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G = C42.423C23order 128 = 27

284th non-split extension by C42 of C23 acting via C23/C2=C22

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

Aliases: C42.423C23, C4.652- 1+4, C8⋊Q819C2, C4⋊C4.135D4, D4.Q825C2, C8.5Q85C2, D42Q811C2, D4⋊Q828C2, C2.32(D4○D8), C4.4D819C2, C4⋊C8.75C22, C22⋊C4.27D4, C4⋊C4.180C23, (C4×C8).117C22, (C2×C8).168C23, (C2×C4).439C24, C23.302(C2×D4), C4⋊Q8.123C22, C8⋊C4.32C22, C4.Q8.89C22, C2.49(D4○SD16), (C4×D4).121C22, (C2×D4).183C23, C4⋊D4.47C22, C41D4.70C22, C22.D824C2, C22⋊C8.66C22, C2.D8.109C22, D4⋊C4.53C22, C23.19D429C2, C23.46D413C2, (C22×C4).312C23, C22.699(C22×D4), C42.C2.26C22, C42.7C2216C2, C23.41C238C2, C42.29C227C2, C42⋊C2.169C22, C22.34C24.3C2, C2.87(C23.38C23), (C2×C4).563(C2×D4), (C2×C4⋊C4).654C22, SmallGroup(128,1973)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C42.423C23
C1C2C4C2×C4C22×C4C2×C4⋊C4C23.41C23 — C42.423C23
C1C2C2×C4 — C42.423C23
C1C22C42⋊C2 — C42.423C23
C1C2C2C2×C4 — C42.423C23

Generators and relations for C42.423C23
 G = < a,b,c,d,e | a4=b4=c2=e2=1, d2=a2, ab=ba, cac=dad-1=a-1, eae=ab2, cbc=dbd-1=b-1, be=eb, dcd-1=bc, ece=a2c, de=ed >

Subgroups: 340 in 166 conjugacy classes, 84 normal (all characteristic)
C1, C2 [×3], C2 [×3], C4 [×2], C4 [×11], C22, C22 [×9], C8 [×4], C2×C4 [×6], C2×C4 [×11], D4 [×8], Q8 [×2], C23, C23 [×2], C42 [×2], C42, C22⋊C4 [×2], C22⋊C4 [×5], C4⋊C4 [×8], C4⋊C4 [×9], C2×C8 [×4], C22×C4, C22×C4 [×3], C2×D4 [×2], C2×D4 [×4], C2×Q8 [×2], C4×C8, C8⋊C4, C22⋊C8 [×2], D4⋊C4 [×8], C4⋊C8 [×2], C4.Q8 [×4], C2.D8 [×4], C2×C4⋊C4, C42⋊C2 [×2], C4×D4 [×2], C4⋊D4 [×2], C4⋊D4 [×2], C22⋊Q8 [×2], C22.D4 [×2], C42.C2 [×3], C42.C2, C41D4, C4⋊Q8 [×2], C4⋊Q8, C42.7C22, D4⋊Q8, D42Q8, D4.Q8 [×2], C22.D8, C23.46D4, C23.19D4 [×2], C4.4D8, C42.29C22, C8.5Q8, C8⋊Q8, C22.34C24, C23.41C23, C42.423C23
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], C2×D4 [×6], C24, C22×D4, 2- 1+4 [×2], C23.38C23, D4○D8, D4○SD16, C42.423C23

Character table of C42.423C23

 class 12A2B2C2D2E2F4A4B4C4D4E4F4G4H4I4J4K4L4M8A8B8C8D8E8F
 size 11114882244444888888444488
ρ111111111111111111111111111    trivial
ρ21111-1-1111-11-1-11-11-1-1111-11-1-11    linear of order 2
ρ311111-1-111111111-11-111-1-1-1-1-1-1    linear of order 2
ρ41111-11-111-11-1-11-1-1-1111-11-111-1    linear of order 2
ρ511111-1-11111111-1-1-1-1-1-1111111    linear of order 2
ρ61111-11-111-11-1-111-111-1-11-11-1-11    linear of order 2
ρ711111111111111-11-11-1-1-1-1-1-1-1-1    linear of order 2
ρ81111-1-1111-11-1-11111-1-1-1-11-111-1    linear of order 2
ρ9111111-111-1-11-1-1-111-1-111-11-11-1    linear of order 2
ρ101111-1-1-1111-1-11-111-11-111111-1-1    linear of order 2
ρ1111111-1111-1-11-1-1-1-111-11-11-11-11    linear of order 2
ρ121111-111111-1-11-11-1-1-1-11-1-1-1-111    linear of order 2
ρ1311111-1111-1-11-1-11-1-111-11-11-11-1    linear of order 2
ρ141111-111111-1-11-1-1-11-11-11111-1-1    linear of order 2
ρ15111111-111-1-11-1-111-1-11-1-11-11-11    linear of order 2
ρ161111-1-1-1111-1-11-1-11111-1-1-1-1-111    linear of order 2
ρ172222-200-2-22-22-22000000000000    orthogonal lifted from D4
ρ182222200-2-222-2-2-2000000000000    orthogonal lifted from D4
ρ192222200-2-2-2-2-222000000000000    orthogonal lifted from D4
ρ202222-200-2-2-2222-2000000000000    orthogonal lifted from D4
ρ214-4-440000000000000000220-22000    orthogonal lifted from D4○D8
ρ224-4-440000000000000000-22022000    orthogonal lifted from D4○D8
ρ234-44-4000-4400000000000000000    symplectic lifted from 2- 1+4, Schur index 2
ρ244-44-40004-400000000000000000    symplectic lifted from 2- 1+4, Schur index 2
ρ2544-4-400000000000000000-2-202-200    complex lifted from D4○SD16
ρ2644-4-4000000000000000002-20-2-200    complex lifted from D4○SD16

Smallest permutation representation of C42.423C23
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 27 53 9)(2 28 54 10)(3 25 55 11)(4 26 56 12)(5 39 52 24)(6 40 49 21)(7 37 50 22)(8 38 51 23)(13 41 31 57)(14 42 32 58)(15 43 29 59)(16 44 30 60)(17 61 36 45)(18 62 33 46)(19 63 34 47)(20 64 35 48)
(1 4)(2 3)(5 21)(6 24)(7 23)(8 22)(9 26)(10 25)(11 28)(12 27)(13 14)(15 16)(17 46)(18 45)(19 48)(20 47)(29 30)(31 32)(33 61)(34 64)(35 63)(36 62)(37 51)(38 50)(39 49)(40 52)(41 58)(42 57)(43 60)(44 59)(53 56)(54 55)
(1 47 3 45)(2 46 4 48)(5 58 7 60)(6 57 8 59)(9 19 11 17)(10 18 12 20)(13 23 15 21)(14 22 16 24)(25 36 27 34)(26 35 28 33)(29 40 31 38)(30 39 32 37)(41 51 43 49)(42 50 44 52)(53 63 55 61)(54 62 56 64)
(1 41)(2 58)(3 43)(4 60)(5 48)(6 61)(7 46)(8 63)(9 13)(10 32)(11 15)(12 30)(14 28)(16 26)(17 21)(18 37)(19 23)(20 39)(22 33)(24 35)(25 29)(27 31)(34 38)(36 40)(42 54)(44 56)(45 49)(47 51)(50 62)(52 64)(53 57)(55 59)

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

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

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

Matrix representation of C42.423C23 in GL8(𝔽17)

071000000
0512120000
5512120000
127000000
000065162
0000611161
00001151112
0000116116
,
10000000
01000000
00100000
00010000
000011500
000011600
000000115
000000116
,
071000000
051250000
1251250000
127000000
000065162
000001101
00001151112
000001606
,
1601500000
00110000
10100000
16161600000
00000700
000012000
00000007
000000120
,
1615000000
01000000
01010000
016100000
00000010
00000001
00001000
00000100

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

C42.423C23 in GAP, Magma, Sage, TeX

C_4^2._{423}C_2^3
% in TeX

G:=Group("C4^2.423C2^3");
// GroupNames label

G:=SmallGroup(128,1973);
// by ID

G=gap.SmallGroup(128,1973);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,120,758,219,100,675,1018,4037,1027,124]);
// 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,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*c,d*e=e*d>;
// generators/relations

Export

Character table of C42.423C23 in TeX

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