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G = C42.60C23  order 128 = 27

60th non-split extension by C42 of C23 acting faithfully

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

Aliases: C42.60C23, C4.812- (1+4), C8⋊Q8⋊26C2, C8⋊8D4⋊22C2, C8⋊D4⋊51C2, C8⋊9D4⋊26C2, C4⋊C4.168D4, D4⋊3Q8⋊8C2, Q8.Q8⋊41C2, D4⋊Q8⋊38C2, (C2×D4).332D4, C8.33(C4○D4), C2.57(D4○D8), C4⋊C8.119C22, C4⋊C4.251C23, (C2×C8).201C23, (C2×C4).538C24, C22⋊C4.177D4, C23.482(C2×D4), C4⋊Q8.170C22, SD16⋊C4⋊40C2, C2.91(D4⋊6D4), C4.Q8.67C22, C8⋊C4.52C22, (C2×D4).256C23, (C4×D4).178C22, C22.D8⋊33C2, C22⋊C8.97C22, (C2×Q8).241C23, (C4×Q8).178C22, M4(2)⋊C4⋊33C2, C2.D8.131C22, D4⋊C4.80C22, C4⋊D4.105C22, C23.48D4⋊33C2, C23.46D4⋊21C2, C23.20D4⋊44C2, (C22×C8).289C22, (C22×C4).342C23, Q8⋊C4.78C22, (C2×SD16).64C22, C22.798(C22×D4), C22.9(C8.C22), C22⋊Q8.104C22, C42.C2.51C22, C42⋊C2.209C22, (C2×M4(2)).131C22, C22.47C24.3C2, (C2×C2.D8)⋊43C2, C4.120(C2×C4○D4), (C2×C4).622(C2×D4), C2.83(C2×C8.C22), (C2×C4⋊C4).687C22, SmallGroup(128,2078)

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

Derived series
C1 — C2×C4 — C42.60C23
Chief series
C1 — C2 — C4 — C2×C4 — C22×C4 — C2×C4⋊C4 — C22.47C24 — C42.60C23
Lower central
C1 — C2 — C2×C4 — C42.60C23
Upper central
C1 — C22 — C4×D4 — C42.60C23
Jennings
C1 — C2 — C2 — C2×C4 — C42.60C23

Subgroups: 336 in 178 conjugacy classes, 88 normal (84 characteristic)
C1, C2 [×3], C2 [×4], C4 [×2], C4 [×11], C22, C22 [×2], C22 [×8], C8 [×2], C8 [×3], C2×C4 [×5], C2×C4 [×15], D4 [×7], Q8 [×3], C23 [×2], C23, C42, C42 [×2], C22⋊C4 [×2], C22⋊C4 [×6], C4⋊C4 [×7], C4⋊C4 [×9], C2×C8 [×4], C2×C8 [×2], M4(2) [×2], SD16 [×2], C22×C4 [×2], C22×C4 [×4], 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 [×3], C2.D8 [×6], C2×C4⋊C4 [×3], C42⋊C2, C4×D4 [×2], C4×D4 [×2], C4×Q8, C4⋊D4 [×2], C4⋊D4, C22⋊Q8 [×2], C22⋊Q8 [×2], C22.D4, C42.C2, C42.C2, C42⋊2C2, C4⋊Q8, C22×C8, C2×M4(2), C2×SD16, C2×C2.D8, M4(2)⋊C4, C8⋊9D4, SD16⋊C4, C8⋊8D4, C8⋊D4, D4⋊Q8, Q8.Q8, C22.D8, C23.46D4, C23.48D4, C23.20D4, C8⋊Q8, C22.47C24, D4⋊3Q8, C42.60C23

Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], C2×D4 [×6], C4○D4 [×2], C24, C8.C22 [×2], C22×D4, C2×C4○D4, 2- (1+4), D4⋊6D4, C2×C8.C22, D4○D8, C42.60C23

Generators and relations
 G = < a,b,c,d,e | a4=b4=e2=1, c2=d2=a2b2, ab=ba, cac-1=eae=a-1, dad-1=ab2, cbc-1=dbd-1=b-1, be=eb, dcd-1=bc, ece=a2c, 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 12 26 49)(2 9 27 50)(3 10 28 51)(4 11 25 52)(5 47 59 62)(6 48 60 63)(7 45 57 64)(8 46 58 61)(13 31 38 36)(14 32 39 33)(15 29 40 34)(16 30 37 35)(17 54 44 23)(18 55 41 24)(19 56 42 21)(20 53 43 22)

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

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

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

Matrix representation ►G ⊆ GL6(𝔽17)

1300000
040000
000010
0000016
001000
0001600
,
100000
010000
000100
0016000
0000016
000010
,
0130000
1300000
00215152
0015151515
00151522
00215215
,
1300000
0130000
000010
000001
0016000
0001600
,
040000
1300000
0000160
000001
0016000
000100

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

Character table of C42.60C23

 class 12A2B2C2D2E2F2G4A4B4C4D4E4F4G4H4I4J4K4L4M4N4O8A8B8C8D8E8F
 size 11112248224444444488888444488
ρ111111111111111111111111111111    trivial
ρ2111111-1-111-1-111-1-1-111-11-11-1-1-1-111    linear of order 2
ρ31111-1-1-1111-11-11-11-11-111-1-11-1-111-1    linear of order 2
ρ41111-1-11-1111-1-111-111-1-111-1-111-11-1    linear of order 2
ρ51111-1-11111-1-1-1-11-1-1-111-1-11-111-11-1    linear of order 2
ρ61111-1-1-1-11111-1-1-111-11-1-1111-1-111-1    linear of order 2
ρ7111111-11111-11-1-1-11-1-11-11-1-1-1-1-111    linear of order 2
ρ81111111-111-111-111-1-1-1-1-1-1-1111111    linear of order 2
ρ91111-1-1-111111-11-1111-1-1-1-11-111-1-11    linear of order 2
ρ101111-1-11-111-1-1-111-1-11-11-1111-1-11-11    linear of order 2
ρ111111111111-111111-111-1-11-1-1-1-1-1-1-1    linear of order 2
ρ12111111-1-1111-111-1-11111-1-1-11111-1-1    linear of order 2
ρ13111111-1111-1-11-1-1-1-1-1-1-11111111-1-1    linear of order 2
ρ141111111-111111-1111-1-111-11-1-1-1-1-1-1    linear of order 2
ρ151111-1-111111-1-1-11-11-11-11-1-11-1-11-11    linear of order 2
ρ161111-1-1-1-111-11-1-1-11-1-11111-1-111-1-11    linear of order 2
ρ172222-2-2-20-2-202202-20000000000000    orthogonal lifted from D4
ρ1822222220-2-202-20-2-20000000000000    orthogonal lifted from D4
ρ19222222-20-2-20-2-20220000000000000    orthogonal lifted from D4
ρ202222-2-220-2-20-220-220000000000000    orthogonal lifted from D4
ρ212-22-200002-22i002i002i2i0000002-2000    complex lifted from C4○D4
ρ222-22-200002-22i002i002i2i0000002-2000    complex lifted from C4○D4
ρ232-22-200002-22i002i002i2i000000-22000    complex lifted from C4○D4
ρ242-22-200002-22i002i002i2i000000-22000    complex lifted from C4○D4
ρ2544-4-400000000000000000002√2002√200    orthogonal lifted from D4○D8
ρ2644-4-400000000000000000002√2002√200    orthogonal lifted from D4○D8
ρ274-44-40000-440000000000000000000    symplectic lifted from 2- (1+4), Schur index 2
ρ284-4-44-4400000000000000000000000    symplectic lifted from C8.C22, Schur index 2
ρ294-4-444-400000000000000000000000    symplectic lifted from C8.C22, Schur index 2

In GAP, Magma, Sage, TeX 

C_4^2._{60}C_2^3
 
% in TeX
 
G:=Group("C4^2.60C2^3");
 
// GroupNames label
 
G:=SmallGroup(128,2078);
 
// by ID
 
G=gap.SmallGroup(128,2078);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,-2,560,253,758,723,436,346,4037,1027,124]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^4=b^4=e^2=1,c^2=d^2=a^2*b^2,a*b=b*a,c*a*c^-1=e*a*e=a^-1,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*c,e*d*e=b^2*d>;
 
// generators/relations
 

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