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G = C42.43C23order 128 = 27

43rd non-split extension by C42 of C23 acting faithfully

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

Aliases: C42.43C23, C4.532+ 1+4, C8⋊D435C2, C89D411C2, C87D437C2, C4⋊C4.154D4, Q86(C4○D4), Q85D49C2, D46D411C2, D4⋊D443C2, C4.Q1634C2, Q8⋊D420C2, (C2×D4).314D4, C4⋊C8.99C22, (C2×C8).95C23, C2.47(Q8○D8), D4.2D441C2, C4⋊C4.232C23, (C2×C4).500C24, C22⋊C4.164D4, (C2×D8).38C22, C23.474(C2×D4), C4⋊Q8.148C22, SD16⋊C431C2, C8⋊C4.40C22, (C4×D4).153C22, (C2×D4).230C23, C22.D825C2, C4⋊D4.79C22, C22⋊C8.77C22, (C2×Q8).396C23, (C4×Q8).154C22, C2.136(D45D4), C2.D8.118C22, C22⋊Q8.79C22, D4⋊C4.69C22, C23.36D415C2, C23.48D425C2, C22.11(C8⋊C22), (C22×C8).305C22, Q8⋊C4.69C22, (C2×SD16).52C22, C4.4D4.63C22, C22.760(C22×D4), (C22×C4).1144C23, (C22×Q8).341C22, C42.28C2213C2, (C2×M4(2)).109C22, C4.225(C2×C4○D4), (C2×C4).597(C2×D4), C2.75(C2×C8⋊C22), (C2×Q8⋊C4)⋊31C2, (C2×C4⋊C4).665C22, (C2×C4○D4).206C22, SmallGroup(128,2040)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C42.43C23
C1C2C4C2×C4C22×C4C22×Q8Q85D4 — C42.43C23
C1C2C2×C4 — C42.43C23
C1C22C4×D4 — C42.43C23
C1C2C2C2×C4 — C42.43C23

Generators and relations for C42.43C23
 G = < a,b,c,d,e | a4=b4=e2=1, c2=a2, d2=b2, ab=ba, cac-1=eae=a-1, dad-1=ab2, cbc-1=dbd-1=b-1, be=eb, dcd-1=bc, ece=a2c, ede=b2d >

Subgroups: 432 in 210 conjugacy classes, 88 normal (84 characteristic)
C1, C2 [×3], C2 [×5], C4 [×2], C4 [×11], C22, C22 [×2], C22 [×11], C8 [×4], C2×C4 [×5], C2×C4 [×20], D4 [×14], Q8 [×2], Q8 [×7], C23 [×2], C23 [×2], C42, C42, C22⋊C4 [×2], C22⋊C4 [×7], C4⋊C4 [×5], C4⋊C4 [×5], C2×C8 [×4], C2×C8, M4(2), D8, SD16 [×3], C22×C4 [×2], C22×C4 [×5], C2×D4 [×3], C2×D4 [×4], C2×Q8 [×2], C2×Q8 [×5], C4○D4 [×7], C8⋊C4, C22⋊C8 [×2], D4⋊C4 [×4], Q8⋊C4 [×6], C4⋊C8, C2.D8 [×3], C2×C4⋊C4 [×2], C4×D4 [×2], C4×D4, C4×Q8, C4⋊D4 [×3], C4⋊D4, C22⋊Q8, C22⋊Q8 [×2], C22.D4 [×2], C4.4D4, C4.4D4, C4⋊Q8, C22×C8, C2×M4(2), C2×D8, C2×SD16 [×2], C22×Q8, C2×C4○D4, C2×C4○D4, C2×Q8⋊C4, C23.36D4, C89D4, SD16⋊C4, Q8⋊D4, D4⋊D4, D4.2D4, C87D4, C8⋊D4, C4.Q16, C22.D8, C23.48D4, C42.28C22, D46D4, Q85D4, C42.43C23
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, D45D4, C2×C8⋊C22, Q8○D8, C42.43C23

Character table of C42.43C23

 class 12A2B2C2D2E2F2G2H4A4B4C4D4E4F4G4H4I4J4K4L4M4N8A8B8C8D8E8F
 size 11112248822444444448888444488
ρ111111111111111111111111111111    trivial
ρ21111-1-1-11-111-11-1-1-1-11-11-11111-1-11-1    linear of order 2
ρ31111-1-111-1111-11-111-11-1-1-1111-1-1-11    linear of order 2
ρ4111111-11111-1-1-11-1-1-1-1-11-111111-1-1    linear of order 2
ρ5111111-1-11111-1-111-1-1-11-1-11-1-1-1-111    linear of order 2
ρ61111-1-11-1-111-1-11-1-11-1111-11-1-1111-1    linear of order 2
ρ71111-1-1-1-1-11111-1-11-11-1-1111-1-111-11    linear of order 2
ρ81111111-1111-1111-1111-1-111-1-1-1-1-1-1    linear of order 2
ρ91111-1-1111111-1-1-111-1-1-1-11-1-1-1111-1    linear of order 2
ρ10111111-11-111-1-111-1-1-11-111-1-1-1-1-111    linear of order 2
ρ1111111111-11111-11111-111-1-1-1-1-1-1-1-1    linear of order 2
ρ121111-1-1-11111-111-1-1-1111-1-1-1-1-111-11    linear of order 2
ρ131111-1-1-1-1111111-11-111-11-1-111-1-11-1    linear of order 2
ρ141111111-1-111-11-11-111-1-1-1-1-1111111    linear of order 2
ρ15111111-1-1-1111-1111-1-111-11-11111-1-1    linear of order 2
ρ161111-1-11-1111-1-1-1-1-11-1-1111-111-1-1-11    linear of order 2
ρ172222-2-2200-2-20-2020-2200000000000    orthogonal lifted from D4
ρ182222-2-2-200-2-2020202-200000000000    orthogonal lifted from D4
ρ19222222200-2-2020-20-2-200000000000    orthogonal lifted from D4
ρ20222222-200-2-20-20-202200000000000    orthogonal lifted from D4
ρ212-22-2000002-220-2i0-2002i0000-2i2i0000    complex lifted from C4○D4
ρ222-22-2000002-2202i0-200-2i00002i-2i0000    complex lifted from C4○D4
ρ232-22-2000002-2-20-2i02002i00002i-2i0000    complex lifted from C4○D4
ρ242-22-2000002-2-202i0200-2i0000-2i2i0000    complex lifted from C4○D4
ρ254-4-444-400000000000000000000000    orthogonal lifted from C8⋊C22
ρ264-44-400000-44000000000000000000    orthogonal lifted from 2+ 1+4
ρ274-4-44-4400000000000000000000000    orthogonal lifted from C8⋊C22
ρ2844-4-400000000000000000000022-2200    symplectic lifted from Q8○D8, Schur index 2
ρ2944-4-4000000000000000000000-222200    symplectic lifted from Q8○D8, Schur index 2

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

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

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

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

Matrix representation of C42.43C23 in GL6(𝔽17)

400000
0130000
00691616
0096016
008930
0061482
,
100000
010000
000100
0016000
00161612
00011616
,
040000
400000
00691616
0081101
005536
00012814
,
100000
010000
000010
00111615
0016000
0011016
,
0130000
400000
00691616
0096016
008930
0061482

G:=sub<GL(6,GF(17))| [4,0,0,0,0,0,0,13,0,0,0,0,0,0,6,9,8,6,0,0,9,6,9,14,0,0,16,0,3,8,0,0,16,16,0,2],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,16,16,0,0,0,1,0,16,1,0,0,0,0,1,16,0,0,0,0,2,16],[0,4,0,0,0,0,4,0,0,0,0,0,0,0,6,8,5,0,0,0,9,11,5,12,0,0,16,0,3,8,0,0,16,1,6,14],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,16,1,0,0,0,1,0,1,0,0,1,16,0,0,0,0,0,15,0,16],[0,4,0,0,0,0,13,0,0,0,0,0,0,0,6,9,8,6,0,0,9,6,9,14,0,0,16,0,3,8,0,0,16,16,0,2] >;

C42.43C23 in GAP, Magma, Sage, TeX

C_4^2._{43}C_2^3
% in TeX

G:=Group("C4^2.43C2^3");
// GroupNames label

G:=SmallGroup(128,2040);
// by ID

G=gap.SmallGroup(128,2040);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,112,253,758,723,352,346,4037,1027,124]);
// Polycyclic

G:=Group<a,b,c,d,e|a^4=b^4=e^2=1,c^2=a^2,d^2=b^2,a*b=b*a,c*a*c^-1=e*a*e=a^-1,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,e*c*e=a^2*c,e*d*e=b^2*d>;
// generators/relations

Export

Character table of C42.43C23 in TeX

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