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G = C42.407C23order 128 = 27

268th non-split extension by C42 of C23 acting via C23/C2=C22

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

Aliases: C42.407C23, C4.1132+ 1+4, C83D414C2, C85D424C2, C4⋊C4.130D4, D4⋊D429C2, Q8⋊D412C2, C4⋊C8.65C22, C22⋊C4.22D4, D4.D411C2, D4.2D426C2, C4⋊C4.160C23, (C2×C8).329C23, (C4×C8).266C22, (C2×C4).419C24, (C2×D8).73C22, C23.291(C2×D4), C4⋊Q8.120C22, C8⋊C4.22C22, C2.45(D4○SD16), (C4×D4).109C22, (C2×D4).168C23, C41D4.67C22, C4⋊D4.44C22, C22⋊C8.54C22, (C2×Q8).156C23, (C22×C4).307C23, (C2×SD16).37C22, C4.4D4.39C22, C22.679(C22×D4), C42.C2.24C22, D4⋊C4.109C22, C42.7C2211C2, C42.30C223C2, C22.34C246C2, Q8⋊C4.102C22, (C22×Q8).325C22, C23.38C2316C2, C42.78C2214C2, C42⋊C2.158C22, C2.90(C22.29C24), (C2×C4).548(C2×D4), (C2×C4○D4).178C22, SmallGroup(128,1953)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C42.407C23
C1C2C4C2×C4C22×C4C22×Q8C23.38C23 — C42.407C23
C1C2C2×C4 — C42.407C23
C1C22C42⋊C2 — C42.407C23
C1C2C2C2×C4 — C42.407C23

Generators and relations for C42.407C23
 G = < a,b,c,d,e | a4=b4=c2=e2=1, d2=b2, ab=ba, cac=dad-1=a-1, eae=ab2, cbc=dbd-1=b-1, be=eb, dcd-1=bc, ece=a2c, de=ed >

Subgroups: 428 in 194 conjugacy classes, 84 normal (32 characteristic)
C1, C2, C2 [×2], C2 [×4], C4 [×2], C4 [×10], C22, C22 [×12], C8 [×4], C2×C4 [×2], C2×C4 [×4], C2×C4 [×13], D4 [×12], Q8 [×8], C23, C23 [×3], C42 [×2], C22⋊C4 [×2], C22⋊C4 [×8], C4⋊C4 [×4], C4⋊C4 [×6], C2×C8 [×4], D8 [×2], SD16 [×6], C22×C4, C22×C4 [×4], C2×D4, C2×D4 [×2], C2×D4 [×5], C2×Q8 [×3], C2×Q8 [×3], C4○D4 [×2], C4×C8, C8⋊C4, C22⋊C8 [×2], D4⋊C4 [×2], Q8⋊C4 [×6], C4⋊C8 [×2], C42⋊C2, C4×D4 [×2], C4⋊D4 [×2], C4⋊D4 [×2], C22⋊Q8 [×2], C22.D4 [×4], C4.4D4 [×2], C42.C2, C41D4, C4⋊Q8 [×2], C2×D8 [×2], C2×SD16 [×6], C22×Q8, C2×C4○D4, C42.7C22, Q8⋊D4 [×2], D4⋊D4 [×2], D4.D4 [×2], D4.2D4 [×2], C42.78C22, C42.30C22, C85D4, C83D4, C23.38C23, C22.34C24, C42.407C23
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], C2×D4 [×6], C24, C22×D4, 2+ 1+4 [×2], C22.29C24, D4○SD16 [×2], C42.407C23

Character table of C42.407C23

 class 12A2B2C2D2E2F2G4A4B4C4D4E4F4G4H4I4J4K4L8A8B8C8D8E8F
 size 11114888224444488888444488
ρ111111111111111111111111111    trivial
ρ211111-1-11111111111-11-1-1-1-1-1-1-1    linear of order 2
ρ3111111-1111-1-11-1-1-111-1-11-11-11-1    linear of order 2
ρ411111-11111-1-11-1-1-11-1-11-11-11-11    linear of order 2
ρ51111-111-1111-1-11-1-11-11-11111-1-1    linear of order 2
ρ61111-1-1-1-1111-1-11-1-11111-1-1-1-111    linear of order 2
ρ71111-11-1-111-11-1-1111-1-111-11-1-11    linear of order 2
ρ81111-1-11-111-11-1-11111-1-1-11-111-1    linear of order 2
ρ91111-1-1-11111-1-11-11-11-111111-1-1    linear of order 2
ρ101111-1111111-1-11-11-1-1-1-1-1-1-1-111    linear of order 2
ρ111111-1-11111-11-1-11-1-111-11-11-1-11    linear of order 2
ρ121111-11-1111-11-1-11-1-1-111-11-111-1    linear of order 2
ρ1311111-1-1-11111111-1-1-1-1-1111111    linear of order 2
ρ141111111-11111111-1-11-11-1-1-1-1-1-1    linear of order 2
ρ1511111-11-111-1-11-1-11-1-1111-11-11-1    linear of order 2
ρ16111111-1-111-1-11-1-11-111-1-11-11-11    linear of order 2
ρ172222-2000-2-2-2222-200000000000    orthogonal lifted from D4
ρ1822222000-2-222-2-2-200000000000    orthogonal lifted from D4
ρ1922222000-2-2-2-2-22200000000000    orthogonal lifted from D4
ρ202222-2000-2-22-22-2200000000000    orthogonal lifted from D4
ρ214-44-40000-440000000000000000    orthogonal lifted from 2+ 1+4
ρ224-44-400004-40000000000000000    orthogonal lifted from 2+ 1+4
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 C42.407C23
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 61 53 45)(2 64 54 48)(3 63 55 47)(4 62 56 46)(5 58 52 42)(6 57 49 41)(7 60 50 44)(8 59 51 43)(9 36 27 17)(10 35 28 20)(11 34 25 19)(12 33 26 18)(13 40 31 21)(14 39 32 24)(15 38 29 23)(16 37 30 22)
(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), (1,61,53,45)(2,64,54,48)(3,63,55,47)(4,62,56,46)(5,58,52,42)(6,57,49,41)(7,60,50,44)(8,59,51,43)(9,36,27,17)(10,35,28,20)(11,34,25,19)(12,33,26,18)(13,40,31,21)(14,39,32,24)(15,38,29,23)(16,37,30,22), (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), (1,61,53,45)(2,64,54,48)(3,63,55,47)(4,62,56,46)(5,58,52,42)(6,57,49,41)(7,60,50,44)(8,59,51,43)(9,36,27,17)(10,35,28,20)(11,34,25,19)(12,33,26,18)(13,40,31,21)(14,39,32,24)(15,38,29,23)(16,37,30,22), (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)], [(1,61,53,45),(2,64,54,48),(3,63,55,47),(4,62,56,46),(5,58,52,42),(6,57,49,41),(7,60,50,44),(8,59,51,43),(9,36,27,17),(10,35,28,20),(11,34,25,19),(12,33,26,18),(13,40,31,21),(14,39,32,24),(15,38,29,23),(16,37,30,22)], [(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 C42.407C23 in GL8(𝔽17)

64080000
136900000
111311130000
4114110000
000001405
000030120
000001203
000050140
,
01000000
160000000
00010000
001600000
00000100
000016000
00000001
000000160
,
64080000
411800000
006130000
0013110000
000001405
000014050
000001203
000012030
,
10161160000
167660000
0016100000
001010000
000051200
0000121200
000000512
0000001212
,
1601500000
0160150000
00100000
00010000
00000010
00000001
00001000
00000100

G:=sub<GL(8,GF(17))| [6,13,11,4,0,0,0,0,4,6,13,11,0,0,0,0,0,9,11,4,0,0,0,0,8,0,13,11,0,0,0,0,0,0,0,0,0,3,0,5,0,0,0,0,14,0,12,0,0,0,0,0,0,12,0,14,0,0,0,0,5,0,3,0],[0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0],[6,4,0,0,0,0,0,0,4,11,0,0,0,0,0,0,0,8,6,13,0,0,0,0,8,0,13,11,0,0,0,0,0,0,0,0,0,14,0,12,0,0,0,0,14,0,12,0,0,0,0,0,0,5,0,3,0,0,0,0,5,0,3,0],[10,16,0,0,0,0,0,0,16,7,0,0,0,0,0,0,11,6,16,10,0,0,0,0,6,6,10,1,0,0,0,0,0,0,0,0,5,12,0,0,0,0,0,0,12,12,0,0,0,0,0,0,0,0,5,12,0,0,0,0,0,0,12,12],[16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,15,0,1,0,0,0,0,0,0,15,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0] >;

C42.407C23 in GAP, Magma, Sage, TeX

C_4^2._{407}C_2^3
% in TeX

G:=Group("C4^2.407C2^3");
// GroupNames label

G:=SmallGroup(128,1953);
// by ID

G=gap.SmallGroup(128,1953);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,120,758,219,352,675,1018,4037,1027,124]);
// 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=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 C42.407C23 in TeX

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