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G = C4.182+ (1+4)  order 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), C8⋊7D4⋊6C2, C8⋊D4⋊19C2, D4⋊D4⋊24C2, (C2×D4).158D4, C2.29(D4○D8), C8.18D4⋊6C2, (C2×Q8).134D4, C2.29(Q8○D8), D4.7D4⋊25C2, C4⋊C4.142C23, (C2×C4).401C24, (C2×C8).157C23, (C2×D8).22C22, C23.282(C2×D4), C2.D8.34C22, (C2×D4).151C23, C22.D8⋊22C2, C4⋊D4.41C22, C22⋊C8.44C22, (C2×Q8).139C23, (C2×Q16).24C22, C22⋊Q8.41C22, D4⋊C4.41C22, C23.48D4⋊22C2, C2.82(C23⋊3D4), (C22×C4).304C23, (C22×C8).152C22, Q8⋊C4.42C22, (C2×SD16).31C22, C22.661(C22×D4), C22.31C24⋊9C2, (C2×M4(2)).85C22, (C2×C4).152(C2×D4), (C22×C8)⋊C2⋊16C2, (C2×C4⋊C4).643C22, (C2×C4○D4).169C22, SmallGroup(128,1935)

Series: Derived ►Chief ►Lower central ►Upper central ►Jennings

Derived series
C1 — C2×C4 — C4.182+ (1+4)
Chief series
C1 — C2 — C4 — C2×C4 — C22×C4 — C2×C4○D4 — C22.31C24 — C4.182+ (1+4)
Lower central
C1 — C2 — C2×C4 — C4.182+ (1+4)
Upper central
C1 — C22 — C2×C4○D4 — C4.182+ (1+4)
Jennings
C1 — C2 — C2 — C2×C4 — C4.182+ (1+4)

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], C8⋊7D4, 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], C23⋊3D4, D4○D8, Q8○D8, C4.182+ (1+4)

Generators and relations
 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 >

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

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

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

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

Matrix representation ►G ⊆ 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] >;
 

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)
ρ2344-4-4000000000000000002√202√200    orthogonal lifted from D4○D8
ρ2444-4-4000000000000000002√202√200    orthogonal lifted from D4○D8
ρ254-4-4400000000000000002√202√2000    symplectic lifted from Q8○D8, Schur index 2
ρ264-4-4400000000000000002√202√2000    symplectic lifted from Q8○D8, Schur index 2

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
 

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