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G = C42.7C23  order 128 = 27

7th non-split extension by C42 of C23 acting faithfully

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

Aliases: C42.7C23, C8⋊Q8⋊2C2, C4⋊C4.42D4, (C2×D4).32D4, C8.2D4⋊2C2, (C2×Q8).32D4, C4⋊Q8.37C22, C2.29(D4⋊4D4), C8⋊C4.94C22, C2.21(D4.9D4), C22.188C22≀C2, C42.C2.5C22, C42.C22⋊4C2, C42.2C22⋊4C2, C4.4D4.11C22, C2.21(D4.10D4), C42.28C22⋊27C2, C22.56C24.1C2, (C2×C4).220(C2×D4), SmallGroup(128,393)

Series: Derived ►Chief ►Lower central ►Upper central ►Jennings

Derived series
C1 — C42 — C42.7C23
Chief series
C1 — C2 — C22 — C2×C4 — C42 — C42.C2 — C22.56C24 — C42.7C23
Lower central
C1 — C22 — C42 — C42.7C23
Upper central
C1 — C22 — C42 — C42.7C23
Jennings
C1 — C22 — C22 — C42 — C42.7C23

Generators and relations for C42.7C23
 G = < a,b,c,d,e | a4=b4=d2=1, c2=a2, e2=a2b2, ab=ba, cac-1=dad=a-1, eae-1=a-1b2, cbc-1=ebe-1=b-1, dbd=a2b-1, dcd=ac, ece-1=bc, de=ed >

Subgroups: 264 in 104 conjugacy classes, 30 normal (all characteristic)
C1, C2, C2, C4, C22, C22, C8, C2×C4, C2×C4, D4, Q8, C23, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, SD16, Q16, C22×C4, C2×D4, C2×D4, C2×Q8, C2×Q8, C8⋊C4, D4⋊C4, Q8⋊C4, C4.Q8, C2.D8, C4⋊D4, C22⋊Q8, C22.D4, C4.4D4, C42.C2, C4⋊Q8, C2×SD16, C2×Q16, C42.C22, C42.2C22, C42.28C22, C8.2D4, C8⋊Q8, C22.56C24, C42.7C23
Quotients: C1, C2, C22, D4, C23, C2×D4, C22≀C2, D4⋊4D4, D4.9D4, D4.10D4, C42.7C23

Character table of C42.7C23

 class 12A2B2C2D2E4A4B4C4D4E4F4G4H8A8B8C8D8E8F
 size 111188444888816888888
ρ111111111111111111111    trivial
ρ21111-1-1111-11-11-111-1-111    linear of order 2
ρ31111-1-1111-11-111-1-111-1-1    linear of order 2
ρ41111111111111-1-1-1-1-1-1-1    linear of order 2
ρ511111-1111-1-11-1-11111-1-1    linear of order 2
ρ61111-111111-1-1-1111-1-1-1-1    linear of order 2
ρ71111-111111-1-1-1-1-1-11111    linear of order 2
ρ811111-1111-1-11-11-1-1-1-111    linear of order 2
ρ9222200-2-22020-20000000    orthogonal lifted from D4
ρ10222220-22-200-200000000    orthogonal lifted from D4
ρ112222022-2-2-20000000000    orthogonal lifted from D4
ρ1222220-22-2-220000000000    orthogonal lifted from D4
ρ13222200-2-220-2020000000    orthogonal lifted from D4
ρ142222-20-22-200200000000    orthogonal lifted from D4
ρ154-4-4400000000000000-22    orthogonal lifted from D4⋊4D4
ρ164-4-44000000000000002-2    orthogonal lifted from D4⋊4D4
ρ1744-4-40000000000002-200    symplectic lifted from D4.10D4, Schur index 2
ρ1844-4-4000000000000-2200    symplectic lifted from D4.10D4, Schur index 2
ρ194-44-400000000002i-2i0000    complex lifted from D4.9D4
ρ204-44-40000000000-2i2i0000    complex lifted from D4.9D4

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

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

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

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

Matrix representation of C42.7C23 ►in GL8(𝔽17)

00100000
00010000
160000000
016000000
000001160
000010016
000020016
000002160
,
01000000
10000000
00010000
00100000
00000100
000016000
000015001
000002160
,
144340000
414430000
343130000
431330000
0000134152
00000022
00000040
000008130
,
10000000
016000000
001600000
00010000
00001000
00000100
000002160
000020016
,
10070000
01700000
071600000
700160000
00001000
000001600
000001510
000020016

G:=sub<GL(8,GF(17))| [0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,2,0,0,0,0,0,1,0,0,2,0,0,0,0,16,0,0,16,0,0,0,0,0,16,16,0],[0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,16,15,0,0,0,0,0,1,0,0,2,0,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0],[14,4,3,4,0,0,0,0,4,14,4,3,0,0,0,0,3,4,3,13,0,0,0,0,4,3,13,3,0,0,0,0,0,0,0,0,13,0,0,0,0,0,0,0,4,0,0,8,0,0,0,0,15,2,4,13,0,0,0,0,2,2,0,0],[1,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,2,0,0,0,0,0,1,2,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16],[1,0,0,7,0,0,0,0,0,1,7,0,0,0,0,0,0,7,16,0,0,0,0,0,7,0,0,16,0,0,0,0,0,0,0,0,1,0,0,2,0,0,0,0,0,16,15,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,16] >;
 

C42.7C23 in GAP, Magma, Sage, TeX 

C_4^2._7C_2^3
 
% in TeX
 
G:=Group("C4^2.7C2^3");
 
// GroupNames label
 
G:=SmallGroup(128,393);
 
// by ID
 
G=gap.SmallGroup(128,393);
 
# by ID
 
G:=PCGroup([7,-2,2,2,-2,2,-2,2,224,141,422,352,1123,570,521,136,3924,1411,998,242]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^4=b^4=d^2=1,c^2=a^2,e^2=a^2*b^2,a*b=b*a,c*a*c^-1=d*a*d=a^-1,e*a*e^-1=a^-1*b^2,c*b*c^-1=e*b*e^-1=b^-1,d*b*d=a^2*b^-1,d*c*d=a*c,e*c*e^-1=b*c,d*e=e*d>;
 
// generators/relations
 

Export

Character table of C42.7C23 in TeX

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