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G = C42.411C23order 128 = 27

272nd non-split extension by C42 of C23 acting via C23/C2=C22

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

Aliases: C42.411C23, C4.1172+ 1+4, C4⋊C4.134D4, Q8⋊D413C2, C42Q1629C2, D4⋊D4.4C2, C8.12D46C2, C8.2D415C2, C4⋊C8.67C22, C22⋊C4.26D4, C2.31(Q8○D8), D4.7D431C2, D4.D412C2, D4.2D428C2, C4⋊C4.164C23, (C4×C8).114C22, (C2×C8).164C23, (C2×C4).423C24, Q8.D428C2, C22⋊Q1624C2, C4.SD1617C2, (C2×D8).26C22, C23.295(C2×D4), C4⋊Q8.122C22, C8⋊C4.24C22, C2.48(D4○SD16), (C2×D4).172C23, (C4×D4).110C22, C4⋊D4.46C22, C22⋊C8.58C22, (C2×Q8).160C23, (C4×Q8).107C22, (C2×Q16).28C22, C22⋊Q8.46C22, D4⋊C4.47C22, (C22×C4).311C23, (C2×SD16).39C22, C4.4D4.42C22, C22.683(C22×D4), C42.28C226C2, C42.7C2215C2, C22.36C248C2, Q8⋊C4.103C22, (C22×Q8).327C22, C42⋊C2.162C22, C23.38C2317C2, C2.94(C22.29C24), (C2×C4).552(C2×D4), (C2×C4○D4).182C22, SmallGroup(128,1957)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C42.411C23
C1C2C4C2×C4C22×C4C2×C4○D4C23.38C23 — C42.411C23
C1C2C2×C4 — C42.411C23
C1C22C42⋊C2 — C42.411C23
C1C2C2C2×C4 — C42.411C23

Generators and relations for C42.411C23
 G = < a,b,c,d,e | a4=b4=c2=d2=1, e2=b2, ab=ba, cac=dad=a-1b2, eae-1=ab2, cbc=dbd=b-1, be=eb, dcd=bc, ece-1=a2b2c, de=ed >

Subgroups: 388 in 188 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 [×13], D4 [×7], Q8 [×11], C23, C23 [×2], C42 [×2], C42, C22⋊C4 [×2], C22⋊C4 [×9], C4⋊C4 [×4], C4⋊C4 [×7], C2×C8 [×4], D8, SD16 [×4], Q16 [×3], C22×C4, C22×C4 [×3], C2×D4 [×2], C2×D4 [×2], C2×Q8 [×4], C2×Q8 [×4], C4○D4 [×2], C4×C8, C8⋊C4, C22⋊C8 [×2], D4⋊C4 [×2], Q8⋊C4 [×6], C4⋊C8 [×2], C42⋊C2, C4×D4, C4×Q8, C4⋊D4, C22⋊Q8, C22⋊Q8 [×3], C22.D4 [×3], C4.4D4 [×3], C4.4D4, C422C2, C4⋊Q8 [×3], C2×D8, C2×SD16 [×4], C2×Q16 [×3], C22×Q8, C2×C4○D4, C42.7C22, Q8⋊D4, D4⋊D4, C22⋊Q16, D4.7D4, D4.D4, C42Q16, D4.2D4, Q8.D4, C4.SD16, C42.28C22, C8.12D4, C8.2D4, C23.38C23, C22.36C24, C42.411C23
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], C2×D4 [×6], C24, C22×D4, 2+ 1+4 [×2], C22.29C24, D4○SD16, Q8○D8, C42.411C23

Character table of C42.411C23

 class 12A2B2C2D2E2F4A4B4C4D4E4F4G4H4I4J4K4L4M8A8B8C8D8E8F
 size 11114882244444888888444488
ρ111111111111111111111111111    trivial
ρ21111-1-1-111-11-11-111-1-1111-11-1-11    linear of order 2
ρ31111-1-1-1111-11-1-11-111-111111-1-1    linear of order 2
ρ4111111111-1-1-1-111-1-1-1-111-11-11-1    linear of order 2
ρ511111-1111111111-1-111-1-1-1-1-1-1-1    linear of order 2
ρ61111-11-111-11-11-11-11-11-1-11-111-1    linear of order 2
ρ71111-11-1111-11-1-111-11-1-1-1-1-1-111    linear of order 2
ρ811111-1111-1-1-1-11111-1-1-1-11-11-11    linear of order 2
ρ9111111-111-1-1-1-11-1-1-1111-11-11-11    linear of order 2
ρ101111-1-11111-11-1-1-1-11-111-1-1-1-111    linear of order 2
ρ111111-1-1111-11-11-1-11-11-11-11-111-1    linear of order 2
ρ12111111-11111111-111-1-11-1-1-1-1-1-1    linear of order 2
ρ1311111-1-111-1-1-1-11-11111-11-11-11-1    linear of order 2
ρ141111-111111-11-1-1-11-1-11-11111-1-1    linear of order 2
ρ151111-11111-11-11-1-1-111-1-11-11-1-11    linear of order 2
ρ1611111-1-11111111-1-1-1-1-1-1111111    linear of order 2
ρ172222-200-2-2-2-2222000000000000    orthogonal lifted from D4
ρ182222200-2-2-222-2-2000000000000    orthogonal lifted from D4
ρ192222-200-2-222-2-22000000000000    orthogonal lifted from D4
ρ202222200-2-22-2-22-2000000000000    orthogonal lifted from D4
ρ214-44-40004-400000000000000000    orthogonal lifted from 2+ 1+4
ρ224-44-4000-4400000000000000000    orthogonal lifted from 2+ 1+4
ρ234-4-440000000000000000-22022000    symplectic lifted from Q8○D8, Schur index 2
ρ244-4-440000000000000000220-22000    symplectic lifted from Q8○D8, 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 C42.411C23
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 29 26 50)(2 30 27 51)(3 31 28 52)(4 32 25 49)(5 60 61 53)(6 57 62 54)(7 58 63 55)(8 59 64 56)(9 35 13 40)(10 36 14 37)(11 33 15 38)(12 34 16 39)(17 43 24 48)(18 44 21 45)(19 41 22 46)(20 42 23 47)
(1 58)(2 54)(3 60)(4 56)(5 31)(6 51)(7 29)(8 49)(9 42)(10 46)(11 44)(12 48)(13 47)(14 41)(15 45)(16 43)(17 39)(18 33)(19 37)(20 35)(21 38)(22 36)(23 40)(24 34)(25 59)(26 55)(27 57)(28 53)(30 62)(32 64)(50 63)(52 61)
(2 25)(4 27)(5 60)(6 56)(7 58)(8 54)(10 16)(12 14)(17 46)(18 44)(19 48)(20 42)(21 45)(22 43)(23 47)(24 41)(29 50)(30 32)(31 52)(33 38)(34 36)(35 40)(37 39)(49 51)(53 61)(55 63)(57 64)(59 62)
(1 9 26 13)(2 14 27 10)(3 11 28 15)(4 16 25 12)(5 23 61 20)(6 17 62 24)(7 21 63 18)(8 19 64 22)(29 35 50 40)(30 37 51 36)(31 33 52 38)(32 39 49 34)(41 56 46 59)(42 60 47 53)(43 54 48 57)(44 58 45 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), (1,29,26,50)(2,30,27,51)(3,31,28,52)(4,32,25,49)(5,60,61,53)(6,57,62,54)(7,58,63,55)(8,59,64,56)(9,35,13,40)(10,36,14,37)(11,33,15,38)(12,34,16,39)(17,43,24,48)(18,44,21,45)(19,41,22,46)(20,42,23,47), (1,58)(2,54)(3,60)(4,56)(5,31)(6,51)(7,29)(8,49)(9,42)(10,46)(11,44)(12,48)(13,47)(14,41)(15,45)(16,43)(17,39)(18,33)(19,37)(20,35)(21,38)(22,36)(23,40)(24,34)(25,59)(26,55)(27,57)(28,53)(30,62)(32,64)(50,63)(52,61), (2,25)(4,27)(5,60)(6,56)(7,58)(8,54)(10,16)(12,14)(17,46)(18,44)(19,48)(20,42)(21,45)(22,43)(23,47)(24,41)(29,50)(30,32)(31,52)(33,38)(34,36)(35,40)(37,39)(49,51)(53,61)(55,63)(57,64)(59,62), (1,9,26,13)(2,14,27,10)(3,11,28,15)(4,16,25,12)(5,23,61,20)(6,17,62,24)(7,21,63,18)(8,19,64,22)(29,35,50,40)(30,37,51,36)(31,33,52,38)(32,39,49,34)(41,56,46,59)(42,60,47,53)(43,54,48,57)(44,58,45,55)>;

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

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

Matrix representation of C42.411C23 in GL8(𝔽17)

000130000
00400000
013000000
40000000
000050120
000005012
0000120120
0000012012
,
01000000
160000000
00010000
001600000
00000100
000016000
00000001
000000160
,
003140000
0014140000
314000000
1414000000
000014300
00003300
000000143
00000033
,
10000000
016000000
00100000
000160000
00001000
000001600
00000010
000000016
,
00100000
00010000
160000000
016000000
00000010
00000001
000016000
000001600

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

C42.411C23 in GAP, Magma, Sage, TeX

C_4^2._{411}C_2^3
% in TeX

G:=Group("C4^2.411C2^3");
// GroupNames label

G:=SmallGroup(128,1957);
// by ID

G=gap.SmallGroup(128,1957);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,568,758,219,675,1018,248,4037,1027,124]);
// Polycyclic

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

Export

Character table of C42.411C23 in TeX

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