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G = C42.64C23  order 128 = 27

64th non-split extension by C42 of C23 acting faithfully

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

Aliases: C42.64C23, C4.852- (1+4), C8⋊Q8⋊30C2, C8⋊9D4⋊29C2, C8⋊8D4⋊24C2, C8⋊D4⋊53C2, C4⋊C4.382D4, Q8.Q8⋊43C2, D4⋊2Q8⋊22C2, (C2×D4).182D4, C8.37(C4○D4), C22⋊C4.65D4, C4⋊C8.123C22, C4⋊C4.255C23, (C2×C8).109C23, (C2×C4).542C24, C23.347(C2×D4), C4⋊Q8.174C22, SD16⋊C4⋊42C2, C2.95(D4⋊6D4), C8⋊C4.56C22, C4.Q8.70C22, C2.92(D4○SD16), (C4×D4).182C22, (C2×D4).259C23, (C4×Q8).181C22, (C2×Q8).244C23, M4(2)⋊C4⋊37C2, C2.D8.225C22, D4⋊C4.83C22, C23.20D4⋊47C2, C23.19D4⋊46C2, C23.25D4⋊33C2, C4⋊D4.108C22, C23.47D4⋊22C2, C22⋊C8.101C22, (C22×C8).293C22, Q8⋊C4.81C22, (C2×SD16).66C22, C22.802(C22×D4), C22⋊Q8.107C22, C42.C2.55C22, C2.97(D8⋊C22), C22.50C24⋊8C2, (C22×C4).1170C23, C42⋊C2.213C22, (C2×M4(2)).135C22, C22.47C24.4C2, C4.124(C2×C4○D4), (C2×C4).626(C2×D4), (C2×C4⋊C4).691C22, SmallGroup(128,2082)

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

Derived series
C1 — C2×C4 — C42.64C23
Chief series
C1 — C2 — C4 — C2×C4 — C22×C4 — C42⋊C2 — C22.50C24 — C42.64C23
Lower central
C1 — C2 — C2×C4 — C42.64C23
Upper central
C1 — C22 — C4×D4 — C42.64C23
Jennings
C1 — C2 — C2 — C2×C4 — C42.64C23

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 [×13], D4 [×6], Q8 [×4], 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], C4×D4 [×2], C4×D4, C4×Q8, C4×Q8, C4⋊D4 [×2], C4⋊D4, C22⋊Q8 [×2], C22.D4, C4.4D4, C42.C2, C42⋊2C2 [×3], C4⋊Q8, C22×C8, C2×M4(2), C2×SD16, C23.25D4, M4(2)⋊C4, C8⋊9D4, SD16⋊C4, C8⋊8D4, C8⋊D4, D4⋊2Q8, Q8.Q8, C23.19D4 [×2], C23.47D4, C23.20D4, C8⋊Q8, C22.47C24, C22.50C24, C42.64C23

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, C42.64C23

Generators and relations
 G = < a,b,c,d,e | a4=b4=e2=1, c2=a2b2, d2=a2, 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 60 30 23)(2 57 31 24)(3 58 32 21)(4 59 29 22)(5 40 47 27)(6 37 48 28)(7 38 45 25)(8 39 46 26)(9 63 50 42)(10 64 51 43)(11 61 52 44)(12 62 49 41)(13 18 36 53)(14 19 33 54)(15 20 34 55)(16 17 35 56)

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

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

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

Matrix representation ►G ⊆ GL6(𝔽17)

400000
0130000
0012044
000121513
004450
00151305
,
100000
010000
00161500
001100
000012
00001616
,
080000
200000
003011
00141480
000161411
009103
,
400000
040000
000010
000001
0016000
0001600
,
090000
200000
003011
0003816
0011140
00816014

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

Character table of C42.64C23

 class 12A2B2C2D2E2F4A4B4C4D4E4F4G4H4I4J4K4L4M4N4O4P8A8B8C8D8E8F
 size 11114482222444444488888444488
ρ111111111111111111111111111111    trivial
ρ21111-11111-1-11-1-111-1-1-1-11-11-1-1111-1    linear of order 2
ρ31111-11111-1-1-1-1-1-11-1-1111-1-111-1-1-11    linear of order 2
ρ411111111111-111-1111-1-111-1-1-1-1-1-1-1    linear of order 2
ρ51111-1-1-111-1-111-11-11-1-1111-1-1-111-11    linear of order 2
ρ611111-1-111111-111-1-111-11-1-11111-1-1    linear of order 2
ρ711111-1-11111-1-11-1-1-11-111-11-1-1-1-111    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
ρ1011111-1111111-1-11-1-1-11-1-11-1-1-1-1-111    linear of order 2
ρ1111111-111111-1-1-1-1-1-1-1-11-1111111-1-1    linear of order 2
ρ121111-1-1111-1-1-111-1-1111-1-1-11-1-111-11    linear of order 2
ρ13111111-1111111-1111-111-1-11-1-1-1-1-1-1    linear of order 2
ρ141111-11-111-1-11-1111-11-1-1-11111-1-1-11    linear of order 2
ρ151111-11-111-1-1-1-11-11-1111-11-1-1-1111-1    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._{64}C_2^3
 
% in TeX
 
G:=Group("C4^2.64C2^3");
 
// GroupNames label
 
G:=SmallGroup(128,2082);
 
// by ID
 
G=gap.SmallGroup(128,2082);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,-2,112,253,758,723,436,346,248,4037,1027,124]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^4=b^4=e^2=1,c^2=a^2*b^2,d^2=a^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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