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G = C4.162+ 1+4order 128 = 27

16th 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: C4.162+ 1+4, C88D44C2, C8⋊D418C2, Q8⋊D410C2, (C2×D4).156D4, C8.D412C2, (C2×Q8).132D4, C2.27(Q8○D8), D4.7D424C2, C8.18D418C2, C4⋊C4.140C23, (C2×C8).322C23, (C2×C4).399C24, C22⋊Q1619C2, C23.281(C2×D4), C4.Q8.81C22, C2.43(D4○SD16), (C2×D4).150C23, C4⋊D4.40C22, C22⋊C8.42C22, (C2×Q8).137C23, (C2×Q16).22C22, Q8⋊C4.1C22, C2.D8.101C22, C22⋊Q8.39C22, D4⋊C4.40C22, C23.47D410C2, C23.48D421C2, C23.19D427C2, C23.20D426C2, C2.80(C233D4), (C22×C8).187C22, (C22×C4).302C23, (C2×SD16).30C22, C22.659(C22×D4), (C2×M4(2)).83C22, (C22×Q8).317C22, C42⋊C2.155C22, C23.38C2315C2, C22.31C24.10C2, (C2×C4).151(C2×D4), (C22×C8)⋊C215C2, (C2×C4⋊C4).642C22, (C2×C4○D4).167C22, SmallGroup(128,1933)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C4.162+ 1+4
C1C2C4C2×C4C22×C4C22×Q8C23.38C23 — C4.162+ 1+4
C1C2C2×C4 — C4.162+ 1+4
C1C22C2×C4○D4 — C4.162+ 1+4
C1C2C2C2×C4 — C4.162+ 1+4

Generators and relations for C4.162+ 1+4
 G = < a,b,c,d,e | a4=c2=e2=1, b4=a2, d2=ab2, dbd-1=ab=ba, ac=ca, dad-1=a-1, ae=ea, cbc=ab3, be=eb, dcd-1=ece=a2c, ede=a-1b2d >

Subgroups: 404 in 195 conjugacy classes, 84 normal (all characteristic)
C1, C2 [×3], C2 [×4], C4 [×2], C4 [×10], C22, C22 [×12], C8 [×4], C2×C4 [×4], C2×C4 [×18], D4 [×9], Q8 [×9], C23 [×3], C23, C42, C22⋊C4 [×9], C4⋊C4 [×4], C4⋊C4 [×7], C2×C8 [×4], C2×C8, M4(2), SD16 [×2], Q16 [×2], C22×C4 [×3], C22×C4 [×3], C2×D4 [×4], C2×D4 [×3], C2×Q8 [×4], C2×Q8 [×3], C4○D4 [×4], C22⋊C8 [×4], D4⋊C4 [×2], Q8⋊C4 [×6], C4.Q8 [×2], C2.D8 [×2], C2×C4⋊C4, C42⋊C2, C4⋊D4 [×2], C4⋊D4 [×3], C22⋊Q8 [×6], C22⋊Q8, C22.D4 [×2], C4.4D4, C4⋊Q8, C22×C8, C2×M4(2), C2×SD16 [×2], C2×Q16 [×2], C22×Q8, C2×C4○D4 [×2], (C22×C8)⋊C2, Q8⋊D4, C22⋊Q16, D4.7D4 [×2], C88D4, C8.18D4, C8⋊D4, C8.D4, C23.19D4, C23.47D4, C23.48D4, C23.20D4, C23.38C23, C22.31C24, C4.162+ 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, Q8○D8, C4.162+ 1+4

Character table of C4.162+ 1+4

 class 12A2B2C2D2E2F2G4A4B4C4D4E4F4G4H4I4J4K4L8A8B8C8D8E8F
 size 11114448224448888888444488
ρ111111111111111111111111111    trivial
ρ211111-1-1-111-11-1-11-11-111-1-1-1-111    linear of order 2
ρ31111-1-11-111-1-11-111-111-1-11-111-1    linear of order 2
ρ41111-11-11111-1-111-1-1-11-11-11-11-1    linear of order 2
ρ51111-1-11-111-1-11111-1-1-111-11-1-11    linear of order 2
ρ61111-11-11111-1-1-11-1-11-11-11-11-11    linear of order 2
ρ71111111111111-1111-1-1-1-1-1-1-1-1-1    linear of order 2
ρ811111-1-1-111-11-111-111-1-11111-1-1    linear of order 2
ρ911111-1-1111-11-11-11-11-1-1-1-1-1-111    linear of order 2
ρ101111111-111111-1-1-1-1-1-1-1111111    linear of order 2
ρ111111-11-1-1111-1-1-1-1111-111-11-11-1    linear of order 2
ρ121111-1-11111-1-111-1-11-1-11-11-111-1    linear of order 2
ρ131111-11-1-1111-1-11-111-11-1-11-11-11    linear of order 2
ρ141111-1-11111-1-11-1-1-1111-11-11-1-11    linear of order 2
ρ1511111-1-1111-11-1-1-11-1-1111111-1-1    linear of order 2
ρ161111111-1111111-1-1-1111-1-1-1-1-1-1    linear of order 2
ρ172222-2-2-20-2-22220000000000000    orthogonal lifted from D4
ρ182222-2220-2-2-22-20000000000000    orthogonal lifted from D4
ρ1922222-220-2-22-2-20000000000000    orthogonal lifted from D4
ρ20222222-20-2-2-2-220000000000000    orthogonal lifted from D4
ρ214-44-40000-440000000000000000    orthogonal lifted from 2+ 1+4
ρ224-44-400004-40000000000000000    orthogonal lifted from 2+ 1+4
ρ2344-4-400000000000000000220-2200    symplectic lifted from Q8○D8, Schur index 2
ρ2444-4-400000000000000000-2202200    symplectic lifted from Q8○D8, Schur index 2
ρ254-4-440000000000000000-2-202-2000    complex lifted from D4○SD16
ρ264-4-4400000000000000002-20-2-2000    complex lifted from D4○SD16

Smallest permutation representation of C4.162+ 1+4
On 64 points
Generators in S64
(1 63 5 59)(2 64 6 60)(3 57 7 61)(4 58 8 62)(9 28 13 32)(10 29 14 25)(11 30 15 26)(12 31 16 27)(17 52 21 56)(18 53 22 49)(19 54 23 50)(20 55 24 51)(33 44 37 48)(34 45 38 41)(35 46 39 42)(36 47 40 43)
(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 55)(2 19)(3 49)(4 21)(5 51)(6 23)(7 53)(8 17)(9 38)(10 44)(11 40)(12 46)(13 34)(14 48)(15 36)(16 42)(18 57)(20 59)(22 61)(24 63)(25 33)(26 47)(27 35)(28 41)(29 37)(30 43)(31 39)(32 45)(50 60)(52 62)(54 64)(56 58)
(1 62 57 6)(2 5 58 61)(3 60 59 4)(7 64 63 8)(9 12 30 25)(10 32 31 11)(13 16 26 29)(14 28 27 15)(17 49 54 20)(18 19 55 56)(21 53 50 24)(22 23 51 52)(33 34 46 47)(35 40 48 45)(36 44 41 39)(37 38 42 43)
(1 37)(2 38)(3 39)(4 40)(5 33)(6 34)(7 35)(8 36)(9 23)(10 24)(11 17)(12 18)(13 19)(14 20)(15 21)(16 22)(25 55)(26 56)(27 49)(28 50)(29 51)(30 52)(31 53)(32 54)(41 64)(42 57)(43 58)(44 59)(45 60)(46 61)(47 62)(48 63)

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

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

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

Matrix representation of C4.162+ 1+4 in GL8(𝔽17)

160000000
016000000
001600000
000160000
000001600
00001000
000000016
00000010
,
07040000
1001300000
0130100000
40700000
0000141400
000031400
0000001414
000000314
,
01000000
10000000
00010000
00100000
00000010
00000001
00001000
00000100
,
07040000
70400000
0130100000
1301000000
000031400
0000141400
000000143
00000033
,
0130100000
1301000000
07040000
70400000
000001100
0000160010
000070016
00000710

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

C4.162+ 1+4 in GAP, Magma, Sage, TeX

C_4._{16}2_+^{1+4}
% in TeX

G:=Group("C4.16ES+(2,2)");
// GroupNames label

G:=SmallGroup(128,1933);
// by ID

G=gap.SmallGroup(128,1933);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,758,219,352,675,1018,4037,1027,124]);
// Polycyclic

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

Export

Character table of C4.162+ 1+4 in TeX

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