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G = C42.480C23  order 128 = 27

341st non-split extension by C42 of C23 acting via C23/C2=C22

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

Aliases: C42.480C23, C4.762+ (1+4), C8⋊6D4⋊18C2, C4⋊C4.376D4, D4⋊3Q8⋊5C2, D4⋊2Q8⋊20C2, C4⋊2Q16⋊41C2, (C4×SD16)⋊59C2, D4⋊6D4.8C2, (C2×D4).326D4, C8.D4⋊30C2, C22⋊C4.59D4, D4.32(C4○D4), D4.7D4⋊50C2, C4⋊C8.113C22, C4⋊C4.423C23, (C4×C8).294C22, (C2×C4).523C24, (C2×C8).358C23, C4.SD16⋊35C2, C23.340(C2×D4), C4⋊Q8.158C22, C4.Q8.63C22, C2.84(D4○SD16), (C4×D4).172C22, (C2×D4).427C23, C4.49(C8.C22), C22⋊C8.91C22, (C2×Q8).231C23, (C2×Q16).87C22, (C4×Q8).167C22, C2.159(D4⋊5D4), C22⋊Q8.94C22, D4⋊C4.77C22, C23.36D4⋊26C2, C23.47D4⋊19C2, (C22×C4).336C23, C22.783(C22×D4), Q8⋊C4.117C22, (C2×SD16).162C22, (C2×M4(2)).125C22, C4.248(C2×C4○D4), (C2×C4).616(C2×D4), C2.80(C2×C8.C22), (C2×C4⋊C4).675C22, (C2×C4○D4).221C22, SmallGroup(128,2063)

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

Derived series
C1 — C2×C4 — C42.480C23
Chief series
C1 — C2 — C4 — C2×C4 — C22×C4 — C2×C4○D4 — D4⋊6D4 — C42.480C23
Lower central
C1 — C2 — C2×C4 — C42.480C23
Upper central
C1 — C22 — C4×D4 — C42.480C23
Jennings
C1 — C2 — C2 — C2×C4 — C42.480C23

Subgroups: 376 in 194 conjugacy classes, 88 normal (38 characteristic)
C1, C2 [×3], C2 [×4], C4 [×2], C4 [×2], C4 [×10], C22, C22 [×10], C8 [×4], C2×C4 [×3], C2×C4 [×2], C2×C4 [×19], D4 [×2], D4 [×7], Q8 [×7], C23 [×2], C23, C42, C42, C22⋊C4 [×2], C22⋊C4 [×5], C4⋊C4 [×3], C4⋊C4 [×2], C4⋊C4 [×10], C2×C8 [×2], C2×C8 [×2], M4(2) [×2], SD16 [×2], Q16 [×2], C22×C4 [×2], C22×C4 [×5], C2×D4 [×2], C2×D4 [×2], C2×Q8, C2×Q8 [×2], C2×Q8, C4○D4 [×6], C4×C8, C22⋊C8 [×2], D4⋊C4, D4⋊C4 [×2], Q8⋊C4, Q8⋊C4 [×6], C4⋊C8, C4.Q8, C4.Q8 [×2], C2×C4⋊C4 [×2], C2×C4⋊C4, C4×D4 [×2], C4×D4, C4×Q8, C4⋊D4, C22⋊Q8 [×4], C22⋊Q8 [×2], C22.D4 [×2], C42.C2, C4⋊Q8 [×2], C2×M4(2) [×2], C2×SD16, C2×Q16 [×2], C2×C4○D4 [×2], C23.36D4 [×2], C8⋊6D4, C4×SD16, D4.7D4 [×2], C4⋊2Q16, C8.D4 [×2], D4⋊2Q8, C23.47D4 [×2], C4.SD16, D4⋊6D4, D4⋊3Q8, C42.480C23

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⋊5D4, C2×C8.C22, D4○SD16, C42.480C23

Generators and relations
 G = < a,b,c,d,e | a4=b4=d2=e2=1, c2=a2, ab=ba, cac-1=a-1, dad=ab2, eae=a-1b2, cbc-1=dbd=b-1, be=eb, dcd=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 48 18 43)(2 45 19 44)(3 46 20 41)(4 47 17 42)(5 36 61 37)(6 33 62 38)(7 34 63 39)(8 35 64 40)(9 21 13 50)(10 22 14 51)(11 23 15 52)(12 24 16 49)(25 56 32 59)(26 53 29 60)(27 54 30 57)(28 55 31 58)

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

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

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

Matrix representation ►G ⊆ GL6(𝔽17)

160000
11160000
00016010
001070
0001001
0070160
,
100000
010000
000100
0016000
000001
0000160
,
040000
400000
00161512
00111212
00512116
0012121616
,
100000
010000
001000
0001600
000010
0000016
,
11160000
160000
0007016
0010010
00016010
001070

G:=sub<GL(6,GF(17))| [1,11,0,0,0,0,6,16,0,0,0,0,0,0,0,1,0,7,0,0,16,0,10,0,0,0,0,7,0,16,0,0,10,0,1,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,16,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,1,0],[0,4,0,0,0,0,4,0,0,0,0,0,0,0,16,1,5,12,0,0,1,1,12,12,0,0,5,12,1,16,0,0,12,12,16,16],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,16],[11,1,0,0,0,0,16,6,0,0,0,0,0,0,0,10,0,1,0,0,7,0,16,0,0,0,0,1,0,7,0,0,16,0,10,0] >;
 

Character table of C42.480C23

 class 12A2B2C2D2E2F2G4A4B4C4D4E4F4G4H4I4J4K4L4M4N4O8A8B8C8D8E8F
 size 11114444222244444888888444488
ρ111111111111111111111111111111    trivial
ρ21111-1111-111-1-1-1-11-11-1-11-11-1-1111-1    linear of order 2
ρ31111-1111-111-11-1-111-1-11-1-1111-1-1-11    linear of order 2
ρ4111111111111-1111-1-11-1-111-1-1-1-1-1-1    linear of order 2
ρ51111-1-11-1-111-1-1-1-11-111-111-111-1-1-11    linear of order 2
ρ611111-11-11111111111-111-1-1-1-1-1-1-1-1    linear of order 2
ρ711111-11-11111-1111-1-1-1-1-1-1-1111111    linear of order 2
ρ81111-1-11-1-111-11-1-111-111-11-1-1-1111-1    linear of order 2
ρ91111-11-1111111-11-1111-1-1-1-11111-1-1    linear of order 2
ρ10111111-11-111-1-11-1-1-11-11-11-1-1-111-11    linear of order 2
ρ11111111-11-111-111-1-11-1-1-111-111-1-11-1    linear of order 2
ρ121111-11-111111-1-11-1-1-1111-1-1-1-1-1-111    linear of order 2
ρ1311111-1-1-1-111-1-11-1-1-1111-1-1111-1-11-1    linear of order 2
ρ141111-1-1-1-111111-11-111-1-1-111-1-1-1-111    linear of order 2
ρ151111-1-1-1-11111-1-11-1-1-1-111111111-1-1    linear of order 2
ρ1611111-1-1-1-111-111-1-11-11-11-11-1-111-11    linear of order 2
ρ17222220-20-2-2-2-20-2220000000000000    orthogonal lifted from D4
ρ18222220202-2-220-2-2-20000000000000    orthogonal lifted from D4
ρ192222-2020-2-2-2-2022-20000000000000    orthogonal lifted from D4
ρ202222-20-202-2-2202-220000000000000    orthogonal lifted from D4
ρ212-22-2020-202-202i0002i0000002i2i0000    complex lifted from C4○D4
ρ222-22-20-20202-202i0002i0000002i2i0000    complex lifted from C4○D4
ρ232-22-20-20202-202i0002i0000002i2i0000    complex lifted from C4○D4
ρ242-22-2020-202-202i0002i0000002i2i0000    complex lifted from C4○D4
ρ254-44-400000-44000000000000000000    orthogonal lifted from 2+ (1+4)
ρ264-4-440000400-400000000000000000    symplectic lifted from C8.C22, Schur index 2
ρ274-4-440000-400400000000000000000    symplectic lifted from C8.C22, Schur index 2
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._{480}C_2^3
 
% in TeX
 
G:=Group("C4^2.480C2^3");
 
// GroupNames label
 
G:=SmallGroup(128,2063);
 
// by ID
 
G=gap.SmallGroup(128,2063);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,-2,560,253,758,723,2019,346,248,4037,1027,124]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^4=b^4=d^2=e^2=1,c^2=a^2,a*b=b*a,c*a*c^-1=a^-1,d*a*d=a*b^2,e*a*e=a^-1*b^2,c*b*c^-1=d*b*d=b^-1,b*e=e*b,d*c*d=b*c,e*c*e=a^2*b^2*c,e*d*e=b^2*d>;
 
// generators/relations
 

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