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G = C42.74C23  order 128 = 27

74th non-split extension by C42 of C23 acting faithfully

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

Aliases: C42.74C23, C4.1502+ (1+4), C8⋊7(C4○D4), C8⋊3D4⋊22C2, C8⋊4D4⋊23C2, C4⋊D8⋊43C2, C4⋊C4.186D4, C8⋊4Q8⋊21C2, D8⋊C4⋊30C2, Q8⋊6D4⋊14C2, C4⋊SD16⋊27C2, (C4×SD16)⋊25C2, C2.72(D4○D8), (C2×Q8).254D4, D4.2D4⋊49C2, C4⋊C8.152C22, C4⋊C4.450C23, C4.81(C8⋊C22), (C4×C8).209C22, (C2×C8).223C23, (C2×C4).591C24, (C2×D8).96C22, C8⋊C4.78C22, C2.45(Q8⋊6D4), (C4×D4).224C22, (C2×D4).285C23, (C4×Q8).214C22, (C2×Q8).270C23, C4.Q8.142C22, D4⋊C4.99C22, C4⋊1D4.109C22, C4.4D4.91C22, C22.851(C22×D4), C22.53C24⋊8C2, Q8⋊C4.169C22, (C2×SD16).126C22, C4.169(C2×C4○D4), (C2×C4).655(C2×D4), C2.93(C2×C8⋊C22), SmallGroup(128,2131)

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

Derived series
C1 — C2×C4 — C42.74C23
Chief series
C1 — C2 — C4 — C2×C4 — C42 — C4×D4 — Q8⋊6D4 — C42.74C23
Lower central
C1 — C2 — C2×C4 — C42.74C23
Upper central
C1 — C22 — C4×Q8 — C42.74C23
Jennings
C1 — C2 — C2 — C2×C4 — C42.74C23

Subgroups: 480 in 209 conjugacy classes, 88 normal (38 characteristic)
C1, C2 [×3], C2 [×5], C4 [×2], C4 [×2], C4 [×8], C22, C22 [×15], C8 [×2], C8 [×3], C2×C4 [×3], C2×C4 [×4], C2×C4 [×11], D4 [×22], Q8 [×4], C23 [×5], C42, C42 [×2], C42, C22⋊C4 [×9], C4⋊C4 [×3], C4⋊C4 [×2], C4⋊C4, C2×C8 [×2], C2×C8 [×2], D8 [×8], SD16 [×4], C22×C4 [×5], C2×D4, C2×D4 [×4], C2×D4 [×8], C2×Q8 [×2], C4○D4 [×4], C4×C8, C8⋊C4 [×2], D4⋊C4, D4⋊C4 [×4], Q8⋊C4, C4⋊C8, C4⋊C8 [×2], C4.Q8, C4×D4, C4×D4 [×4], C4×D4, C4×Q8 [×2], C4⋊D4 [×3], C22.D4 [×2], C4.4D4 [×2], C4.4D4, C4⋊1D4 [×2], C4⋊1D4 [×2], C2×D8 [×6], C2×SD16, C2×SD16 [×2], C2×C4○D4, C4×SD16, D8⋊C4 [×2], C8⋊4Q8, C4⋊D8, C4⋊D8 [×2], C4⋊SD16, D4.2D4 [×2], C8⋊4D4, C8⋊3D4 [×2], Q8⋊6D4, C22.53C24, C42.74C23

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), Q8⋊6D4, C2×C8⋊C22, D4○D8, C42.74C23

Generators and relations
 G = < a,b,c,d,e | a4=b4=1, c2=b2, d2=e2=a2b2, ab=ba, cac-1=eae-1=a-1b2, dad-1=ab2, cbc-1=dbd-1=b-1, be=eb, dcd-1=bc, ce=ec, de=ed >

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

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

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

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

Matrix representation ►G ⊆ GL6(𝔽17)

16150000
110000
000010
00116115
0016000
0016101
,
100000
010000
000100
0016000
00116115
0010116
,
100000
16160000
00611161
001110016
00117013
006161
,
1300000
0130000
0016000
000100
000010
00016116
,
400000
13130000
0016000
0001600
0000160
0000016

G:=sub<GL(6,GF(17))| [16,1,0,0,0,0,15,1,0,0,0,0,0,0,0,1,16,16,0,0,0,16,0,1,0,0,1,1,0,0,0,0,0,15,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,16,1,1,0,0,1,0,16,0,0,0,0,0,1,1,0,0,0,0,15,16],[1,16,0,0,0,0,0,16,0,0,0,0,0,0,6,11,11,6,0,0,11,10,7,1,0,0,16,0,0,6,0,0,1,16,13,1],[13,0,0,0,0,0,0,13,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,16,0,0,0,0,1,1,0,0,0,0,0,16],[4,13,0,0,0,0,0,13,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16] >;
 

Character table of C42.74C23

 class 12A2B2C2D2E2F2G2H4A4B4C4D4E4F4G4H4I4J4K4L4M4N8A8B8C8D8E8F
 size 11118888822224444444448444488
ρ111111111111111111111111111111    trivial
ρ2111111-1-1111-1-11-1-1-1-1-11-1-11-11-111-1    linear of order 2
ρ31111-11-1-1111-1-1-111-111-111-1-11-11-11    linear of order 2
ρ41111-111111111-1-1-11-1-1-1-1-1-11111-1-1    linear of order 2
ρ51111-111-1-11111-1-111-1-1-11-11-1-1-1-111    linear of order 2
ρ61111-11-11-111-1-1-11-1-111-1-1111-11-11-1    linear of order 2
ρ7111111-11-111-1-11-11-1-1-111-1-11-11-1-11    linear of order 2
ρ81111111-1-1111111-11111-11-1-1-1-1-1-1-1    linear of order 2
ρ91111-1-1-11111111111-1111-11-1-1-1-1-1-1    linear of order 2
ρ101111-1-11-1111-1-11-1-1-11-11-1111-11-1-11    linear of order 2
ρ1111111-11-1111-1-1-111-1-11-11-1-11-11-11-1    linear of order 2
ρ1211111-1-1111111-1-1-111-1-1-11-1-1-1-1-111    linear of order 2
ρ1311111-1-1-1-11111-1-1111-1-11111111-1-1    linear of order 2
ρ1411111-111-111-1-1-11-1-1-11-1-1-11-11-11-11    linear of order 2
ρ151111-1-111-111-1-11-11-11-1111-1-11-111-1    linear of order 2
ρ161111-1-1-1-1-1111111-11-111-1-1-1111111    linear of order 2
ρ17222200000-2-2-2-2-22020-22000000000    orthogonal lifted from D4
ρ18222200000-2-222220-20-2-2000000000    orthogonal lifted from D4
ρ19222200000-2-2-2-22-20202-2000000000    orthogonal lifted from D4
ρ20222200000-2-222-2-20-2022000000000    orthogonal lifted from D4
ρ212-22-2000002-200002i02i002i2i0020-200    complex lifted from C4○D4
ρ222-22-2000002-200002i02i002i2i00-20200    complex lifted from C4○D4
ρ232-22-2000002-200002i02i002i2i00-20200    complex lifted from C4○D4
ρ242-22-2000002-200002i02i002i2i0020-200    complex lifted from C4○D4
ρ254-4-440000000-440000000000000000    orthogonal lifted from C8⋊C22
ρ264-44-400000-44000000000000000000    orthogonal lifted from 2+ (1+4)
ρ274-4-4400000004-40000000000000000    orthogonal lifted from C8⋊C22
ρ2844-4-400000000000000000002√202√2000    orthogonal lifted from D4○D8
ρ2944-4-400000000000000000002√202√2000    orthogonal lifted from D4○D8

In GAP, Magma, Sage, TeX 

C_4^2._{74}C_2^3
 
% in TeX
 
G:=Group("C4^2.74C2^3");
 
// GroupNames label
 
G:=SmallGroup(128,2131);
 
// by ID
 
G=gap.SmallGroup(128,2131);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,-2,448,253,568,758,723,100,346,80,4037,1027,124]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^4=b^4=1,c^2=b^2,d^2=e^2=a^2*b^2,a*b=b*a,c*a*c^-1=e*a*e^-1=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,c*e=e*c,d*e=e*d>;
 
// generators/relations
 

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