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G = C4.192+ 1+4order 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, C82D414C2, C88D428C2, D4⋊D425C2, (C2×D4).159D4, C8.D414C2, (C2×Q8).135D4, D4.7D426C2, C4⋊C4.143C23, (C2×C8).323C23, (C2×C4).402C24, (C2×D8).69C22, C23.283(C2×D4), C4.Q8.82C22, C2.44(D4○SD16), (C2×D4).152C23, C4⋊D4.42C22, C22⋊C8.45C22, (C2×Q8).140C23, (C2×Q16).69C22, C22⋊Q8.42C22, C23.47D411C2, C23.46D411C2, C2.83(C233D4), (C22×C8).350C22, (C22×C4).305C23, Q8⋊C4.43C22, (C2×SD16).82C22, C22.662(C22×D4), D4⋊C4.105C22, (C2×M4(2)).86C22, C22.31C2410C2, (C2×C4).153(C2×D4), (C22×C8)⋊C217C2, (C2×C4⋊C4).644C22, (C2×C4○D4).170C22, SmallGroup(128,1936)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C4.192+ 1+4
C1C2C4C2×C4C22×C4C2×C4○D4C22.31C24 — C4.192+ 1+4
C1C2C2×C4 — C4.192+ 1+4
C1C22C2×C4○D4 — C4.192+ 1+4
C1C2C2C2×C4 — C4.192+ 1+4

Generators and relations for C4.192+ 1+4
 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 >

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], C88D4 [×2], C82D4, 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], C233D4, D4○SD16 [×2], C4.192+ 1+4

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
ρ234-4-440000000000000000-2-202-2000    complex lifted from D4○SD16
ρ2444-4-400000000000000000-2-202-200    complex lifted from D4○SD16
ρ254-4-4400000000000000002-20-2-2000    complex lifted from D4○SD16
ρ2644-4-4000000000000000002-20-2-200    complex lifted from D4○SD16

Smallest permutation representation of C4.192+ 1+4
On 64 points
Generators in S64
(1 57 5 61)(2 58 6 62)(3 59 7 63)(4 60 8 64)(9 31 13 27)(10 32 14 28)(11 25 15 29)(12 26 16 30)(17 52 21 56)(18 53 22 49)(19 54 23 50)(20 55 24 51)(33 45 37 41)(34 46 38 42)(35 47 39 43)(36 48 40 44)
(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 40 13 36)(10 43 14 47)(11 34 15 38)(12 45 16 41)(18 59 22 63)(20 61 24 57)(25 46 29 42)(26 37 30 33)(27 48 31 44)(28 39 32 35)(50 58 54 62)(52 60 56 64)
(1 47 63 37)(2 40 64 42)(3 45 57 35)(4 38 58 48)(5 43 59 33)(6 36 60 46)(7 41 61 39)(8 34 62 44)(9 52 29 19)(10 22 30 55)(11 50 31 17)(12 20 32 53)(13 56 25 23)(14 18 26 51)(15 54 27 21)(16 24 28 49)
(1 33 5 37)(2 34 6 38)(3 35 7 39)(4 36 8 40)(9 21 13 17)(10 22 14 18)(11 23 15 19)(12 24 16 20)(25 50 29 54)(26 51 30 55)(27 52 31 56)(28 53 32 49)(41 57 45 61)(42 58 46 62)(43 59 47 63)(44 60 48 64)

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

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

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

Matrix representation of C4.192+ 1+4 in 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] >;

C4.192+ 1+4 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

Export

Character table of C4.192+ 1+4 in TeX

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