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G = C42.62C23  order 128 = 27

62nd non-split extension by C42 of C23 acting faithfully

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

Aliases: C42.62C23, C4.832- (1+4), C8⋊Q8⋊28C2, C8⋊8D4⋊23C2, C8⋊9D4⋊28C2, C8⋊D4⋊52C2, C4⋊C4.380D4, D4.Q8⋊43C2, Q8⋊Q8⋊22C2, (C2×D4).180D4, C8.35(C4○D4), C22⋊C4.64D4, C4⋊C8.121C22, C4⋊C4.253C23, (C2×C8).107C23, (C2×C4).540C24, C23.345(C2×D4), C4⋊Q8.172C22, SD16⋊C4⋊41C2, C2.93(D4⋊6D4), C8⋊C4.54C22, C4.Q8.68C22, C2.91(D4○SD16), (C2×D4).258C23, (C4×D4).180C22, C22⋊C8.99C22, (C4×Q8).179C22, (C2×Q8).242C23, M4(2)⋊C4⋊35C2, C2.D8.224C22, D4⋊C4.82C22, C23.20D4⋊45C2, C23.19D4⋊45C2, C23.25D4⋊31C2, C4⋊D4.107C22, C23.46D4⋊22C2, (C22×C8).291C22, Q8⋊C4.79C22, (C2×SD16).65C22, C22.800(C22×D4), C22⋊Q8.105C22, C42.C2.53C22, C2.95(D8⋊C22), (C22×C4).1168C23, C42⋊C2.211C22, C22.46C24⋊10C2, (C2×M4(2)).133C22, C22.49C24.5C2, C4.122(C2×C4○D4), (C2×C4).624(C2×D4), (C2×C4⋊C4).689C22, SmallGroup(128,2080)

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

Derived series
C1 — C2×C4 — C42.62C23
Chief series
C1 — C2 — C4 — C2×C4 — C22×C4 — C42⋊C2 — C22.49C24 — C42.62C23
Lower central
C1 — C2 — C2×C4 — C42.62C23
Upper central
C1 — C22 — C4×D4 — C42.62C23
Jennings
C1 — C2 — C2 — C2×C4 — C42.62C23

Subgroups: 320 in 174 conjugacy classes, 86 normal (84 characteristic)
C1, C2 [×3], C2 [×3], C4 [×2], C4 [×11], C22, C22 [×9], C8 [×2], C8 [×3], C2×C4 [×5], C2×C4 [×15], D4 [×5], Q8 [×3], C23 [×2], C23, C42, C42 [×4], C22⋊C4 [×2], C22⋊C4 [×8], C4⋊C4 [×7], C4⋊C4 [×7], C2×C8 [×4], C2×C8 [×2], M4(2) [×2], SD16 [×2], C22×C4 [×2], C22×C4 [×2], C2×D4 [×2], C2×D4 [×2], C2×Q8, C2×Q8, C8⋊C4, C22⋊C8 [×2], D4⋊C4 [×3], Q8⋊C4 [×3], C4⋊C8, C4.Q8 [×5], C2.D8 [×4], C2×C4⋊C4, C42⋊C2 [×3], C42⋊C2 [×2], C4×D4 [×2], C4×Q8, C4⋊D4 [×2], C4⋊D4, C22⋊Q8 [×2], C22.D4, C4.4D4 [×2], C42.C2, C42.C2, C42⋊2C2, C4⋊Q8, C22×C8, C2×M4(2), C2×SD16, C23.25D4, M4(2)⋊C4, C8⋊9D4, SD16⋊C4, C8⋊8D4, C8⋊D4, Q8⋊Q8, D4.Q8, C23.46D4, C23.19D4, C23.20D4 [×2], C8⋊Q8, C22.46C24, C22.49C24, C42.62C23

Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], C2×D4 [×6], C4○D4 [×2], C24, C22×D4, C2×C4○D4, 2- (1+4), D4⋊6D4, D8⋊C22, D4○SD16, C42.62C23

Generators and relations
 G = < a,b,c,d,e | a4=b4=e2=1, c2=a2, d2=a2b2, ab=ba, cac-1=eae=a-1b2, dad-1=ab2, cbc-1=dbd-1=b-1, be=eb, dcd-1=bc, ece=a2b2c, ede=b2d >

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

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

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

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

Matrix representation ►G ⊆ GL6(𝔽17)

1300000
040000
00160150
00016015
000010
000001
,
100000
010000
000100
0016000
000001
0000160
,
0130000
1300000
00512107
00121277
00125125
005555
,
1300000
0130000
00160150
000102
001010
00016016
,
040000
1300000
0016000
0001600
001010
000101

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

Character table of C42.62C23

 class 12A2B2C2D2E2F4A4B4C4D4E4F4G4H4I4J4K4L4M4N4O4P8A8B8C8D8E8F
 size 11114482222444444488888444488
ρ111111111111111111111111111111    trivial
ρ21111-1-1-111-1-1-111-1-11111-1-11-1-111-11    linear of order 2
ρ31111-11111-1-1-1-1-1-11-1-1111-1-111-1-1-11    linear of order 2
ρ411111-1-111111-1-11-1-1-111-11-1-1-1-1-111    linear of order 2
ρ51111-11111-1-1-1-11-11-111-1-11-1-1-1111-1    linear of order 2
ρ611111-1-111111-111-1-111-11-1-11111-1-1    linear of order 2
ρ71111111111111-1111-11-1-1-11-1-1-1-1-1-1    linear of order 2
ρ81111-1-1-111-1-1-11-1-1-11-11-111111-1-11-1    linear of order 2
ρ91111-1-1111-1-11111-111-11-1-1-111-1-11-1    linear of order 2
ρ10111111-11111-111-1111-1111-1-1-1-1-1-1-1    linear of order 2
ρ1111111-111111-1-1-1-1-1-1-1-11-1111111-1-1    linear of order 2
ρ121111-11-111-1-11-1-111-1-1-111-11-1-1111-1    linear of order 2
ρ1311111-111111-1-11-1-1-11-1-11-11-1-1-1-111    linear of order 2
ρ141111-11-111-1-11-1111-11-1-1-11111-1-1-11    linear of order 2
ρ151111-1-1111-1-111-11-11-1-1-111-1-1-111-11    linear of order 2
ρ16111111-11111-11-1-111-1-1-1-1-1-1111111    linear of order 2
ρ1722222-20-2-2-2-202002-2000000000000    orthogonal lifted from D4
ρ182222-220-2-2220200-2-2000000000000    orthogonal lifted from D4
ρ192222220-2-2-2-20-200-22000000000000    orthogonal lifted from D4
ρ202222-2-20-2-2220-20022000000000000    orthogonal lifted from D4
ρ212-22-20002-2002i02i2i002i000002-20000    complex lifted from C4○D4
ρ222-22-20002-2002i02i2i002i000002-20000    complex lifted from C4○D4
ρ232-22-20002-2002i02i2i002i00000-220000    complex lifted from C4○D4
ρ242-22-20002-2002i02i2i002i00000-220000    complex lifted from C4○D4
ρ254-44-4000-4400000000000000000000    symplectic lifted from 2- (1+4), Schur index 2
ρ264-4-44000004i4i000000000000000000    complex lifted from D8⋊C22
ρ274-4-44000004i4i000000000000000000    complex lifted from D8⋊C22
ρ2844-4-40000000000000000000002√-22√-200    complex lifted from D4○SD16
ρ2944-4-40000000000000000000002√-22√-200    complex lifted from D4○SD16

In GAP, Magma, Sage, TeX 

C_4^2._{62}C_2^3
 
% in TeX
 
G:=Group("C4^2.62C2^3");
 
// GroupNames label
 
G:=SmallGroup(128,2080);
 
// by ID
 
G=gap.SmallGroup(128,2080);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,-2,560,253,758,723,100,346,248,4037,1027,124]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^4=b^4=e^2=1,c^2=a^2,d^2=a^2*b^2,a*b=b*a,c*a*c^-1=e*a*e=a^-1*b^2,d*a*d^-1=a*b^2,c*b*c^-1=d*b*d^-1=b^-1,b*e=e*b,d*c*d^-1=b*c,e*c*e=a^2*b^2*c,e*d*e=b^2*d>;
 
// generators/relations
 

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