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G = C4.172+ (1+4)  order 128 = 27

17th 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: C4.172+ (1+4), (C2×D4).157D4, C8.D4⋊13C2, C8.18D4⋊5C2, (C2×Q8).133D4, C2.28(Q8○D8), C23.91(C2×D4), C4⋊C4.141C23, (C2×C4).400C24, (C2×C8).156C23, C22⋊Q16⋊20C2, C4.Q8.35C22, C2.D8.33C22, C22⋊C8.43C22, (C2×Q16).23C22, (C2×Q8).138C23, C22⋊Q8.40C22, C23.20D4⋊27C2, C2.81(C23⋊3D4), (C22×C8).151C22, (C22×C4).303C23, Q8⋊C4.41C22, C22.660(C22×D4), (C2×M4(2)).84C22, (C22×Q8).318C22, C42⋊C2.156C22, C23.38C23.14C2, (C2×C4).536(C2×D4), (C22×C8)⋊C2.4C2, (C2×C4○D4).168C22, SmallGroup(128,1934)

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

Derived series
C1 — C2×C4 — C4.172+ (1+4)
Chief series
C1 — C2 — C4 — C2×C4 — C22×C4 — C22×Q8 — C23.38C23 — C4.172+ (1+4)
Lower central
C1 — C2 — C2×C4 — C4.172+ (1+4)
Upper central
C1 — C22 — C2×C4○D4 — C4.172+ (1+4)
Jennings
C1 — C2 — C2 — C2×C4 — C4.172+ (1+4)

Subgroups: 364 in 189 conjugacy classes, 84 normal (16 characteristic)
C1, C2, C2 [×2], C2 [×3], C4 [×2], C4 [×11], C22, C22 [×9], C8 [×4], C2×C4 [×2], C2×C4 [×2], C2×C4 [×18], D4 [×3], Q8 [×13], C23, C23 [×2], C42 [×2], C22⋊C4 [×10], C4⋊C4 [×4], C4⋊C4 [×8], C2×C8 [×4], C2×C8, M4(2), Q16 [×4], C22×C4, C22×C4 [×2], C22×C4 [×2], C2×D4, C2×D4 [×2], C2×Q8, C2×Q8 [×4], C2×Q8 [×6], C4○D4 [×2], C22⋊C8 [×4], Q8⋊C4 [×8], C4.Q8 [×2], C2.D8 [×2], C42⋊C2 [×2], C22⋊Q8 [×8], C22.D4 [×4], C4.4D4 [×2], C4⋊Q8 [×2], C22×C8, C2×M4(2), C2×Q16 [×4], C22×Q8 [×2], C2×C4○D4, (C22×C8)⋊C2, C22⋊Q16 [×4], C8.18D4 [×2], C8.D4 [×2], C23.20D4 [×4], C23.38C23 [×2], C4.172+ (1+4)

Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], C2×D4 [×6], C24, C22×D4, 2+ (1+4) [×2], C23⋊3D4, Q8○D8 [×2], C4.172+ (1+4)

Generators and relations
 G = < a,b,c,d,e | a4=c2=1, b4=e2=a2, d2=a-1b2, dbd-1=ab=ba, ac=ca, dad-1=a-1, ae=ea, cbc=ab3, be=eb, dcd-1=ece-1=a2c, ede-1=a-1b2d >

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

(2 62)(4 64)(6 58)(8 60)(9 33)(10 14)(11 35)(12 16)(13 37)(15 39)(17 21)(18 46)(19 23)(20 48)(22 42)(24 44)(25 50)(27 52)(29 54)(31 56)(34 38)(36 40)(41 45)(43 47)

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

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

Matrix representation ►G ⊆ GL8(𝔽17)

10000000
01000000
00100000
00010000
000001600
00001000
000037162
0000155161
,
901660000
901680000
014830000
98000000
0000141400
000031400
000094116
00001115140
,
10000000
01000000
101600000
000160000
00001000
00000100
000000160
00001410016
,
811000000
89000000
014830000
80190000
000013263
00001410815
000016700
00004161411
,
910000000
98000000
95850000
80190000
00000010
00001410115
000016000
0000108127

G:=sub<GL(8,GF(17))| [1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,3,15,0,0,0,0,16,0,7,5,0,0,0,0,0,0,16,16,0,0,0,0,0,0,2,1],[9,9,0,9,0,0,0,0,0,0,14,8,0,0,0,0,16,16,8,0,0,0,0,0,6,8,3,0,0,0,0,0,0,0,0,0,14,3,9,11,0,0,0,0,14,14,4,15,0,0,0,0,0,0,11,14,0,0,0,0,0,0,6,0],[1,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,1,0,0,14,0,0,0,0,0,1,0,10,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16],[8,8,0,8,0,0,0,0,11,9,14,0,0,0,0,0,0,0,8,1,0,0,0,0,0,0,3,9,0,0,0,0,0,0,0,0,13,14,16,4,0,0,0,0,2,10,7,16,0,0,0,0,6,8,0,14,0,0,0,0,3,15,0,11],[9,9,9,8,0,0,0,0,10,8,5,0,0,0,0,0,0,0,8,1,0,0,0,0,0,0,5,9,0,0,0,0,0,0,0,0,0,14,16,10,0,0,0,0,0,10,0,8,0,0,0,0,1,1,0,12,0,0,0,0,0,15,0,7] >;
 

Character table of C4.172+ (1+4)

 class 12A2B2C2D2E2F4A4B4C4D4E4F4G4H4I4J4K4L4M8A8B8C8D8E8F
 size 11114442244488888888444488
ρ111111111111111111111111111    trivial
ρ21111-11-111-1-11-111-1-111-11-11-1-11    linear of order 2
ρ31111-1-1111-11-1-1111-1-1-11-11-11-11    linear of order 2
ρ411111-1-1111-1-1111-11-1-1-1-1-1-1-111    linear of order 2
ρ51111-11-111-1-11-11-111-11-1-11-111-1    linear of order 2
ρ611111111111111-1-1-1-111-1-1-1-1-1-1    linear of order 2
ρ711111-1-1111-1-111-11-11-1-11111-1-1    linear of order 2
ρ81111-1-1111-11-1-11-1-111-111-11-11-1    linear of order 2
ρ911111-1-1111-1-1-1-11-11-1111111-1-1    linear of order 2
ρ101111-1-1111-11-11-111-1-11-11-11-11-1    linear of order 2
ρ111111-11-111-1-111-11-1-11-11-11-111-1    linear of order 2
ρ12111111111111-1-11111-1-1-1-1-1-1-1-1    linear of order 2
ρ131111-1-1111-11-11-1-1-1111-1-11-11-11    linear of order 2
ρ1411111-1-1111-1-1-1-1-11-1111-1-1-1-111    linear of order 2
ρ15111111111111-1-1-1-1-1-1-1-1111111    linear of order 2
ρ161111-11-111-1-111-1-111-1-111-11-1-11    linear of order 2
ρ172222-222-2-22-2-200000000000000    orthogonal lifted from D4
ρ182222-2-2-2-2-222200000000000000    orthogonal lifted from D4
ρ19222222-2-2-2-22-200000000000000    orthogonal lifted from D4
ρ2022222-22-2-2-2-2200000000000000    orthogonal lifted from D4
ρ214-44-40004-400000000000000000    orthogonal lifted from 2+ (1+4)
ρ224-44-4000-4400000000000000000    orthogonal lifted from 2+ (1+4)
ρ2344-4-4000000000000000002√202√200    symplectic lifted from Q8○D8, Schur index 2
ρ2444-4-4000000000000000002√202√200    symplectic lifted from Q8○D8, Schur index 2
ρ254-4-4400000000000000002√202√2000    symplectic lifted from Q8○D8, Schur index 2
ρ264-4-4400000000000000002√202√2000    symplectic lifted from Q8○D8, Schur index 2

In GAP, Magma, Sage, TeX 

C_4._{17}2_+^{(1+4)}
 
% in TeX
 
G:=Group("C4.17ES+(2,2)");
 
// GroupNames label
 
G:=SmallGroup(128,1934);
 
// by ID
 
G=gap.SmallGroup(128,1934);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,-2,448,253,456,758,219,352,675,1018,4037,1027,124]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^4=c^2=1,b^4=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=a*b^3,b*e=e*b,d*c*d^-1=e*c*e^-1=a^2*c,e*d*e^-1=a^-1*b^2*d>;
 
// generators/relations
 

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