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G = C4.2- (1+4)  order 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), C8⋊8D4⋊6C2, C8⋊7D4⋊20C2, C8⋊D4⋊23C2, C4⋊D8⋊31C2, C8⋊2D4⋊17C2, C4⋊C4.139D4, Q8.Q8⋊30C2, D4.Q8⋊30C2, D4⋊2Q8⋊15C2, D4⋊Q8⋊32C2, C4⋊SD16⋊15C2, C2.34(D4○D8), C4⋊C8.85C22, C22⋊C4.31D4, C23.92(C2×D4), D4.2D4⋊31C2, C4⋊C4.196C23, (C2×C8).174C23, (C2×C4).455C24, (C2×D8).30C22, C4⋊Q8.127C22, C4.Q8.95C22, C2.D8.48C22, C2.52(D4○SD16), (C2×D4).196C23, (C4×D4).134C22, C4⋊D4.50C22, C4⋊1D4.71C22, (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, C22.34C24⋊7C2, C22.36C24⋊9C2, (C22×C4).1110C23, C42.6C22⋊12C2, (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

Derived series
C1 — C2×C4 — C4.2- (1+4)
Chief series
C1 — C2 — C4 — C2×C4 — C42 — C4×D4 — C22.36C24 — C4.2- (1+4)
Lower central
C1 — C2 — C2×C4 — C4.2- (1+4)
Upper central
C1 — C22 — C42⋊C2 — C4.2- (1+4)
Jennings
C1 — C2 — C2 — C2×C4 — C4.2- (1+4)

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, C42⋊2C2, C4⋊1D4, C4⋊Q8, C22×C8, C2×M4(2), C2×D8 [×2], C2×SD16 [×2], C42.6C22, C4⋊D8, C4⋊SD16, D4.2D4 [×2], C8⋊8D4, C8⋊7D4, C8⋊D4, C8⋊2D4, D4⋊Q8, D4⋊2Q8, 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)

Generators and relations
 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 >

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

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

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

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

Matrix representation ►G ⊆ 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] >;
 

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
ρ214-4-4400000000000000002√202√2000    orthogonal lifted from D4○D8
ρ224-44-400004-40000000000000000    orthogonal lifted from 2+ (1+4)
ρ234-4-4400000000000000002√202√2000    orthogonal lifted from D4○D8
ρ244-44-40000-440000000000000000    symplectic lifted from 2- (1+4), Schur index 2
ρ2544-4-4000000000000000002√-202√-200    complex lifted from D4○SD16
ρ2644-4-4000000000000000002√-202√-200    complex lifted from D4○SD16

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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