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G = C42.300D4order 128 = 27

282nd non-split extension by C42 of D4 acting via D4/C2=C22

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

Aliases: C42.300D4, C4.82- 1+4, C42.434C23, C4.272+ 1+4, C8⋊D4.2C2, Q8.Q828C2, C4.Q1630C2, D42Q813C2, C42Q1630C2, C8.D415C2, C4⋊C8.80C22, (C2×C8).76C23, D4.D413C2, C4⋊C4.191C23, (C2×C4).450C24, Q8.D429C2, C23.307(C2×D4), (C22×C4).527D4, C4⋊Q8.328C22, C4⋊M4(2)⋊11C2, C4.Q8.46C22, (C2×D4).192C23, (C4×D4).130C22, (C2×Q8).180C23, (C2×Q16).74C22, (C4×Q8).127C22, C2.D8.112C22, D4⋊C4.57C22, C4⋊D4.212C22, C4.122(C8.C22), (C2×C42).907C22, Q8⋊C4.55C22, (C2×SD16).41C22, C22.710(C22×D4), C22⋊Q8.217C22, C2.73(D8⋊C22), (C22×C4).1105C23, C4.4D4.167C22, (C2×M4(2)).88C22, C42.C2.144C22, C23.37C2326C2, C23.36C23.31C2, C2.69(C22.31C24), (C2×C4).574(C2×D4), C2.67(C2×C8.C22), SmallGroup(128,1984)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C42.300D4
C1C2C4C2×C4C42C4×D4C23.36C23 — C42.300D4
C1C2C2×C4 — C42.300D4
C1C22C2×C42 — C42.300D4
C1C2C2C2×C4 — C42.300D4

Generators and relations for C42.300D4
 G = < a,b,c,d | a4=b4=1, c4=d2=a2, ab=ba, cac-1=a-1, dad-1=ab2, cbc-1=a2b-1, dbd-1=b-1, dcd-1=c3 >

Subgroups: 316 in 174 conjugacy classes, 86 normal (44 characteristic)
C1, C2 [×3], C2 [×2], C4 [×2], C4 [×2], C4 [×11], C22, C22 [×6], C8 [×4], C2×C4 [×6], C2×C4 [×14], D4 [×4], Q8 [×8], C23, C23, C42 [×4], C42 [×3], C22⋊C4 [×7], C4⋊C4 [×2], C4⋊C4 [×2], C4⋊C4 [×11], C2×C8 [×4], M4(2) [×2], SD16 [×2], Q16 [×2], C22×C4 [×3], C22×C4, C2×D4, C2×D4, C2×Q8, C2×Q8 [×2], C2×Q8, D4⋊C4 [×2], Q8⋊C4 [×6], C4⋊C8 [×2], C4⋊C8 [×2], C4.Q8 [×2], C2.D8 [×2], C2×C42, C42⋊C2 [×2], C4×D4, C4×D4, C4×Q8, C4×Q8 [×2], C4×Q8, C4⋊D4, C22⋊Q8, C22⋊Q8 [×2], C22⋊Q8, C22.D4, C4.4D4, C42.C2, C42.C2, C422C2, C4⋊Q8 [×2], C2×M4(2) [×2], C2×SD16 [×2], C2×Q16 [×2], C4⋊M4(2), D4.D4, C42Q16, Q8.D4 [×2], C8⋊D4 [×2], C8.D4 [×2], D42Q8, C4.Q16, Q8.Q8 [×2], C23.36C23, C23.37C23, C42.300D4
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], C2×D4 [×6], C24, C8.C22 [×2], C22×D4, 2+ 1+4, 2- 1+4, C22.31C24, C2×C8.C22, D8⋊C22, C42.300D4

Character table of C42.300D4

 class 12A2B2C2D2E4A4B4C4D4E4F4G4H4I4J4K4L4M4N4O4P8A8B8C8D
 size 11114822222244488888888888
ρ111111111111111111111111111    trivial
ρ211111-1111111111-111-1-111-1-1-1-1    linear of order 2
ρ31111-11-111-1111-1-1111-1-1-1-111-1-1    linear of order 2
ρ41111-1-1-111-1111-1-1-11111-1-1-1-111    linear of order 2
ρ51111-111-111-11-1-11-11-1-11-111-1-11    linear of order 2
ρ61111-1-11-111-11-1-1111-11-1-11-111-1    linear of order 2
ρ7111111-1-11-1-11-11-1-11-11-11-11-11-1    linear of order 2
ρ811111-1-1-11-1-11-11-111-1-111-1-11-11    linear of order 2
ρ91111111111111111-1-111-1-1-1-1-1-1    linear of order 2
ρ1011111-1111111111-1-1-1-1-1-1-11111    linear of order 2
ρ111111-11-111-1111-1-11-1-1-1-111-1-111    linear of order 2
ρ121111-1-1-111-1111-1-1-1-1-1111111-1-1    linear of order 2
ρ131111-111-111-11-1-11-1-11-111-1-111-1    linear of order 2
ρ141111-1-11-111-11-1-111-111-11-11-1-11    linear of order 2
ρ15111111-1-11-1-11-11-1-1-111-1-11-11-11    linear of order 2
ρ1611111-1-1-11-1-11-11-11-11-11-111-11-1    linear of order 2
ρ172222-202-2-22-2-222-200000000000    orthogonal lifted from D4
ρ18222220-2-2-2-2-2-22-2200000000000    orthogonal lifted from D4
ρ192222-20-22-2-22-2-22200000000000    orthogonal lifted from D4
ρ2022222022-222-2-2-2-200000000000    orthogonal lifted from D4
ρ214-44-40000-400400000000000000    orthogonal lifted from 2+ 1+4
ρ224-4-44000400-4000000000000000    symplectic lifted from C8.C22, Schur index 2
ρ234-4-44000-4004000000000000000    symplectic lifted from C8.C22, Schur index 2
ρ244-44-40000400-400000000000000    symplectic lifted from 2- 1+4, Schur index 2
ρ2544-4-4004i00-4i0000000000000000    complex lifted from D8⋊C22
ρ2644-4-400-4i004i0000000000000000    complex lifted from D8⋊C22

Smallest permutation representation of C42.300D4
On 64 points
Generators in S64
(1 57 5 61)(2 62 6 58)(3 59 7 63)(4 64 8 60)(9 40 13 36)(10 37 14 33)(11 34 15 38)(12 39 16 35)(17 28 21 32)(18 25 22 29)(19 30 23 26)(20 27 24 31)(41 52 45 56)(42 49 46 53)(43 54 47 50)(44 51 48 55)
(1 25 55 11)(2 16 56 30)(3 27 49 13)(4 10 50 32)(5 29 51 15)(6 12 52 26)(7 31 53 9)(8 14 54 28)(17 64 37 43)(18 48 38 61)(19 58 39 45)(20 42 40 63)(21 60 33 47)(22 44 34 57)(23 62 35 41)(24 46 36 59)
(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 4 5 8)(2 7 6 3)(9 12 13 16)(10 15 14 11)(17 38 21 34)(18 33 22 37)(19 36 23 40)(20 39 24 35)(25 32 29 28)(26 27 30 31)(41 63 45 59)(42 58 46 62)(43 61 47 57)(44 64 48 60)(49 56 53 52)(50 51 54 55)

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

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

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

Matrix representation of C42.300D4 in GL8(𝔽17)

001600000
000160000
160000000
016000000
0000131348
00000040
00000400
000001344
,
016000000
10000000
000160000
00100000
0000111615
000000160
000001600
0000011616
,
001670000
00710000
167000000
71000000
0000401313
00000004
0000041313
00004000
,
001670000
00710000
110000000
1016000000
0000401313
000000013
0000041313
000001300

G:=sub<GL(8,GF(17))| [0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,0,0,13,0,0,0,0,0,0,0,13,0,4,13,0,0,0,0,4,4,0,4,0,0,0,0,8,0,0,4],[0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,16,1,0,0,0,0,16,16,0,16,0,0,0,0,15,0,0,16],[0,0,16,7,0,0,0,0,0,0,7,1,0,0,0,0,16,7,0,0,0,0,0,0,7,1,0,0,0,0,0,0,0,0,0,0,4,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,13,0,13,0,0,0,0,0,13,4,13,0],[0,0,1,10,0,0,0,0,0,0,10,16,0,0,0,0,16,7,0,0,0,0,0,0,7,1,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,4,13,0,0,0,0,13,0,13,0,0,0,0,0,13,13,13,0] >;

C42.300D4 in GAP, Magma, Sage, TeX

C_4^2._{300}D_4
% in TeX

G:=Group("C4^2.300D4");
// GroupNames label

G:=SmallGroup(128,1984);
// by ID

G=gap.SmallGroup(128,1984);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,448,253,758,891,675,80,4037,1027,124]);
// Polycyclic

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

Export

Character table of C42.300D4 in TeX

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