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G = C42.26C23order 128 = 27

26th non-split extension by C42 of C23 acting faithfully

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

Aliases: C42.26C23, C4.342+ 1+4, C4.152- 1+4, C87D48C2, C4⋊D832C2, C82D418C2, C4⋊C4.141D4, D4.Q832C2, C2.35(D4○D8), C4⋊C8.87C22, C22⋊C4.33D4, C23.94(C2×D4), C4⋊C4.198C23, (C2×C8).175C23, (C2×C4).457C24, (C2×D8).77C22, C2.D8.50C22, C4.Q8.50C22, (C4×D4).136C22, (C2×D4).198C23, C41D4.72C22, C4⋊D4.52C22, D4⋊C4.62C22, (C22×C8).156C22, C22.717(C22×D4), C42.C2.32C22, (C22×C4).1112C23, C42.6C2214C2, C22.34C248C2, (C2×M4(2)).95C22, C42⋊C2.175C22, C2.76(C22.31C24), (C2×C4).581(C2×D4), SmallGroup(128,1991)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C42.26C23
C1C2C4C2×C4C42C4×D4C22.34C24 — C42.26C23
C1C2C2×C4 — C42.26C23
C1C22C42⋊C2 — C42.26C23
C1C2C2C2×C4 — C42.26C23

Generators and relations for C42.26C23
 G = < a,b,c,d,e | a4=b4=c2=e2=1, d2=b2, ab=ba, cac=a-1, dad-1=ab2, ae=ea, cbc=ebe=b-1, bd=db, dcd-1=a2c, ece=bc, ede=a2d >

Subgroups: 428 in 185 conjugacy classes, 84 normal (14 characteristic)
C1, C2, C2 [×2], C2 [×5], C4 [×2], C4 [×9], C22, C22 [×15], C8 [×4], C2×C4 [×2], C2×C4 [×4], C2×C4 [×10], D4 [×16], C23, C23 [×4], C42 [×2], C22⋊C4 [×2], C22⋊C4 [×8], C4⋊C4 [×6], C4⋊C4 [×4], C2×C8 [×4], C2×C8, M4(2), D8 [×4], C22×C4, C22×C4 [×4], C2×D4 [×4], C2×D4 [×8], D4⋊C4 [×8], C4⋊C8 [×4], C4.Q8 [×2], C2.D8 [×2], C42⋊C2, C4×D4 [×4], C4⋊D4 [×4], C4⋊D4 [×4], C22.D4 [×4], C42.C2 [×2], C41D4 [×2], C22×C8, C2×M4(2), C2×D8 [×4], C42.6C22, C4⋊D8 [×4], C87D4 [×2], C82D4 [×2], D4.Q8 [×4], C22.34C24 [×2], C42.26C23
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], C2×D4 [×6], C24, C22×D4, 2+ 1+4, 2- 1+4, C22.31C24, D4○D8 [×2], C42.26C23

Character table of C42.26C23

 class 12A2B2C2D2E2F2G2H4A4B4C4D4E4F4G4H4I4J4K8A8B8C8D8E8F
 size 11114888822444448888444488
ρ111111111111111111111111111    trivial
ρ211111-1-11111111111-11-1-1-1-1-1-1-1    linear of order 2
ρ31111-1-111111-1-1-111-11-1-11-11-1-11    linear of order 2
ρ41111-11-11111-1-1-111-1-1-11-11-111-1    linear of order 2
ρ51111-11-1-1-111-1-1-1111-1111-11-1-11    linear of order 2
ρ61111-1-11-1-111-1-1-111111-1-11-111-1    linear of order 2
ρ711111-1-1-1-11111111-1-1-1-1111111    linear of order 2
ρ81111111-1-11111111-11-11-1-1-1-1-1-1    linear of order 2
ρ9111111-1-1111-1-11-1-111-1-11111-1-1    linear of order 2
ρ1011111-11-1111-1-11-1-11-1-11-1-1-1-111    linear of order 2
ρ111111-1-1-1-111111-1-1-1-11111-11-11-1    linear of order 2
ρ121111-111-111111-1-1-1-1-11-1-11-11-11    linear of order 2
ρ131111-1111-11111-1-1-11-1-1-11-11-11-1    linear of order 2
ρ141111-1-1-11-11111-1-1-111-11-11-11-11    linear of order 2
ρ1511111-111-111-1-11-1-1-1-1111111-1-1    linear of order 2
ρ16111111-11-111-1-11-1-1-111-1-1-1-1-111    linear of order 2
ρ172222-20000-2-2-2222-20000000000    orthogonal lifted from D4
ρ18222220000-2-22-2-22-20000000000    orthogonal lifted from D4
ρ192222-20000-2-22-22-220000000000    orthogonal lifted from D4
ρ20222220000-2-2-22-2-220000000000    orthogonal lifted from D4
ρ214-4-440000000000000000-22022000    orthogonal lifted from D4○D8
ρ224-44-400000-44000000000000000    orthogonal lifted from 2+ 1+4
ρ2344-4-400000000000000000220-2200    orthogonal lifted from D4○D8
ρ2444-4-400000000000000000-2202200    orthogonal lifted from D4○D8
ρ254-4-440000000000000000220-22000    orthogonal lifted from D4○D8
ρ264-44-4000004-4000000000000000    symplectic lifted from 2- 1+4, Schur index 2

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

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

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

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

Matrix representation of C42.26C23 in GL8(𝔽17)

00100000
1161150000
160000000
161010000
00000100
000016000
0000116115
000010116
,
01000000
160000000
1161150000
101160000
00001000
00000100
00000010
00000001
,
00100000
1611620000
10000000
101160000
00000100
00001000
0000116115
000000016
,
1161150000
001600000
01000000
101160000
0000116115
000000160
000001600
0000016116
,
1414000000
143000000
1431160000
1401460000
00000010
0000116115
00001000
00000001

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

C42.26C23 in GAP, Magma, Sage, TeX

C_4^2._{26}C_2^3
% in TeX

G:=Group("C4^2.26C2^3");
// GroupNames label

G:=SmallGroup(128,1991);
// by ID

G=gap.SmallGroup(128,1991);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,456,758,219,675,1018,80,4037,1027,124]);
// Polycyclic

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

Export

Character table of C42.26C23 in TeX

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