Copied to
clipboard

?

G = C4.192+ (1+4)  order 128 = 27

19th 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: C4.192+ (1+4), C8⋊8D4⋊28C2, C8⋊2D4⋊14C2, D4⋊D4⋊25C2, (C2×D4).159D4, C8.D4⋊14C2, (C2×Q8).135D4, D4.7D4⋊26C2, C4⋊C4.143C23, (C2×C4).402C24, (C2×C8).323C23, (C2×D8).69C22, C23.283(C2×D4), C4.Q8.82C22, C2.44(D4○SD16), (C2×D4).152C23, C4⋊D4.42C22, C22⋊C8.45C22, (C2×Q16).69C22, (C2×Q8).140C23, C22⋊Q8.42C22, C23.47D4⋊11C2, C23.46D4⋊11C2, C2.83(C23⋊3D4), (C22×C4).305C23, (C22×C8).350C22, Q8⋊C4.43C22, (C2×SD16).82C22, C22.662(C22×D4), D4⋊C4.105C22, (C2×M4(2)).86C22, C22.31C24⋊10C2, (C2×C4).153(C2×D4), (C22×C8)⋊C2⋊17C2, (C2×C4⋊C4).644C22, (C2×C4○D4).170C22, SmallGroup(128,1936)

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

Derived series
C1 — C2×C4 — C4.192+ (1+4)
Chief series
C1 — C2 — C4 — C2×C4 — C22×C4 — C2×C4○D4 — C22.31C24 — C4.192+ (1+4)
Lower central
C1 — C2 — C2×C4 — C4.192+ (1+4)
Upper central
C1 — C22 — C2×C4○D4 — C4.192+ (1+4)
Jennings
C1 — C2 — C2 — C2×C4 — C4.192+ (1+4)

Subgroups: 444 in 201 conjugacy classes, 84 normal (24 characteristic)
C1, C2, C2 [×2], 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], C4.Q8 [×2], C4.Q8 [×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⋊8D4 [×2], C8⋊2D4, C8.D4, C23.46D4 [×2], C23.47D4 [×2], C22.31C24 [×2], C4.192+ (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○SD16 [×2], C4.192+ (1+4)

Generators and relations
 G = < a,b,c,d,e | a4=1, b4=c2=e2=a2, d2=a-1b2, dbd-1=ab=ba, ac=ca, dad-1=a-1, ae=ea, cbc-1=a-1b3, be=eb, dcd-1=ece-1=a2c, ede-1=ab2d >

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

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

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

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

Matrix representation ►G ⊆ GL8(𝔽17)

016000000
10000000
1161620000
101610000
00000100
000016000
00000001
000000160
,
55000000
125000000
1251070000
120500000
00008998
00008899
00009898
00009999
,
00100000
1611150000
160000000
1610160000
00000010
00000001
000016000
000001600
,
1161620000
001600000
016000000
000160000
000050120
000001205
0000120120
00000505
,
1161620000
00100000
016000000
1601160000
000050120
000005012
0000120120
0000012012

G:=sub<GL(8,GF(17))| [0,1,1,1,0,0,0,0,16,0,16,0,0,0,0,0,0,0,16,16,0,0,0,0,0,0,2,1,0,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0],[5,12,12,12,0,0,0,0,5,5,5,0,0,0,0,0,0,0,10,5,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,8,8,9,9,0,0,0,0,9,8,8,9,0,0,0,0,9,9,9,9,0,0,0,0,8,9,8,9],[0,16,16,16,0,0,0,0,0,1,0,1,0,0,0,0,1,1,0,0,0,0,0,0,0,15,0,16,0,0,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0],[1,0,0,0,0,0,0,0,16,0,16,0,0,0,0,0,16,16,0,0,0,0,0,0,2,0,0,16,0,0,0,0,0,0,0,0,5,0,12,0,0,0,0,0,0,12,0,5,0,0,0,0,12,0,12,0,0,0,0,0,0,5,0,5],[1,0,0,16,0,0,0,0,16,0,16,0,0,0,0,0,16,1,0,1,0,0,0,0,2,0,0,16,0,0,0,0,0,0,0,0,5,0,12,0,0,0,0,0,0,5,0,12,0,0,0,0,12,0,12,0,0,0,0,0,0,12,0,12] >;
 

Character table of C4.192+ (1+4)

 class 12A2B2C2D2E2F2G2H4A4B4C4D4E4F4G4H4I4J4K8A8B8C8D8E8F
 size 11114448822444888888444488
ρ111111111111111111111111111    trivial
ρ21111-1-111-111-1-111-11-11-1-11-111-1    linear of order 2
ρ31111-1-11-1-111-1-11111-1-111-11-1-11    linear of order 2
ρ41111111-11111111-111-1-1-1-1-1-1-1-1    linear of order 2
ρ51111-11-1-1-111-11-111-111-1-1-1-1-111    linear of order 2
ρ611111-1-1-11111-1-11-1-1-1111-11-11-1    linear of order 2
ρ711111-1-111111-1-111-1-1-1-1-11-11-11    linear of order 2
ρ81111-11-11-111-11-11-1-11-111111-1-1    linear of order 2
ρ911111-1-1-1-1111-1-1-1-11111-11-11-11    linear of order 2
ρ101111-11-1-1111-11-1-111-11-11111-1-1    linear of order 2
ρ111111-11-11111-11-1-1-11-1-11-1-1-1-111    linear of order 2
ρ1211111-1-11-1111-1-1-1111-1-11-11-11-1    linear of order 2
ρ131111-1-111111-1-11-1-1-111-11-11-1-11    linear of order 2
ρ1411111111-111111-11-1-111-1-1-1-1-1-1    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)
ρ234-4-4400000000000000002√-202√-2000    complex lifted from D4○SD16
ρ2444-4-4000000000000000002√-202√-200    complex lifted from D4○SD16
ρ254-4-4400000000000000002√-202√-2000    complex lifted from D4○SD16
ρ2644-4-4000000000000000002√-202√-200    complex lifted from D4○SD16

In GAP, Magma, Sage, TeX 

C_4._{19}2_+^{(1+4)}
 
% in TeX
 
G:=Group("C4.19ES+(2,2)");
 
// GroupNames label
 
G:=SmallGroup(128,1936);
 
// by ID
 
G=gap.SmallGroup(128,1936);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,-2,448,253,456,758,219,675,1018,248,4037,1027,124]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^4=1,b^4=c^2=e^2=a^2,d^2=a^-1*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^-1*b^3,b*e=e*b,d*c*d^-1=e*c*e^-1=a^2*c,e*d*e^-1=a*b^2*d>;
 
// generators/relations
 

׿
×
⋊
ℤ
𝔽
○
≀