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G = C42.272D4order 128 = 27

254th non-split extension by C42 of D4 acting via D4/C2=C22

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

Aliases: C42.272D4, C42.400C23, C4.1062+ 1+4, C83D410C2, C4⋊D825C2, D4⋊D426C2, D4.D49C2, C4⋊C8.61C22, (C2×C8).62C23, C4⋊C4.153C23, (C2×C4).412C24, (C2×D8).71C22, C23.287(C2×D4), (C22×C4).501D4, C4⋊Q8.304C22, C4.104(C8⋊C22), C42.6C410C2, C8⋊C4.18C22, (C4×D4).106C22, (C2×D4).161C23, C22⋊C8.47C22, (C2×Q8).149C23, D4⋊C4.43C22, C41D4.165C22, C4⋊D4.192C22, (C2×C42).879C22, Q8⋊C4.44C22, (C2×SD16).32C22, C22.672(C22×D4), C2.57(D8⋊C22), (C22×C4).1083C23, C42.28C222C2, C22.26C2420C2, C4.4D4.153C22, C2.83(C22.29C24), (C2×C4).541(C2×D4), C2.57(C2×C8⋊C22), (C2×C4○D4).174C22, SmallGroup(128,1946)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C42.272D4
C1C2C4C2×C4C22×C4C2×C4○D4C22.26C24 — C42.272D4
C1C2C2×C4 — C42.272D4
C1C22C2×C42 — C42.272D4
C1C2C2C2×C4 — C42.272D4

Generators and relations for C42.272D4
 G = < a,b,c,d | a4=b4=d2=1, c4=b2, ab=ba, cac-1=ab2, dad=a-1, cbc-1=dbd=a2b, dcd=b2c3 >

Subgroups: 492 in 217 conjugacy classes, 86 normal (28 characteristic)
C1, C2 [×3], C2 [×5], C4 [×4], C4 [×8], C22, C22 [×15], C8 [×4], C2×C4 [×6], C2×C4 [×17], D4 [×24], Q8 [×6], C23, C23 [×4], C42 [×4], C22⋊C4 [×8], C4⋊C4 [×2], C4⋊C4 [×3], C2×C8 [×4], D8 [×4], SD16 [×4], C22×C4 [×3], C22×C4 [×4], C2×D4 [×4], C2×D4 [×8], C2×Q8 [×2], C2×Q8, C4○D4 [×8], C8⋊C4 [×2], C22⋊C8 [×2], D4⋊C4 [×4], Q8⋊C4 [×4], C4⋊C8 [×2], C2×C42, C4×D4 [×2], C4×D4 [×3], C4⋊D4 [×2], C4⋊D4 [×3], C4.4D4 [×2], C4.4D4, C41D4 [×2], C4⋊Q8 [×2], C2×D8 [×4], C2×SD16 [×4], C2×C4○D4 [×2], C2×C4○D4, C42.6C4, D4⋊D4 [×4], C4⋊D8 [×2], D4.D4 [×2], C42.28C22 [×2], C83D4 [×2], C22.26C24 [×2], C42.272D4
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, C42.272D4

Character table of C42.272D4

 class 12A2B2C2D2E2F2G2H4A4B4C4D4E4F4G4H4I4J4K4L4M8A8B8C8D
 size 11114888822222244488888888
ρ111111111111111111111111111    trivial
ρ21111-11-1111-1111-1-11-1-1-1-111-1-11    linear of order 2
ρ31111-1-1111-1-11-11-11-11-11-1-11-11-1    linear of order 2
ρ411111-1-111-111-111-1-1-11-11-111-1-1    linear of order 2
ρ511111-1-1-1-1111111111-1-1-1-11111    linear of order 2
ρ61111-1-11-1-11-1111-1-11-1111-11-1-11    linear of order 2
ρ71111-11-1-1-1-1-11-11-11-111-1111-11-1    linear of order 2
ρ81111111-1-1-111-111-1-1-1-11-1111-1-1    linear of order 2
ρ91111-1111-1-1-11-11-11-111-1-1-1-11-11    linear of order 2
ρ10111111-11-1-111-111-1-1-1-111-1-1-111    linear of order 2
ρ1111111-111-1111111111-1-111-1-1-1-1    linear of order 2
ρ121111-1-1-11-11-1111-1-11-111-11-111-1    linear of order 2
ρ131111-1-1-1-11-1-11-11-11-11-1111-11-11    linear of order 2
ρ1411111-11-11-111-111-1-1-11-1-11-1-111    linear of order 2
ρ15111111-1-1111111111111-1-1-1-1-1-1    linear of order 2
ρ161111-111-111-1111-1-11-1-1-11-1-111-1    linear of order 2
ρ172222-2000022-22-22-2-2200000000    orthogonal lifted from D4
ρ182222200002-2-22-2-22-2-200000000    orthogonal lifted from D4
ρ192222-20000-22-2-2-2222-200000000    orthogonal lifted from D4
ρ20222220000-2-2-2-2-2-2-22200000000    orthogonal lifted from D4
ρ214-44-40000000-404000000000000    orthogonal lifted from 2+ 1+4
ρ224-44-4000000040-4000000000000    orthogonal lifted from 2+ 1+4
ρ2344-4-400000-40040000000000000    orthogonal lifted from C8⋊C22
ρ2444-4-400000400-40000000000000    orthogonal lifted from C8⋊C22
ρ254-4-44000000-4i0004i00000000000    complex lifted from D8⋊C22
ρ264-4-440000004i000-4i00000000000    complex lifted from D8⋊C22

Smallest permutation representation of C42.272D4
On 64 points
Generators in S64
(1 32 59 15)(2 29 60 12)(3 26 61 9)(4 31 62 14)(5 28 63 11)(6 25 64 16)(7 30 57 13)(8 27 58 10)(17 37 50 47)(18 34 51 44)(19 39 52 41)(20 36 53 46)(21 33 54 43)(22 38 55 48)(23 35 56 45)(24 40 49 42)
(1 17 5 21)(2 51 6 55)(3 19 7 23)(4 53 8 49)(9 41 13 45)(10 40 14 36)(11 43 15 47)(12 34 16 38)(18 64 22 60)(20 58 24 62)(25 48 29 44)(26 39 30 35)(27 42 31 46)(28 33 32 37)(50 63 54 59)(52 57 56 61)
(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 36)(2 35)(3 34)(4 33)(5 40)(6 39)(7 38)(8 37)(9 51)(10 50)(11 49)(12 56)(13 55)(14 54)(15 53)(16 52)(17 27)(18 26)(19 25)(20 32)(21 31)(22 30)(23 29)(24 28)(41 64)(42 63)(43 62)(44 61)(45 60)(46 59)(47 58)(48 57)

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

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

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

Matrix representation of C42.272D4 in GL8(𝔽17)

000160000
001600000
01000000
10000000
000000160
000000016
00001000
00000100
,
01000000
10000000
00010000
00100000
00000010
00000001
000016000
000001600
,
130000000
04000000
00400000
000130000
000014300
0000141400
000000314
00000033
,
00400000
000130000
130000000
04000000
000014300
00003300
000000314
0000001414

G:=sub<GL(8,GF(17))| [0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0],[0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0],[13,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,13,0,0,0,0,0,0,0,0,14,14,0,0,0,0,0,0,3,14,0,0,0,0,0,0,0,0,3,3,0,0,0,0,0,0,14,3],[0,0,13,0,0,0,0,0,0,0,0,4,0,0,0,0,4,0,0,0,0,0,0,0,0,13,0,0,0,0,0,0,0,0,0,0,14,3,0,0,0,0,0,0,3,3,0,0,0,0,0,0,0,0,3,14,0,0,0,0,0,0,14,14] >;

C42.272D4 in GAP, Magma, Sage, TeX

C_4^2._{272}D_4
% in TeX

G:=Group("C4^2.272D4");
// GroupNames label

G:=SmallGroup(128,1946);
// by ID

G=gap.SmallGroup(128,1946);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,120,758,891,675,248,4037,1027,124]);
// Polycyclic

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

Export

Character table of C42.272D4 in TeX

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