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G = C42.426C23order 128 = 27

287th non-split extension by C42 of C23 acting via C23/C2=C22

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

Aliases: C42.426C23, C4.682- 1+4, C8⋊Q822C2, C4⋊C4.138D4, C83Q824C2, D4.Q827C2, Q8.Q827C2, Q8⋊Q812C2, D42Q812C2, C4⋊C8.78C22, C22⋊C4.30D4, C4⋊C4.183C23, (C2×C8).337C23, (C4×C8).270C22, (C2×C4).442C24, C23.305(C2×D4), C4⋊Q8.126C22, C8⋊C4.35C22, C4.Q8.90C22, C2.51(D4○SD16), (C4×D4).123C22, (C2×D4).185C23, C4⋊D4.49C22, C22⋊C8.69C22, (C4×Q8).120C22, (C2×Q8).173C23, C2.D8.110C22, C22⋊Q8.49C22, D4⋊C4.55C22, C23.47D413C2, C23.20D430C2, (C22×C4).315C23, C4.4D4.44C22, C23.19D4.5C2, C23.46D4.3C2, C22.702(C22×D4), C42.C2.29C22, C42.7C2218C2, Q8⋊C4.107C22, C42.28C2211C2, C23.41C2310C2, C42.78C2216C2, C42⋊C2.172C22, C22.36C24.4C2, C2.90(C23.38C23), (C2×C4).566(C2×D4), (C2×C4⋊C4).657C22, SmallGroup(128,1976)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C42.426C23
C1C2C4C2×C4C22×C4C2×C4⋊C4C23.41C23 — C42.426C23
C1C2C2×C4 — C42.426C23
C1C22C42⋊C2 — C42.426C23
C1C2C2C2×C4 — C42.426C23

Generators and relations for C42.426C23
 G = < a,b,c,d,e | a4=b4=c2=e2=1, d2=a2, ab=ba, cac=dad-1=a-1b2, eae=ab2, cbc=dbd-1=b-1, be=eb, dcd-1=bc, ece=a2b2c, de=ed >

Subgroups: 300 in 160 conjugacy classes, 84 normal (all characteristic)
C1, C2 [×3], C2 [×2], C4 [×2], C4 [×12], C22, C22 [×6], C8 [×4], C2×C4 [×6], C2×C4 [×11], D4 [×3], Q8 [×5], C23, C23, C42 [×2], C42 [×2], C22⋊C4 [×2], C22⋊C4 [×6], C4⋊C4 [×8], C4⋊C4 [×10], C2×C8 [×4], C22×C4, C22×C4 [×2], C2×D4, C2×D4, C2×Q8, C2×Q8 [×3], C4×C8, C8⋊C4, C22⋊C8 [×2], D4⋊C4 [×4], Q8⋊C4 [×4], C4⋊C8 [×2], C4.Q8 [×6], C2.D8 [×2], C2×C4⋊C4, C42⋊C2 [×2], C4×D4, C4×Q8, C4⋊D4, C22⋊Q8, C22⋊Q8 [×3], C22.D4, C4.4D4, C4.4D4, C42.C2 [×2], C42.C2, C422C2, C4⋊Q8 [×3], C4⋊Q8, C42.7C22, Q8⋊Q8, D42Q8, D4.Q8, Q8.Q8, C23.46D4, C23.19D4, C23.47D4, C23.20D4, C42.78C22, C42.28C22, C83Q8, C8⋊Q8, C22.36C24, C23.41C23, C42.426C23
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], C2×D4 [×6], C24, C22×D4, 2- 1+4 [×2], C23.38C23, D4○SD16 [×2], C42.426C23

Character table of C42.426C23

 class 12A2B2C2D2E4A4B4C4D4E4F4G4H4I4J4K4L4M4N8A8B8C8D8E8F
 size 11114822444448888888444488
ρ111111111111111111111111111    trivial
ρ21111-11111-1-11-11-11-11-1-1-1-1-1-111    linear of order 2
ρ311111111-1-11-1-1-11-1-11-11-11-11-11    linear of order 2
ρ41111-1111-11-1-11-1-1-1111-11-11-1-11    linear of order 2
ρ51111-1111-11-1-111-1-11-1-11-11-111-1    linear of order 2
ρ611111111-1-11-1-111-1-1-11-11-11-11-1    linear of order 2
ρ71111-11111-1-11-1-1-11-1-1111111-1-1    linear of order 2
ρ81111111111111-1111-1-1-1-1-1-1-1-1-1    linear of order 2
ρ911111-111-1-11-1-11-111-11-1-11-11-11    linear of order 2
ρ101111-1-111-11-1-11111-1-1-111-11-1-11    linear of order 2
ρ1111111-11111111-1-1-1-1-1-1-1111111    linear of order 2
ρ121111-1-1111-1-11-1-11-11-111-1-1-1-111    linear of order 2
ρ131111-1-1111-1-11-111-111-1-11111-1-1    linear of order 2
ρ1411111-111111111-1-1-1111-1-1-1-1-1-1    linear of order 2
ρ151111-1-111-11-1-11-111-111-1-11-111-1    linear of order 2
ρ1611111-111-1-11-1-1-1-1111-111-11-11-1    linear of order 2
ρ17222220-2-222-2-2-20000000000000    orthogonal lifted from D4
ρ182222-20-2-2-2222-20000000000000    orthogonal lifted from D4
ρ19222220-2-2-2-2-2220000000000000    orthogonal lifted from D4
ρ202222-20-2-22-22-220000000000000    orthogonal lifted from D4
ρ214-44-4004-4000000000000000000    symplectic lifted from 2- 1+4, Schur index 2
ρ224-44-400-44000000000000000000    symplectic lifted from 2- 1+4, Schur index 2
ρ234-4-440000000000000000-2-202-2000    complex lifted from D4○SD16
ρ2444-4-400000000000000000-2-202-200    complex lifted from D4○SD16
ρ254-4-4400000000000000002-20-2-2000    complex lifted from D4○SD16
ρ2644-4-4000000000000000002-20-2-200    complex lifted from D4○SD16

Smallest permutation representation of C42.426C23
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 53 12 26)(2 54 9 27)(3 55 10 28)(4 56 11 25)(5 43 39 23)(6 44 40 24)(7 41 37 21)(8 42 38 22)(13 31 50 60)(14 32 51 57)(15 29 52 58)(16 30 49 59)(17 62 48 33)(18 63 45 34)(19 64 46 35)(20 61 47 36)
(2 11)(4 9)(5 41)(6 24)(7 43)(8 22)(13 52)(15 50)(17 35)(18 63)(19 33)(20 61)(21 39)(23 37)(25 27)(26 53)(28 55)(29 31)(30 59)(32 57)(34 45)(36 47)(38 42)(40 44)(46 62)(48 64)(54 56)(58 60)
(1 45 3 47)(2 17 4 19)(5 58 7 60)(6 32 8 30)(9 48 11 46)(10 20 12 18)(13 23 15 21)(14 42 16 44)(22 49 24 51)(25 64 27 62)(26 34 28 36)(29 37 31 39)(33 56 35 54)(38 59 40 57)(41 50 43 52)(53 63 55 61)
(1 50)(2 14)(3 52)(4 16)(5 63)(6 35)(7 61)(8 33)(9 51)(10 15)(11 49)(12 13)(17 42)(18 23)(19 44)(20 21)(22 48)(24 46)(25 59)(26 31)(27 57)(28 29)(30 56)(32 54)(34 39)(36 37)(38 62)(40 64)(41 47)(43 45)(53 60)(55 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), (1,53,12,26)(2,54,9,27)(3,55,10,28)(4,56,11,25)(5,43,39,23)(6,44,40,24)(7,41,37,21)(8,42,38,22)(13,31,50,60)(14,32,51,57)(15,29,52,58)(16,30,49,59)(17,62,48,33)(18,63,45,34)(19,64,46,35)(20,61,47,36), (2,11)(4,9)(5,41)(6,24)(7,43)(8,22)(13,52)(15,50)(17,35)(18,63)(19,33)(20,61)(21,39)(23,37)(25,27)(26,53)(28,55)(29,31)(30,59)(32,57)(34,45)(36,47)(38,42)(40,44)(46,62)(48,64)(54,56)(58,60), (1,45,3,47)(2,17,4,19)(5,58,7,60)(6,32,8,30)(9,48,11,46)(10,20,12,18)(13,23,15,21)(14,42,16,44)(22,49,24,51)(25,64,27,62)(26,34,28,36)(29,37,31,39)(33,56,35,54)(38,59,40,57)(41,50,43,52)(53,63,55,61), (1,50)(2,14)(3,52)(4,16)(5,63)(6,35)(7,61)(8,33)(9,51)(10,15)(11,49)(12,13)(17,42)(18,23)(19,44)(20,21)(22,48)(24,46)(25,59)(26,31)(27,57)(28,29)(30,56)(32,54)(34,39)(36,37)(38,62)(40,64)(41,47)(43,45)(53,60)(55,58)>;

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

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

Matrix representation of C42.426C23 in GL8(𝔽17)

0101600000
700160000
10070000
011000000
000031160
000011306
0000156146
0000615614
,
01000000
160000000
00010000
001600000
000016000
000001600
000000160
000000016
,
10000000
016000000
001600000
00010000
00001000
000001600
0000160160
00000101
,
115120000
11612120000
512110000
12121160000
0000159130
0000915013
000013828
000081382
,
00100000
00010000
10000000
01000000
00000100
00001000
00000001
00000010

G:=sub<GL(8,GF(17))| [0,7,1,0,0,0,0,0,10,0,0,1,0,0,0,0,16,0,0,10,0,0,0,0,0,16,7,0,0,0,0,0,0,0,0,0,3,11,15,6,0,0,0,0,11,3,6,15,0,0,0,0,6,0,14,6,0,0,0,0,0,6,6,14],[0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16],[1,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,16,0,0,0,0,0,0,16,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,1],[1,1,5,12,0,0,0,0,1,16,12,12,0,0,0,0,5,12,1,1,0,0,0,0,12,12,1,16,0,0,0,0,0,0,0,0,15,9,13,8,0,0,0,0,9,15,8,13,0,0,0,0,13,0,2,8,0,0,0,0,0,13,8,2],[0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0] >;

C42.426C23 in GAP, Magma, Sage, TeX

C_4^2._{426}C_2^3
% in TeX

G:=Group("C4^2.426C2^3");
// GroupNames label

G:=SmallGroup(128,1976);
// by ID

G=gap.SmallGroup(128,1976);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,568,758,219,436,675,1018,4037,1027,124]);
// Polycyclic

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

Export

Character table of C42.426C23 in TeX

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