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G = C42.481C23  order 128 = 27

342nd non-split extension by C42 of C23 acting via C23/C2=C22

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

Aliases: C42.481C23, C4.772+ (1+4), C8⋊2D4⋊30C2, C8⋊6D4⋊19C2, C4⋊D8⋊40C2, C4⋊C4.377D4, Q8⋊7(C4○D4), Q8⋊6D4⋊9C2, D4⋊6D4⋊13C2, D4⋊D4⋊50C2, Q8⋊Q8⋊20C2, (C4×SD16)⋊60C2, (C2×D4).327D4, C4.4D8⋊35C2, C22⋊C4.60D4, C4⋊C4.424C23, C4⋊C8.114C22, C4.76(C8⋊C22), (C2×C8).359C23, (C2×C4).524C24, (C4×C8).295C22, (C2×D8).88C22, C23.341(C2×D4), C4⋊Q8.159C22, C4.Q8.64C22, C2.85(D4○SD16), (C4×D4).173C22, (C2×D4).246C23, C4⋊D4.95C22, C4⋊1D4.92C22, C22⋊C8.92C22, (C4×Q8).168C22, (C2×Q8).401C23, C2.160(D4⋊5D4), C23.46D4⋊19C2, C23.36D4⋊27C2, (C22×C4).337C23, Q8⋊C4.74C22, C22.784(C22×D4), D4⋊C4.123C22, (C2×SD16).163C22, (C2×M4(2)).126C22, C4.249(C2×C4○D4), (C2×C4).617(C2×D4), C2.81(C2×C8⋊C22), (C2×C4⋊C4).676C22, (C2×C4○D4).222C22, SmallGroup(128,2064)

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

Derived series
C1 — C2×C4 — C42.481C23
Chief series
C1 — C2 — C4 — C2×C4 — C22×C4 — C2×C4○D4 — Q8⋊6D4 — C42.481C23
Lower central
C1 — C2 — C2×C4 — C42.481C23
Upper central
C1 — C22 — C4×D4 — C42.481C23
Jennings
C1 — C2 — C2 — C2×C4 — C42.481C23

Subgroups: 472 in 215 conjugacy classes, 88 normal (38 characteristic)
C1, C2 [×3], C2 [×5], C4 [×2], C4 [×2], C4 [×9], C22, C22 [×15], C8 [×4], C2×C4 [×3], C2×C4 [×2], C2×C4 [×19], D4 [×21], Q8 [×2], Q8 [×3], C23 [×2], C23 [×3], C42, C42, C22⋊C4 [×2], C22⋊C4 [×5], C4⋊C4 [×3], C4⋊C4 [×2], C4⋊C4 [×4], C2×C8 [×2], C2×C8 [×2], M4(2) [×2], D8 [×2], SD16 [×2], C22×C4 [×2], C22×C4 [×5], C2×D4 [×2], C2×D4 [×2], C2×D4 [×8], C2×Q8, C2×Q8, C4○D4 [×10], C4×C8, C22⋊C8 [×2], D4⋊C4, D4⋊C4 [×6], Q8⋊C4, Q8⋊C4 [×2], C4⋊C8, C4.Q8, C4.Q8 [×2], C2×C4⋊C4 [×2], C4×D4 [×2], C4×D4, C4×Q8, C4⋊D4 [×4], C4⋊D4 [×2], C22⋊Q8, C22.D4 [×2], C4⋊1D4, C4⋊1D4, C4⋊Q8, C2×M4(2) [×2], C2×D8 [×2], C2×SD16, C2×C4○D4 [×2], C2×C4○D4, C23.36D4 [×2], C8⋊6D4, C4×SD16, D4⋊D4 [×2], C4⋊D8, C8⋊2D4 [×2], Q8⋊Q8, C23.46D4 [×2], C4.4D8, D4⋊6D4, Q8⋊6D4, C42.481C23

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, C42.481C23

Generators and relations
 G = < a,b,c,d,e | a4=b4=e2=1, c2=a2b2, d2=b2, ab=ba, cac-1=a-1, dad-1=ab2, eae=a-1b2, 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 52 9 43)(2 49 10 44)(3 50 11 41)(4 51 12 42)(5 39 31 55)(6 40 32 56)(7 37 29 53)(8 38 30 54)(13 21 20 46)(14 22 17 47)(15 23 18 48)(16 24 19 45)(25 57 62 33)(26 58 63 34)(27 59 64 35)(28 60 61 36)

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

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

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

Matrix representation ►G ⊆ GL6(𝔽17)

16150000
110000
0016200
0016100
0000162
0000161
,
100000
010000
0011500
0011600
0000115
0000116
,
400000
13130000
00710215
001210115
00215107
0011557
,
100000
010000
0000160
0000161
001000
0011600
,
16150000
010000
0000115
0000116
0016200
0016100

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

Character table of C42.481C23

 class 12A2B2C2D2E2F2G2H4A4B4C4D4E4F4G4H4I4J4K4L4M4N8A8B8C8D8E8F
 size 11114488822224444444888444488
ρ111111111111111111111111111111    trivial
ρ211111-11-11-111-1-1-11-1-1-1-111-11-11-11-1    linear of order 2
ρ31111-1-1-1111111-11-1-1-1-1-11-111111-1-1    linear of order 2
ρ41111-11-1-11-111-11-1-111111-1-11-11-1-11    linear of order 2
ρ51111-1-11111111-11-1-1-111-1-1-1-1-1-1-111    linear of order 2
ρ61111-111-11-111-11-1-111-1-1-1-11-11-111-1    linear of order 2
ρ7111111-111111111111-1-1-11-1-1-1-1-1-1-1    linear of order 2
ρ811111-1-1-11-111-1-1-11-1-111-111-11-11-11    linear of order 2
ρ91111-1-11-1-1111111-11-111-11-11111-1-1    linear of order 2
ρ101111-1111-1-111-1-1-1-1-11-1-1-1111-11-1-11    linear of order 2
ρ11111111-1-1-11111-111-11-1-1-1-1-1111111    linear of order 2
ρ1211111-1-11-1-111-11-111-111-1-111-11-11-1    linear of order 2
ρ131111111-1-11111-111-11111-11-1-1-1-1-1-1    linear of order 2
ρ1411111-111-1-111-11-111-1-1-11-1-1-11-11-11    linear of order 2
ρ151111-1-1-1-1-1111111-11-1-1-1111-1-1-1-111    linear of order 2
ρ161111-11-11-1-111-1-1-1-1-111111-1-11-111-1    linear of order 2
ρ172222220002-2-220-2-20-200000000000    orthogonal lifted from D4
ρ182222-22000-2-2-2-20220-200000000000    orthogonal lifted from D4
ρ192222-2-20002-2-220-220200000000000    orthogonal lifted from D4
ρ2022222-2000-2-2-2-202-20200000000000    orthogonal lifted from D4
ρ212-22-2000000-220200-202i2i0002i02i000    complex lifted from C4○D4
ρ222-22-2000000-220-200202i2i0002i02i000    complex lifted from C4○D4
ρ232-22-2000000-220-200202i2i0002i02i000    complex lifted from C4○D4
ρ242-22-2000000-220200-202i2i0002i02i000    complex lifted from C4○D4
ρ254-4-4400000-40040000000000000000    orthogonal lifted from C8⋊C22
ρ264-4-4400000400-40000000000000000    orthogonal lifted from C8⋊C22
ρ274-44-40000004-400000000000000000    orthogonal lifted from 2+ (1+4)
ρ2844-4-4000000000000000000002√-202√-200    complex lifted from D4○SD16
ρ2944-4-4000000000000000000002√-202√-200    complex lifted from D4○SD16

In GAP, Magma, Sage, TeX 

C_4^2._{481}C_2^3
 
% in TeX
 
G:=Group("C4^2.481C2^3");
 
// GroupNames label
 
G:=SmallGroup(128,2064);
 
// by ID
 
G=gap.SmallGroup(128,2064);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,-2,112,253,758,723,352,2019,346,248,4037,1027,124]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^4=b^4=e^2=1,c^2=a^2*b^2,d^2=b^2,a*b=b*a,c*a*c^-1=a^-1,d*a*d^-1=a*b^2,e*a*e=a^-1*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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