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G = C4.2- 1+4order 128 = 27

13rd non-split extension by C4 of 2- 1+4 acting via 2- 1+4/C2×Q8=C2

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

Aliases: C42.24C23, C4.132- 1+4, C4.322+ 1+4, C88D46C2, C4⋊D831C2, C8⋊D423C2, C82D417C2, C87D420C2, C4⋊C4.139D4, Q8.Q830C2, D4.Q830C2, D4⋊Q832C2, C4⋊SD1615C2, D42Q815C2, C2.34(D4○D8), C4⋊C8.85C22, C22⋊C4.31D4, C23.92(C2×D4), D4.2D431C2, C4⋊C4.196C23, (C2×C4).455C24, (C2×C8).174C23, (C2×D8).30C22, C4⋊Q8.127C22, C4.Q8.95C22, C2.D8.48C22, C2.52(D4○SD16), (C4×D4).134C22, (C2×D4).196C23, C4⋊D4.50C22, C41D4.71C22, (C2×Q8).184C23, (C4×Q8).131C22, C22⋊Q8.50C22, D4⋊C4.61C22, (C22×C8).190C22, Q8⋊C4.59C22, (C2×SD16).44C22, C4.4D4.45C22, C22.715(C22×D4), C42.C2.30C22, C22.34C247C2, (C22×C4).1110C23, C22.36C249C2, C42.6C2212C2, (C2×M4(2)).93C22, C42⋊C2.173C22, C2.74(C22.31C24), (C2×C4).579(C2×D4), SmallGroup(128,1989)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C4.2- 1+4
C1C2C4C2×C4C42C4×D4C22.36C24 — C4.2- 1+4
C1C2C2×C4 — C4.2- 1+4
C1C22C42⋊C2 — C4.2- 1+4
C1C2C2C2×C4 — C4.2- 1+4

Generators and relations for C4.2- 1+4
 G = < a,b,c,d,e | a4=b4=c2=1, d2=ab2, e2=b2, bab-1=a-1, ac=ca, ad=da, ae=ea, cbc=b-1, dbd-1=ab, ebe-1=a2b, cd=dc, ece-1=a2c, ede-1=a2b2d >

Subgroups: 388 in 179 conjugacy classes, 84 normal (all characteristic)
C1, C2 [×3], C2 [×4], C4 [×2], C4 [×10], C22, C22 [×12], C8 [×4], C2×C4 [×6], C2×C4 [×10], D4 [×11], Q8 [×3], C23, C23 [×3], C42 [×2], C42, C22⋊C4 [×2], C22⋊C4 [×9], C4⋊C4 [×6], C4⋊C4 [×5], C2×C8 [×4], C2×C8, M4(2), D8 [×2], SD16 [×2], C22×C4, C22×C4 [×3], C2×D4 [×3], C2×D4 [×5], C2×Q8, C2×Q8, D4⋊C4 [×6], Q8⋊C4 [×2], C4⋊C8 [×4], C4.Q8 [×2], C2.D8 [×2], C42⋊C2, C4×D4 [×3], C4×Q8, C4⋊D4 [×3], C4⋊D4 [×2], C22⋊Q8, C22⋊Q8, C22.D4 [×3], C4.4D4, C4.4D4, C42.C2, C422C2, C41D4, C4⋊Q8, C22×C8, C2×M4(2), C2×D8 [×2], C2×SD16 [×2], C42.6C22, C4⋊D8, C4⋊SD16, D4.2D4 [×2], C88D4, C87D4, C8⋊D4, C82D4, D4⋊Q8, D42Q8, D4.Q8, Q8.Q8, C22.34C24, C22.36C24, C4.2- 1+4
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], C2×D4 [×6], C24, C22×D4, 2+ 1+4, 2- 1+4, C22.31C24, D4○D8, D4○SD16, C4.2- 1+4

Character table of C4.2- 1+4

 class 12A2B2C2D2E2F2G4A4B4C4D4E4F4G4H4I4J4K4L8A8B8C8D8E8F
 size 11114888224444488888444488
ρ111111111111111111111111111    trivial
ρ21111111-111-1-11-1-1-11-1-11-1-1-1-111    linear of order 2
ρ3111111-1111-1-11-1-1-11-11-11111-1-1    linear of order 2
ρ4111111-1-11111111111-1-1-1-1-1-1-1-1    linear of order 2
ρ51111-11-11111-1-11-1-1-11-111-11-1-11    linear of order 2
ρ61111-11-1-111-11-1-111-1-111-11-11-11    linear of order 2
ρ71111-111111-11-1-111-1-1-1-11-11-11-1    linear of order 2
ρ81111-111-1111-1-11-1-1-111-1-11-111-1    linear of order 2
ρ91111-1-1-1-111-11-1-11-111111-11-11-1    linear of order 2
ρ101111-1-1-11111-1-11-111-1-11-11-111-1    linear of order 2
ρ111111-1-11-1111-1-11-111-11-11-11-1-11    linear of order 2
ρ121111-1-11111-11-1-11-111-1-1-11-11-11    linear of order 2
ρ1311111-11-111-1-11-1-11-11-111111-1-1    linear of order 2
ρ1411111-1111111111-1-1-111-1-1-1-1-1-1    linear of order 2
ρ1511111-1-1-11111111-1-1-1-1-1111111    linear of order 2
ρ1611111-1-1111-1-11-1-11-111-1-1-1-1-111    linear of order 2
ρ1722222000-2-22-2-2-2200000000000    orthogonal lifted from D4
ρ182222-2000-2-2-2-222200000000000    orthogonal lifted from D4
ρ192222-2000-2-2222-2-200000000000    orthogonal lifted from D4
ρ2022222000-2-2-22-22-200000000000    orthogonal lifted from D4
ρ214-4-440000000000000000220-22000    orthogonal lifted from D4○D8
ρ224-44-400004-40000000000000000    orthogonal lifted from 2+ 1+4
ρ234-4-440000000000000000-22022000    orthogonal lifted from D4○D8
ρ244-44-40000-440000000000000000    symplectic lifted from 2- 1+4, Schur index 2
ρ2544-4-400000000000000000-2-202-200    complex lifted from D4○SD16
ρ2644-4-4000000000000000002-20-2-200    complex lifted from D4○SD16

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

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

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

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

Matrix representation of C4.2- 1+4 in GL8(𝔽17)

115000000
116000000
016010000
1161600000
00001000
00000100
00000010
00000001
,
160020000
001610000
016010000
00010000
000010150
000001015
000010160
000001016
,
101500000
001610000
001600000
011600000
000010150
000001015
000000160
000000016
,
110660000
140600000
014330000
1414330000
00007100
000011000
00000071
000000110
,
160000000
016000000
160100000
160010000
00000100
000016000
00000001
000000160

G:=sub<GL(8,GF(17))| [1,1,0,1,0,0,0,0,15,16,16,16,0,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[16,0,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,2,1,1,1,0,0,0,0,0,0,0,0,1,0,1,0,0,0,0,0,0,1,0,1,0,0,0,0,15,0,16,0,0,0,0,0,0,15,0,16],[1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,15,16,16,16,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,15,0,16,0,0,0,0,0,0,15,0,16],[11,14,0,14,0,0,0,0,0,0,14,14,0,0,0,0,6,6,3,3,0,0,0,0,6,0,3,3,0,0,0,0,0,0,0,0,7,1,0,0,0,0,0,0,1,10,0,0,0,0,0,0,0,0,7,1,0,0,0,0,0,0,1,10],[16,0,16,16,0,0,0,0,0,16,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0] >;

C4.2- 1+4 in GAP, Magma, Sage, TeX

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

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

G:=SmallGroup(128,1989);
// by ID

G=gap.SmallGroup(128,1989);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,758,219,675,1018,80,4037,1027,124]);
// Polycyclic

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

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Character table of C4.2- 1+4 in TeX

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