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G = C42.29C23  order 128 = 27

29th non-split extension by C42 of C23 acting faithfully

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

Aliases: C42.29C23, C4.182- (1+4), C4.372+ (1+4), C8⋊D4⋊25C2, C4⋊C4.144D4, C8⋊7D4.8C2, D4⋊Q8⋊33C2, C4.Q16⋊33C2, C2.36(D4○D8), C8.18D4⋊8C2, C4⋊C8.90C22, C22⋊C4.36D4, C2.36(Q8○D8), C23.97(C2×D4), D4.2D4⋊32C2, C4⋊C4.201C23, (C2×C8).177C23, (C2×C4).460C24, Q8.D4⋊32C2, (C2×D8).31C22, C4⋊Q8.131C22, C2.D8.52C22, (C2×D4).200C23, (C4×D4).138C22, C4⋊D4.54C22, (C2×Q8).188C23, (C4×Q8).135C22, (C2×Q16).32C22, C22⋊Q8.54C22, D4⋊C4.63C22, (C22×C8).158C22, Q8⋊C4.63C22, (C2×SD16).46C22, C4.4D4.47C22, C22.720(C22×D4), (C22×C4).1115C23, C42.6C22⋊16C2, (C2×M4(2)).98C22, C42⋊C2.178C22, C22.36C24⋊10C2, C2.79(C22.31C24), (C2×C4).584(C2×D4), SmallGroup(128,1994)

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

Derived series
C1 — C2×C4 — C42.29C23
Chief series
C1 — C2 — C4 — C2×C4 — C42 — C4×D4 — C22.36C24 — C42.29C23
Lower central
C1 — C2 — C2×C4 — C42.29C23
Upper central
C1 — C22 — C42⋊C2 — C42.29C23
Jennings
C1 — C2 — C2 — C2×C4 — C42.29C23

Subgroups: 348 in 173 conjugacy classes, 84 normal (26 characteristic)
C1, C2 [×3], C2 [×3], C4 [×2], C4 [×11], C22, C22 [×9], C8 [×4], C2×C4 [×2], C2×C4 [×4], C2×C4 [×10], D4 [×6], Q8 [×6], C23, C23 [×2], C42 [×2], C42 [×2], C22⋊C4 [×2], C22⋊C4 [×10], C4⋊C4 [×2], 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], C2×D4 [×2], C2×D4 [×2], C2×Q8 [×2], C2×Q8 [×2], D4⋊C4 [×4], Q8⋊C4 [×4], C4⋊C8 [×4], C2.D8 [×2], C2.D8 [×2], C42⋊C2, C4×D4 [×2], C4×Q8 [×2], C4⋊D4 [×2], C22⋊Q8 [×2], C22⋊Q8 [×2], C22.D4 [×2], C4.4D4 [×2], C4.4D4 [×2], C42⋊2C2 [×2], C4⋊Q8 [×2], C22×C8, C2×M4(2), C2×D8, C2×SD16 [×2], C2×Q16, C42.6C22, D4.2D4 [×2], Q8.D4 [×2], C8⋊7D4, C8.18D4, C8⋊D4 [×2], D4⋊Q8 [×2], C4.Q16 [×2], C22.36C24 [×2], C42.29C23

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, Q8○D8, C42.29C23

Generators and relations
 G = < a,b,c,d,e | a4=b4=c2=e2=1, d2=b2, ab=ba, cac=a-1b2, dad-1=ab2, ae=ea, cbc=ebe=b-1, bd=db, dcd-1=a2b2c, ece=bc, ede=a2b2d >

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

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

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

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

Matrix representation ►G ⊆ GL8(𝔽17)

011500000
100150000
000160000
001600000
00000010
00000001
000016000
000001600
,
01000000
160000000
160010000
011600000
00000100
000016000
00000001
000000160
,
98890000
1616990000
124810000
412910000
000014300
00003300
000000143
00000033
,
74030000
13101400000
007130000
004100000
000000130
000000013
000013000
000001300
,
10000000
016000000
016100000
100160000
00001000
000001600
00000010
000000016

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

Character table of C42.29C23

 class 12A2B2C2D2E2F4A4B4C4D4E4F4G4H4I4J4K4L4M8A8B8C8D8E8F
 size 11114882244444888888444488
ρ111111111111111111111111111    trivial
ρ21111-1-1111-1-111-1-11-11-11-11-111-1    linear of order 2
ρ311111-1111111111-1-111-1-1-1-1-1-1-1    linear of order 2
ρ41111-11111-1-111-1-1-111-1-11-11-1-11    linear of order 2
ρ51111-1-1-111-1-111-111-1-1111-11-1-11    linear of order 2
ρ6111111-11111111-111-1-11-1-1-1-1-1-1    linear of order 2
ρ71111-11-111-1-111-11-11-11-1-11-111-1    linear of order 2
ρ811111-1-11111111-1-1-1-1-1-1111111    linear of order 2
ρ911111-1-1111-1-1-1-11-111-111111-1-1    linear of order 2
ρ101111-11-111-11-1-11-1-1-1111-11-11-11    linear of order 2
ρ11111111-1111-1-1-1-111-11-1-1-1-1-1-111    linear of order 2
ρ121111-1-1-111-11-1-11-11111-11-11-11-1    linear of order 2
ρ131111-11111-11-1-111-1-1-1-111-11-11-1    linear of order 2
ρ1411111-11111-1-1-1-1-1-11-111-1-1-1-111    linear of order 2
ρ151111-1-1111-11-1-11111-1-1-1-11-11-11    linear of order 2
ρ161111111111-1-1-1-1-11-1-11-11111-1-1    linear of order 2
ρ172222-200-2-22-22-22000000000000    orthogonal lifted from D4
ρ182222200-2-2-222-2-2000000000000    orthogonal lifted from D4
ρ192222-200-2-222-22-2000000000000    orthogonal lifted from D4
ρ202222200-2-2-2-2-222000000000000    orthogonal lifted from D4
ρ214-44-4000-4400000000000000000    orthogonal lifted from 2+ (1+4)
ρ2244-4-4000000000000000002√202√200    orthogonal lifted from D4○D8
ρ2344-4-4000000000000000002√202√200    orthogonal lifted from D4○D8
ρ244-44-40004-400000000000000000    symplectic lifted from 2- (1+4), 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_4^2._{29}C_2^3
 
% in TeX
 
G:=Group("C4^2.29C2^3");
 
// GroupNames label
 
G:=SmallGroup(128,1994);
 
// by ID
 
G=gap.SmallGroup(128,1994);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,456,758,219,675,1018,304,4037,1027,124]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^4=b^4=c^2=e^2=1,d^2=b^2,a*b=b*a,c*a*c=a^-1*b^2,d*a*d^-1=a*b^2,a*e=e*a,c*b*c=e*b*e=b^-1,b*d=d*b,d*c*d^-1=a^2*b^2*c,e*c*e=b*c,e*d*e=a^2*b^2*d>;
 
// generators/relations
 

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