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G = C42.72C23order 128 = 27

72nd non-split extension by C42 of C23 acting faithfully

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

Aliases: C42.72C23, C4.1482+ 1+4, (C4×D8)⋊35C2, C86(C4○D4), C4⋊D842C2, C83D421C2, C85D411C2, C4⋊C4.184D4, C84Q819C2, C4⋊SD1626C2, Q86D413C2, (C2×Q8).252D4, D4.2D448C2, D4.D427C2, C4⋊C4.448C23, C4⋊C8.150C22, C4.53(C8⋊C22), (C2×C4).589C24, (C4×C8).207C22, (C2×C8).380C23, (C2×D8).95C22, C4⋊Q8.216C22, SD16⋊C449C2, C8⋊C4.76C22, C2.43(Q86D4), (C4×D4).222C22, (C2×D4).283C23, (C2×Q8).268C23, (C4×Q8).212C22, C2.D8.228C22, C2.113(D4○SD16), D4⋊C4.98C22, C41D4.108C22, Q8⋊C4.94C22, (C2×SD16).75C22, C4.4D4.89C22, C22.849(C22×D4), C22.50C2415C2, C4.167(C2×C4○D4), (C2×C4).653(C2×D4), C2.92(C2×C8⋊C22), SmallGroup(128,2129)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C42.72C23
C1C2C4C2×C4C42C4×D4C22.50C24 — C42.72C23
C1C2C2×C4 — C42.72C23
C1C22C4×Q8 — C42.72C23
C1C2C2C2×C4 — C42.72C23

Generators and relations for C42.72C23
 G = < a,b,c,d,e | a4=b4=c2=1, d2=e2=a2, ab=ba, cac=eae-1=a-1, dad-1=ab2, cbc=dbd-1=b-1, be=eb, dcd-1=bc, ce=ec, de=ed >

Subgroups: 440 in 203 conjugacy classes, 88 normal (38 characteristic)
C1, C2 [×3], C2 [×4], C4 [×2], C4 [×2], C4 [×9], C22, C22 [×12], C8 [×2], C8 [×3], C2×C4 [×3], C2×C4 [×4], C2×C4 [×12], D4 [×17], Q8 [×6], C23 [×4], C42, C42 [×2], C42 [×2], C22⋊C4 [×8], C4⋊C4 [×3], C4⋊C4 [×2], C4⋊C4 [×4], C2×C8 [×2], C2×C8 [×2], D8 [×4], SD16 [×8], C22×C4 [×4], C2×D4 [×2], C2×D4 [×2], C2×D4 [×6], C2×Q8, C2×Q8 [×2], C4○D4 [×4], C4×C8, C8⋊C4 [×2], D4⋊C4 [×2], D4⋊C4 [×2], Q8⋊C4 [×2], C4⋊C8, C4⋊C8 [×2], C2.D8, C42⋊C2, C4×D4 [×2], C4×D4 [×2], C4×Q8, C4×Q8 [×2], C4⋊D4 [×3], C22⋊Q8, C4.4D4 [×2], C422C2 [×2], C41D4, C41D4 [×2], C4⋊Q8, C2×D8, C2×D8 [×2], C2×SD16 [×6], C2×C4○D4, C4×D8, SD16⋊C4 [×2], C84Q8, C4⋊D8, C4⋊SD16 [×2], D4.D4, D4.2D4 [×2], C85D4, C83D4 [×2], Q86D4, C22.50C24, C42.72C23
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, Q86D4, C2×C8⋊C22, D4○SD16, C42.72C23

Character table of C42.72C23

 class 12A2B2C2D2E2F2G4A4B4C4D4E4F4G4H4I4J4K4L4M4N4O8A8B8C8D8E8F
 size 11118888222244444444488444488
ρ111111111111111111111111111111    trivial
ρ21111111-111-1-1-11-1-1-1-11-1-1-1111-1-1-11    linear of order 2
ρ3111111-1-111111-1-1111-1-11-1-11111-1-1    linear of order 2
ρ4111111-1111-1-1-1-11-1-1-1-11-11-111-1-11-1    linear of order 2
ρ511111-11111111-1-1-111-1-1-1-1-1-1-1-1-111    linear of order 2
ρ611111-11-111-1-1-1-111-1-1-1111-1-1-111-11    linear of order 2
ρ711111-1-1-11111111-11111-111-1-1-1-1-1-1    linear of order 2
ρ811111-1-1111-1-1-11-11-1-11-11-11-1-1111-1    linear of order 2
ρ91111-1-11-111-1-11-111-11-111-1111-1-11-1    linear of order 2
ρ101111-1-1111111-1-1-1-11-1-1-1-1111111-1-1    linear of order 2
ρ111111-1-1-1111-1-111-11-111-111-111-1-1-11    linear of order 2
ρ121111-1-1-1-11111-111-11-111-1-1-1111111    linear of order 2
ρ131111-111-111-1-111-1-1-111-1-11-1-1-1111-1    linear of order 2
ρ141111-11111111-11111-1111-1-1-1-1-1-1-1-1    linear of order 2
ρ151111-11-1111-1-11-11-1-11-11-1-11-1-111-11    linear of order 2
ρ161111-11-1-11111-1-1-111-1-1-1111-1-1-1-111    linear of order 2
ρ1722220000-2-2-2-202-2020-22000000000    orthogonal lifted from D4
ρ1822220000-2-2220220-20-2-2000000000    orthogonal lifted from D4
ρ1922220000-2-2220-2-20-2022000000000    orthogonal lifted from D4
ρ2022220000-2-2-2-20-220202-2000000000    orthogonal lifted from D4
ρ212-22-200002-2002i00-2i0-2i002i002-20000    complex lifted from C4○D4
ρ222-22-200002-2002i002i0-2i00-2i00-220000    complex lifted from C4○D4
ρ232-22-200002-200-2i002i02i00-2i002-20000    complex lifted from C4○D4
ρ242-22-200002-200-2i00-2i02i002i00-220000    complex lifted from C4○D4
ρ254-4-440000004-400000000000000000    orthogonal lifted from C8⋊C22
ρ264-4-44000000-4400000000000000000    orthogonal lifted from C8⋊C22
ρ274-44-40000-440000000000000000000    orthogonal lifted from 2+ 1+4
ρ2844-4-40000000000000000000002-2-2-200    complex lifted from D4○SD16
ρ2944-4-4000000000000000000000-2-22-200    complex lifted from D4○SD16

Smallest permutation representation of C42.72C23
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 59 11 41)(2 60 12 42)(3 57 9 43)(4 58 10 44)(5 62 50 20)(6 63 51 17)(7 64 52 18)(8 61 49 19)(13 25 31 47)(14 26 32 48)(15 27 29 45)(16 28 30 46)(21 55 40 34)(22 56 37 35)(23 53 38 36)(24 54 39 33)
(1 28)(2 27)(3 26)(4 25)(5 36)(6 35)(7 34)(8 33)(9 48)(10 47)(11 46)(12 45)(13 58)(14 57)(15 60)(16 59)(17 22)(18 21)(19 24)(20 23)(29 42)(30 41)(31 44)(32 43)(37 63)(38 62)(39 61)(40 64)(49 54)(50 53)(51 56)(52 55)
(1 63 3 61)(2 18 4 20)(5 42 7 44)(6 57 8 59)(9 19 11 17)(10 62 12 64)(13 23 15 21)(14 39 16 37)(22 32 24 30)(25 36 27 34)(26 54 28 56)(29 40 31 38)(33 46 35 48)(41 51 43 49)(45 55 47 53)(50 60 52 58)
(1 27 3 25)(2 26 4 28)(5 37 7 39)(6 40 8 38)(9 47 11 45)(10 46 12 48)(13 41 15 43)(14 44 16 42)(17 55 19 53)(18 54 20 56)(21 49 23 51)(22 52 24 50)(29 57 31 59)(30 60 32 58)(33 62 35 64)(34 61 36 63)

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

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

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

Matrix representation of C42.72C23 in GL6(𝔽17)

400000
4130000
004089
0013480
00413013
0081349
,
100000
010000
0011500
0011600
00116016
0001610
,
1380000
1340000
004089
0081309
0013440
0004813
,
400000
040000
0010150
0000161
0000160
0001160
,
1150000
1160000
001000
000100
000010
000001

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

C42.72C23 in GAP, Magma, Sage, TeX

C_4^2._{72}C_2^3
% in TeX

G:=Group("C4^2.72C2^3");
// GroupNames label

G:=SmallGroup(128,2129);
// by ID

G=gap.SmallGroup(128,2129);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,120,758,723,100,346,80,4037,1027,124]);
// Polycyclic

G:=Group<a,b,c,d,e|a^4=b^4=c^2=1,d^2=e^2=a^2,a*b=b*a,c*a*c=e*a*e^-1=a^-1,d*a*d^-1=a*b^2,c*b*c=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

Export

Character table of C42.72C23 in TeX

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