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G = C42.181C23order 128 = 27

42nd non-split extension by C42 of C23 acting via C23/C2=C22

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

Aliases: C42.181C23, Q8⋊C814C2, C4⋊C4.289D4, (C2×D4).23D4, C84D4.3C2, (C2×Q8).41D4, C4⋊SD1629C2, C4.79(C4○D8), C4.D811C2, (C4×C8).40C22, C4⋊C8.157C22, C4.55(C8⋊C22), C41D4.5C22, (C4×Q8).17C22, C2.13(D44D4), C2.12(D4⋊D4), C22.147C22≀C2, C22.53C241C2, (C2×C4).938(C2×D4), SmallGroup(128,352)

Series: Derived Chief Lower central Upper central Jennings

C1C42 — C42.181C23
C1C2C22C2×C4C42C4×Q8C22.53C24 — C42.181C23
C1C22C42 — C42.181C23
C1C22C42 — C42.181C23
C1C22C22C42 — C42.181C23

Generators and relations for C42.181C23
 G = < a,b,c,d,e | a4=b4=c2=d2=1, e2=a2, ab=ba, cac=dad=a-1, eae-1=ab2, cbc=dbd=ebe-1=b-1, dcd=ac, ece-1=bc, de=ed >

Subgroups: 328 in 120 conjugacy classes, 34 normal (12 characteristic)
C1, C2, C2 [×2], C2 [×3], C4 [×4], C4 [×5], C22, C22 [×9], C8 [×4], C2×C4, C2×C4 [×2], C2×C4 [×6], D4 [×12], Q8 [×4], C23 [×3], C42, C42 [×2], C22⋊C4 [×6], C4⋊C4 [×2], C4⋊C4 [×2], C2×C8 [×3], D8 [×4], SD16 [×4], C22×C4 [×2], C2×D4 [×2], C2×D4 [×5], C2×Q8 [×2], C4×C8, D4⋊C4 [×2], C4⋊C8 [×2], C4×D4 [×2], C4×Q8 [×2], C22.D4 [×2], C4.4D4 [×2], C41D4 [×2], C2×D8 [×2], C2×SD16 [×2], Q8⋊C8 [×2], C4.D8, C4⋊SD16 [×2], C84D4, C22.53C24, C42.181C23
Quotients: C1, C2 [×7], C22 [×7], D4 [×6], C23, C2×D4 [×3], C22≀C2, C4○D8 [×2], C8⋊C22 [×2], D4⋊D4 [×2], D44D4, C42.181C23

Character table of C42.181C23

 class 12A2B2C2D2E2F4A4B4C4D4E4F4G4H4I4J4K8A8B8C8D8E8F8G8H
 size 111188162222444448844448888
ρ111111111111111111111111111    trivial
ρ21111-1-1111111-11-111-1-1-1-1-11-1-11    linear of order 2
ρ31111-1-1-11111-11-111-1111111-1-11    linear of order 2
ρ4111111-11111-1-1-1-11-1-1-1-1-1-11111    linear of order 2
ρ51111-1-1-111111-11-111-11111-111-1    linear of order 2
ρ6111111-111111111111-1-1-1-1-1-1-1-1    linear of order 2
ρ711111111111-1-1-1-11-1-11111-1-1-1-1    linear of order 2
ρ81111-1-111111-11-111-11-1-1-1-1-111-1    linear of order 2
ρ92222-220-2-2-2-2000020000000000    orthogonal lifted from D4
ρ10222200022-2-20202-20-200000000    orthogonal lifted from D4
ρ112222000-2-222-20-20-22000000000    orthogonal lifted from D4
ρ122222000-2-2222020-2-2000000000    orthogonal lifted from D4
ρ1322222-20-2-2-2-2000020000000000    orthogonal lifted from D4
ρ14222200022-2-20-20-2-20200000000    orthogonal lifted from D4
ρ1522-2-20002-2000-2i02i00022-2-2--200-2    complex lifted from C4○D8
ρ1622-2-20002-20002i0-2i00022-2-2-200--2    complex lifted from C4○D8
ρ172-2-2200000-22-2i02i0000-22-220-2--20    complex lifted from C4○D8
ρ182-2-2200000-22-2i02i00002-22-20--2-20    complex lifted from C4○D8
ρ192-2-2200000-222i0-2i0000-22-220--2-20    complex lifted from C4○D8
ρ202-2-2200000-222i0-2i00002-22-20-2--20    complex lifted from C4○D8
ρ2122-2-20002-2000-2i02i000-2-222-200--2    complex lifted from C4○D8
ρ2222-2-20002-20002i0-2i000-2-222--200-2    complex lifted from C4○D8
ρ234-44-400000000000000-222-20000    orthogonal lifted from D44D4
ρ244-4-44000004-4000000000000000    orthogonal lifted from C8⋊C22
ρ2544-4-4000-4400000000000000000    orthogonal lifted from C8⋊C22
ρ264-44-4000000000000002-2-220000    orthogonal lifted from D44D4

Smallest permutation representation of C42.181C23
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 18 15 5)(2 19 16 6)(3 20 13 7)(4 17 14 8)(9 62 56 59)(10 63 53 60)(11 64 54 57)(12 61 55 58)(21 31 33 26)(22 32 34 27)(23 29 35 28)(24 30 36 25)(37 48 49 44)(38 45 50 41)(39 46 51 42)(40 47 52 43)
(2 4)(5 18)(6 17)(7 20)(8 19)(9 61)(10 64)(11 63)(12 62)(14 16)(21 26)(22 25)(23 28)(24 27)(29 35)(30 34)(31 33)(32 36)(37 52)(38 51)(39 50)(40 49)(41 42)(43 44)(45 46)(47 48)(53 57)(54 60)(55 59)(56 58)
(1 41)(2 44)(3 43)(4 42)(5 38)(6 37)(7 40)(8 39)(9 36)(10 35)(11 34)(12 33)(13 47)(14 46)(15 45)(16 48)(17 51)(18 50)(19 49)(20 52)(21 55)(22 54)(23 53)(24 56)(25 59)(26 58)(27 57)(28 60)(29 63)(30 62)(31 61)(32 64)
(1 29 3 31)(2 25 4 27)(5 35 7 33)(6 24 8 22)(9 51 11 49)(10 40 12 38)(13 26 15 28)(14 32 16 30)(17 34 19 36)(18 23 20 21)(37 56 39 54)(41 63 43 61)(42 57 44 59)(45 60 47 58)(46 64 48 62)(50 53 52 55)

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

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

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

Matrix representation of C42.181C23 in GL4(𝔽17) generated by

0100
16000
00115
00116
,
1000
0100
00115
00116
,
1000
01600
00160
00161
,
141400
14300
00116
00146
,
4000
0400
00107
0057
G:=sub<GL(4,GF(17))| [0,16,0,0,1,0,0,0,0,0,1,1,0,0,15,16],[1,0,0,0,0,1,0,0,0,0,1,1,0,0,15,16],[1,0,0,0,0,16,0,0,0,0,16,16,0,0,0,1],[14,14,0,0,14,3,0,0,0,0,11,14,0,0,6,6],[4,0,0,0,0,4,0,0,0,0,10,5,0,0,7,7] >;

C42.181C23 in GAP, Magma, Sage, TeX

C_4^2._{181}C_2^3
% in TeX

G:=Group("C4^2.181C2^3");
// GroupNames label

G:=SmallGroup(128,352);
// by ID

G=gap.SmallGroup(128,352);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,2,141,422,184,1123,570,521,136,2804,1411,718,172]);
// Polycyclic

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

Export

Character table of C42.181C23 in TeX

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