000000000000000000    symplectic lifted from C8.C22, Schur index 2
ρ2544-4-4000-4i04i0000000000000000    complex lifted from D8⋊C22
ρ2644-4-40004i0-4i0000000000000000    complex lifted from D8⋊C22

Smallest permutation representation of C42.288D4
On 64 points
Generators in S64
(1 40 24 41)(2 37 17 46)(3 34 18 43)(4 39 19 48)(5 36 20 45)(6 33 21 42)(7 38 22 47)(8 35 23 44)(9 64 50 28)(10 61 51 25)(11 58 52 30)(12 63 53 27)(13 60 54 32)(14 57 55 29)(15 62 56 26)(16 59 49 31)
(1 22 5 18)(2 4 6 8)(3 24 7 20)(9 11 13 15)(10 49 14 53)(12 51 16 55)(17 19 21 23)(25 59 29 63)(26 28 30 32)(27 61 31 57)(33 35 37 39)(34 41 38 45)(36 43 40 47)(42 44 46 48)(50 52 54 56)(58 60 62 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 54 5 50)(2 12 6 16)(3 52 7 56)(4 10 8 14)(9 24 13 20)(11 22 15 18)(17 53 21 49)(19 51 23 55)(25 35 29 39)(26 43 30 47)(27 33 31 37)(28 41 32 45)(34 58 38 62)(36 64 40 60)(42 59 46 63)(44 57 48 61)

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

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

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

Matrix representation of C42.288D4 in GL8(𝔽17)

0010100000
120700000
01212120000
55550000
00009300
00001800
0000161066
00001514811
,
1615000000
11000000
110160000
016100000
00001000
00000100
0000150160
000057016
,
1216700000
1415120000
841390000
08380000
0000114813
0000161042
00000594
0000146014
,
6101160000
8121600000
841390000
614930000
0000160160
000001052
00000010
0000010157

G:=sub<GL(8,GF(17))| [0,12,0,5,0,0,0,0,0,0,12,5,0,0,0,0,10,7,12,5,0,0,0,0,10,0,12,5,0,0,0,0,0,0,0,0,9,1,16,15,0,0,0,0,3,8,10,14,0,0,0,0,0,0,6,8,0,0,0,0,0,0,6,11],[16,1,1,0,0,0,0,0,15,1,1,16,0,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,0,1,0,15,5,0,0,0,0,0,1,0,7,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16],[12,14,8,0,0,0,0,0,16,1,4,8,0,0,0,0,7,5,13,3,0,0,0,0,0,12,9,8,0,0,0,0,0,0,0,0,1,16,0,14,0,0,0,0,14,10,5,6,0,0,0,0,8,4,9,0,0,0,0,0,13,2,4,14],[6,8,8,6,0,0,0,0,10,12,4,14,0,0,0,0,1,16,13,9,0,0,0,0,16,0,9,3,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,10,0,10,0,0,0,0,16,5,1,15,0,0,0,0,0,2,0,7] >;

C42.288D4 in GAP, Magma, Sage, TeX

C_4^2._{288}D_4
% in TeX

G:=Group("C4^2.288D4");
// GroupNames label

G:=SmallGroup(128,1968);
// by ID

G=gap.SmallGroup(128,1968);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,448,253,120,758,891,100,675,4037,1027,124]);
// Polycyclic

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

Export

Character table of C42.288D4 in TeX

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