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G = C4.182+ 1+4order 128 = 27

18th non-split extension by C4 of 2+ 1+4 acting via 2+ 1+4/C2×D4=C2

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

Aliases: C4.182+ 1+4, C87D46C2, C8⋊D419C2, D4⋊D424C2, (C2×D4).158D4, C2.29(D4○D8), C8.18D46C2, (C2×Q8).134D4, C2.29(Q8○D8), D4.7D425C2, C4⋊C4.142C23, (C2×C4).401C24, (C2×C8).157C23, (C2×D8).22C22, C23.282(C2×D4), C2.D8.34C22, (C2×D4).151C23, C4⋊D4.41C22, C22.D822C2, C22⋊C8.44C22, (C2×Q16).24C22, (C2×Q8).139C23, C22⋊Q8.41C22, D4⋊C4.41C22, C23.48D422C2, C2.82(C233D4), (C22×C8).152C22, (C22×C4).304C23, Q8⋊C4.42C22, (C2×SD16).31C22, C22.661(C22×D4), C22.31C249C2, (C2×M4(2)).85C22, (C2×C4).152(C2×D4), (C22×C8)⋊C216C2, (C2×C4⋊C4).643C22, (C2×C4○D4).169C22, SmallGroup(128,1935)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C4.182+ 1+4
C1C2C4C2×C4C22×C4C2×C4○D4C22.31C24 — C4.182+ 1+4
C1C2C2×C4 — C4.182+ 1+4
C1C22C2×C4○D4 — C4.182+ 1+4
C1C2C2C2×C4 — C4.182+ 1+4

Generators and relations for C4.182+ 1+4
 G = < a,b,c,d,e | a4=1, b4=c2=e2=a2, d2=ab2, dbd-1=ab=ba, ac=ca, dad-1=a-1, ae=ea, cbc-1=ab3, be=eb, dcd-1=ece-1=a2c, ede-1=a-1b2d >

Subgroups: 444 in 201 conjugacy classes, 84 normal (26 characteristic)
C1, C2 [×3], C2 [×5], C4 [×2], C4 [×9], C22, C22 [×15], C8 [×4], C2×C4 [×2], C2×C4 [×2], C2×C4 [×18], D4 [×15], Q8 [×5], C23, C23 [×2], C23 [×2], C22⋊C4 [×8], C4⋊C4 [×4], C4⋊C4 [×6], C2×C8 [×2], C2×C8 [×2], C2×C8, M4(2), D8, SD16 [×2], Q16, C22×C4, C22×C4 [×2], C22×C4 [×4], C2×D4, C2×D4 [×4], C2×D4 [×6], C2×Q8, C2×Q8 [×2], C4○D4 [×6], C22⋊C8 [×4], D4⋊C4 [×4], Q8⋊C4 [×4], C2.D8 [×2], C2.D8 [×2], C2×C4⋊C4 [×2], C4⋊D4 [×4], C4⋊D4 [×6], C22⋊Q8 [×4], C22⋊Q8 [×2], C22×C8, C2×M4(2), C2×D8, C2×SD16 [×2], C2×Q16, C2×C4○D4, C2×C4○D4 [×2], (C22×C8)⋊C2, D4⋊D4 [×2], D4.7D4 [×2], C87D4, C8.18D4, C8⋊D4 [×2], C22.D8 [×2], C23.48D4 [×2], C22.31C24 [×2], C4.182+ 1+4
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], C2×D4 [×6], C24, C22×D4, 2+ 1+4 [×2], C233D4, D4○D8, Q8○D8, C4.182+ 1+4

Character table of C4.182+ 1+4

 class 12A2B2C2D2E2F2G2H4A4B4C4D4E4F4G4H4I4J4K8A8B8C8D8E8F
 size 11114448822444888888444488
ρ111111111111111111111111111    trivial
ρ21111-1-111-111-1-111-11-11-1-11-111-1    linear of order 2
ρ311111-1-1-1-1111-1-1-11111-11-11-11-1    linear of order 2
ρ41111-11-1-1111-11-1-1-11-111-1-1-1-111    linear of order 2
ρ511111-1-11-1111-1-1-1-1-1111-11-11-11    linear of order 2
ρ61111-11-11111-11-1-11-1-11-11111-1-1    linear of order 2
ρ71111111-11111111-1-111-1-1-1-1-1-1-1    linear of order 2
ρ81111-1-11-1-111-1-1111-1-1111-11-1-11    linear of order 2
ρ911111111-111111-111-1-11-1-1-1-1-1-1    linear of order 2
ρ101111-1-111111-1-11-1-111-1-11-11-1-11    linear of order 2
ρ1111111-1-1-11111-1-1111-1-1-1-11-11-11    linear of order 2
ρ121111-11-1-1-111-11-11-111-111111-1-1    linear of order 2
ρ1311111-1-111111-1-11-1-1-1-111-11-11-1    linear of order 2
ρ141111-11-11-111-11-111-11-1-1-1-1-1-111    linear of order 2
ρ151111111-1-111111-1-1-1-1-1-1111111    linear of order 2
ρ161111-1-11-1111-1-11-11-11-11-11-111-1    linear of order 2
ρ172222-2-2-200-2-2222000000000000    orthogonal lifted from D4
ρ1822222-2200-2-2-22-2000000000000    orthogonal lifted from D4
ρ192222-22200-2-22-2-2000000000000    orthogonal lifted from D4
ρ20222222-200-2-2-2-22000000000000    orthogonal lifted from D4
ρ214-44-4000004-4000000000000000    orthogonal lifted from 2+ 1+4
ρ224-44-400000-44000000000000000    orthogonal lifted from 2+ 1+4
ρ2344-4-400000000000000000220-2200    orthogonal lifted from D4○D8
ρ2444-4-400000000000000000-2202200    orthogonal lifted from D4○D8
ρ254-4-440000000000000000-22022000    symplectic lifted from Q8○D8, Schur index 2
ρ264-4-440000000000000000220-22000    symplectic lifted from Q8○D8, Schur index 2

Smallest permutation representation of C4.182+ 1+4
On 64 points
Generators in S64
(1 61 5 57)(2 62 6 58)(3 63 7 59)(4 64 8 60)(9 25 13 29)(10 26 14 30)(11 27 15 31)(12 28 16 32)(17 55 21 51)(18 56 22 52)(19 49 23 53)(20 50 24 54)(33 42 37 46)(34 43 38 47)(35 44 39 48)(36 45 40 41)
(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 23 6 19)(3 56 7 52)(4 17 8 21)(9 39 13 35)(10 43 14 47)(11 33 15 37)(12 45 16 41)(18 63 22 59)(20 57 24 61)(25 48 29 44)(26 38 30 34)(27 42 31 46)(28 40 32 36)(49 62 53 58)(51 64 55 60)
(1 43 63 40)(2 39 64 42)(3 41 57 38)(4 37 58 48)(5 47 59 36)(6 35 60 46)(7 45 61 34)(8 33 62 44)(9 55 27 23)(10 22 28 54)(11 53 29 21)(12 20 30 52)(13 51 31 19)(14 18 32 50)(15 49 25 17)(16 24 26 56)
(1 36 5 40)(2 37 6 33)(3 38 7 34)(4 39 8 35)(9 21 13 17)(10 22 14 18)(11 23 15 19)(12 24 16 20)(25 51 29 55)(26 52 30 56)(27 53 31 49)(28 54 32 50)(41 61 45 57)(42 62 46 58)(43 63 47 59)(44 64 48 60)

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

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

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

Matrix representation of C4.182+ 1+4 in GL8(𝔽17)

01000000
160000000
00010000
001600000
000001600
00001000
000000016
00000010
,
1343140000
1313330000
3144130000
33440000
0000141400
000031400
0000001414
000000314
,
00100000
00010000
160000000
016000000
00000010
00000001
000016000
000001600
,
701600000
010010000
1601000000
01070000
0000013011
0000130110
000001104
000011040
,
701600000
070160000
1601000000
0160100000
0000013011
00004060
000001104
000060130

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

C4.182+ 1+4 in GAP, Magma, Sage, TeX

C_4._{18}2_+^{1+4}
% in TeX

G:=Group("C4.18ES+(2,2)");
// GroupNames label

G:=SmallGroup(128,1935);
// by ID

G=gap.SmallGroup(128,1935);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,456,758,219,675,1018,248,4037,1027,124]);
// Polycyclic

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

Export

Character table of C4.182+ 1+4 in TeX

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