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G = C42.274D4order 128 = 27

256th non-split extension by C42 of D4 acting via D4/C2=C22

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

Aliases: C42.274D4, C42.402C23, C4.1082+ 1+4, C4⋊SD169C2, C42Q1626C2, C8.2D410C2, C4⋊C8.62C22, (C2×C8).64C23, D4.7D427C2, C4⋊C4.155C23, (C2×C4).414C24, (C22×C4).503D4, C23.288(C2×D4), C4⋊Q8.305C22, C8⋊C4.19C22, C42.6C412C2, (C2×D4).163C23, C22⋊C8.49C22, (C4×Q8).103C22, (C2×Q16).70C22, (C2×Q8).151C23, D4⋊C4.45C22, C41D4.166C22, C4.100(C8.C22), (C2×C42).881C22, Q8⋊C4.45C22, (C2×SD16).34C22, C22.674(C22×D4), C22⋊Q8.197C22, C2.59(D8⋊C22), (C22×C4).1085C23, C42.28C224C2, C4.4D4.155C22, C23.37C2319C2, C2.85(C22.29C24), C22.26C24.42C2, (C2×C4).543(C2×D4), C2.57(C2×C8.C22), (C2×C4○D4).175C22, SmallGroup(128,1948)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C42.274D4
C1C2C4C2×C4C22×C4C2×C4○D4C22.26C24 — C42.274D4
C1C2C2×C4 — C42.274D4
C1C22C2×C42 — C42.274D4
C1C2C2C2×C4 — C42.274D4

Generators and relations for C42.274D4
 G = < a,b,c,d | a4=b4=1, c4=d2=b2, ab=ba, cac-1=ab2, dad-1=a-1, cbc-1=dbd-1=a2b, dcd-1=c3 >

Subgroups: 396 in 195 conjugacy classes, 86 normal (28 characteristic)
C1, C2 [×3], C2 [×3], C4 [×4], C4 [×10], C22, C22 [×9], C8 [×4], C2×C4 [×6], C2×C4 [×15], D4 [×12], Q8 [×10], C23, C23 [×2], C42 [×4], C42 [×2], C22⋊C4 [×6], C4⋊C4 [×2], C4⋊C4 [×9], C2×C8 [×4], SD16 [×4], Q16 [×4], C22×C4 [×3], C22×C4 [×2], C2×D4 [×2], C2×D4 [×4], C2×Q8 [×4], C2×Q8, C4○D4 [×4], C8⋊C4 [×2], C22⋊C8 [×2], D4⋊C4 [×4], Q8⋊C4 [×4], C4⋊C8 [×2], C2×C42, C42⋊C2, C4×D4 [×2], C4×Q8 [×2], C4×Q8, C4⋊D4 [×2], C22⋊Q8 [×2], C22⋊Q8, C4.4D4 [×2], C42.C2, C41D4, C4⋊Q8 [×3], C2×SD16 [×4], C2×Q16 [×4], C2×C4○D4 [×2], C42.6C4, D4.7D4 [×4], C4⋊SD16 [×2], C42Q16 [×2], C42.28C22 [×2], C8.2D4 [×2], C22.26C24, C23.37C23, C42.274D4
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], C2×D4 [×6], C24, C8.C22 [×2], C22×D4, 2+ 1+4 [×2], C22.29C24, C2×C8.C22, D8⋊C22, C42.274D4

Character table of C42.274D4

 class 12A2B2C2D2E2F4A4B4C4D4E4F4G4H4I4J4K4L4M4N4O8A8B8C8D
 size 11114882222224448888888888
ρ111111111111111111111111111    trivial
ρ21111-1111-1111-1-11-1-1-11-1-11-111-1    linear of order 2
ρ311111-11-111-111-1-1-11-1-1-11111-1-1    linear of order 2
ρ41111-1-11-1-11-11-11-11-11-11-11-11-11    linear of order 2
ρ51111-1111-1111-1-11-1-1-1-111-11-1-11    linear of order 2
ρ6111111111111111111-1-1-1-1-1-1-1-1    linear of order 2
ρ71111-1-11-1-11-11-11-11-111-11-11-11-1    linear of order 2
ρ811111-11-111-111-1-1-11-111-1-1-1-111    linear of order 2
ρ9111111-1-111-111-1-1-1-1111-1-111-1-1    linear of order 2
ρ101111-11-1-1-11-11-11-111-11-11-1-11-11    linear of order 2
ρ1111111-1-1111111111-1-1-1-1-1-11111    linear of order 2
ρ121111-1-1-11-1111-1-11-111-111-1-111-1    linear of order 2
ρ131111-11-1-1-11-11-11-111-1-11-111-11-1    linear of order 2
ρ14111111-1-111-111-1-1-1-11-1-111-1-111    linear of order 2
ρ151111-1-1-11-1111-1-11-1111-1-111-1-11    linear of order 2
ρ1611111-1-1111111111-1-11111-1-1-1-1    linear of order 2
ρ172222200-2-2-2-2-2-2-2220000000000    orthogonal lifted from D4
ρ182222-200-22-2-2-2222-20000000000    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
ρ214-44-40000040-400000000000000    orthogonal lifted from 2+ 1+4
ρ224-44-400000-40400000000000000    orthogonal lifted from 2+ 1+4
ρ2344-4-4000-4004000000000000000    symplectic lifted from C8.C22, Schur index 2
ρ2444-4-4000400-4000000000000000    symplectic lifted from C8.C22, Schur index 2
ρ254-4-440000-4i0004i0000000000000    complex lifted from D8⋊C22
ρ264-4-4400004i000-4i0000000000000    complex lifted from D8⋊C22

Smallest permutation representation of C42.274D4
On 64 points
Generators in S64
(1 32 63 11)(2 29 64 16)(3 26 57 13)(4 31 58 10)(5 28 59 15)(6 25 60 12)(7 30 61 9)(8 27 62 14)(17 36 50 42)(18 33 51 47)(19 38 52 44)(20 35 53 41)(21 40 54 46)(22 37 55 43)(23 34 56 48)(24 39 49 45)
(1 22 5 18)(2 56 6 52)(3 24 7 20)(4 50 8 54)(9 41 13 45)(10 36 14 40)(11 43 15 47)(12 38 16 34)(17 62 21 58)(19 64 23 60)(25 44 29 48)(26 39 30 35)(27 46 31 42)(28 33 32 37)(49 61 53 57)(51 63 55 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 40 5 36)(2 35 6 39)(3 38 7 34)(4 33 8 37)(9 56 13 52)(10 51 14 55)(11 54 15 50)(12 49 16 53)(17 32 21 28)(18 27 22 31)(19 30 23 26)(20 25 24 29)(41 60 45 64)(42 63 46 59)(43 58 47 62)(44 61 48 57)

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

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

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

Matrix representation of C42.274D4 in GL8(𝔽17)

11000000
1516000000
0016160000
00210000
00000040
000000013
000013000
00000400
,
10000000
01000000
001600000
000160000
00004000
00000400
00000040
00000004
,
0016160000
00010000
1616000000
01000000
0000991515
000089152
000015289
00002288
,
00100000
00010000
10000000
01000000
000015289
00002288
0000991515
000089152

G:=sub<GL(8,GF(17))| [1,15,0,0,0,0,0,0,1,16,0,0,0,0,0,0,0,0,16,2,0,0,0,0,0,0,16,1,0,0,0,0,0,0,0,0,0,0,13,0,0,0,0,0,0,0,0,4,0,0,0,0,4,0,0,0,0,0,0,0,0,13,0,0],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4],[0,0,16,0,0,0,0,0,0,0,16,1,0,0,0,0,16,0,0,0,0,0,0,0,16,1,0,0,0,0,0,0,0,0,0,0,9,8,15,2,0,0,0,0,9,9,2,2,0,0,0,0,15,15,8,8,0,0,0,0,15,2,9,8],[0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,15,2,9,8,0,0,0,0,2,2,9,9,0,0,0,0,8,8,15,15,0,0,0,0,9,8,15,2] >;

C42.274D4 in GAP, Magma, Sage, TeX

C_4^2._{274}D_4
% in TeX

G:=Group("C4^2.274D4");
// GroupNames label

G:=SmallGroup(128,1948);
// by ID

G=gap.SmallGroup(128,1948);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,448,253,120,758,891,675,248,4037,1027,124]);
// 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=d*b*d^-1=a^2*b,d*c*d^-1=c^3>;
// generators/relations

Export

Character table of C42.274D4 in TeX

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