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G = C42.532C23  order 128 = 27

393rd non-split extension by C42 of C23 acting via C23/C2=C22

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

Aliases: C42.532C23, C4.1532+ (1+4), C4⋊C4.189D4, (C4×Q16)⋊36C2, C8⋊4Q8⋊24C2, C4⋊2Q16⋊46C2, C8.23(C4○D4), C8.2D4⋊23C2, (C2×Q8).144D4, C2.73(Q8○D8), C4⋊C4.453C23, C4⋊C8.155C22, (C2×C8).123C23, (C4×C8).212C22, (C2×C4).594C24, Q8.D4⋊50C2, D4.2D4.4C2, C8.12D4.7C2, (C2×D8).43C22, C4⋊Q8.220C22, SD16⋊C4⋊51C2, C8⋊C4.81C22, C2.48(Q8⋊6D4), (C2×D4).288C23, (C4×D4).227C22, (C4×Q8).217C22, (C2×Q8).273C23, C2.D8.230C22, Q8⋊C4.97C22, (C2×Q16).166C22, (C2×SD16).78C22, C4.4D4.94C22, C22.854(C22×D4), D4⋊C4.102C22, C2.109(D8⋊C22), C22.50C24⋊17C2, C4.172(C2×C4○D4), (C2×C4).658(C2×D4), SmallGroup(128,2134)

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

Derived series
C1 — C2×C4 — C42.532C23
Chief series
C1 — C2 — C4 — C2×C4 — C42 — C4×Q8 — C22.50C24 — C42.532C23
Lower central
C1 — C2 — C2×C4 — C42.532C23
Upper central
C1 — C22 — C4×Q8 — C42.532C23
Jennings
C1 — C2 — C2 — C2×C4 — C42.532C23

Subgroups: 328 in 178 conjugacy classes, 86 normal (26 characteristic)
C1, C2 [×3], C2 [×2], C4 [×2], C4 [×12], C22, C22 [×6], C8 [×2], C8 [×3], C2×C4 [×3], C2×C4 [×4], C2×C4 [×10], D4 [×4], Q8 [×9], C23 [×2], C42, C42 [×2], C42 [×4], C22⋊C4 [×10], C4⋊C4, C4⋊C4 [×4], C4⋊C4 [×8], C2×C8 [×2], C2×C8 [×2], D8, SD16 [×6], Q16 [×5], C22×C4 [×2], C2×D4 [×2], C2×Q8, C2×Q8 [×4], C4×C8, C8⋊C4 [×2], D4⋊C4 [×2], Q8⋊C4 [×4], C4⋊C8, C4⋊C8 [×2], C2.D8, C42⋊C2 [×2], C4×D4 [×2], C4×Q8, C4×Q8 [×4], C22⋊Q8 [×2], C4.4D4 [×4], C42⋊2C2 [×4], C4⋊Q8 [×2], C2×D8, C2×SD16 [×4], C2×Q16 [×2], C2×Q16 [×2], C4×Q16, SD16⋊C4 [×2], C8⋊4Q8, C4⋊2Q16 [×2], D4.2D4 [×2], Q8.D4 [×2], C8.12D4, C8.2D4 [×2], C22.50C24 [×2], C42.532C23

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), Q8⋊6D4, D8⋊C22, Q8○D8, C42.532C23

Generators and relations
 G = < a,b,c,d,e | a4=b4=1, c2=b2, d2=e2=a2b2, ab=ba, ac=ca, dad-1=a-1b2, ae=ea, cbc-1=ebe-1=b-1, bd=db, dcd-1=a2b2c, ece-1=bc, ede-1=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 28 20 23)(2 25 17 24)(3 26 18 21)(4 27 19 22)(5 12 15 63)(6 9 16 64)(7 10 13 61)(8 11 14 62)(29 36 37 41)(30 33 38 42)(31 34 39 43)(32 35 40 44)(45 51 56 60)(46 52 53 57)(47 49 54 58)(48 50 55 59)

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

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

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

Matrix representation ►G ⊆ GL6(𝔽17)

0160000
100000
0013000
0001300
0000130
0000013
,
100000
010000
000100
0016000
0000016
000010
,
0130000
400000
00160161
00011616
0011016
00161160
,
1300000
040000
0040413
00041313
00134130
0044013
,
1300000
0130000
000010
000001
001000
000100

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

Character table of C42.532C23

 class 12A2B2C2D2E4A4B4C4D4E4F4G4H4I4J4K4L4M4N4O4P4Q8A8B8C8D8E8F
 size 11118822224444444448888444488
ρ111111111111111111111111111111    trivial
ρ21111-1-1-11-111-1-111-1-11-1111-11-11-11-1    linear of order 2
ρ31111-1-11111-1111-1-111-1-11-111111-1-1    linear of order 2
ρ4111111-11-11-1-1-11-11-111-11-1-11-11-1-11    linear of order 2
ρ511111-11111111-1111-1111-1-1-1-1-1-1-1-1    linear of order 2
ρ61111-11-11-111-1-1-11-1-1-1-111-11-11-11-11    linear of order 2
ρ71111-111111-111-1-1-11-1-1-111-1-1-1-1-111    linear of order 2
ρ811111-1-11-11-1-1-1-1-11-1-11-1111-11-111-1    linear of order 2
ρ911111-1-11-1111-111-111-1-1-11-1-11-11-11    linear of order 2
ρ101111-1111111-11111-111-1-111-1-1-1-1-1-1    linear of order 2
ρ111111-11-11-11-11-11-111111-1-1-1-11-111-1    linear of order 2
ρ1211111-11111-1-111-1-1-11-11-1-11-1-1-1-111    linear of order 2
ρ13111111-11-1111-1-11-11-1-1-1-1-111-11-11-1    linear of order 2
ρ141111-1-111111-11-111-1-11-1-1-1-1111111    linear of order 2
ρ151111-1-1-11-11-11-1-1-111-111-1111-11-1-11    linear of order 2
ρ161111111111-1-11-1-1-1-1-1-11-11-11111-1-1    linear of order 2
ρ172222002-22-220-20-2-20020000000000    orthogonal lifted from D4
ρ182222002-22-2-20-202200-20000000000    orthogonal lifted from D4
ρ19222200-2-2-2-22020-2200-20000000000    orthogonal lifted from D4
ρ20222200-2-2-2-2-20202-20020000000000    orthogonal lifted from D4
ρ212-22-200020-202i02i002i2i000000-20200    complex lifted from C4○D4
ρ222-22-200020-202i02i002i2i00000020-200    complex lifted from C4○D4
ρ232-22-200020-202i02i002i2i000000-20200    complex lifted from C4○D4
ρ242-22-200020-202i02i002i2i00000020-200    complex lifted from C4○D4
ρ254-44-4000-4040000000000000000000    orthogonal lifted from 2+ (1+4)
ρ2644-4-400000000000000000002√202√2000    symplectic lifted from Q8○D8, Schur index 2
ρ2744-4-400000000000000000002√202√2000    symplectic lifted from Q8○D8, Schur index 2
ρ284-4-44004i04i00000000000000000000    complex lifted from D8⋊C22
ρ294-4-44004i04i00000000000000000000    complex lifted from D8⋊C22

In GAP, Magma, Sage, TeX 

C_4^2._{532}C_2^3
 
% in TeX
 
G:=Group("C4^2.532C2^3");
 
// GroupNames label
 
G:=SmallGroup(128,2134);
 
// by ID
 
G=gap.SmallGroup(128,2134);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,-2,448,253,120,758,723,100,346,304,4037,1027,124]);
 
// 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,a*c=c*a,d*a*d^-1=a^-1*b^2,a*e=e*a,c*b*c^-1=e*b*e^-1=b^-1,b*d=d*b,d*c*d^-1=a^2*b^2*c,e*c*e^-1=b*c,e*d*e^-1=b^2*d>;
 
// generators/relations
 

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