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G = C42.C23order 128 = 27

1st non-split extension by C42 of C23 acting faithfully

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

Aliases: C42.1C23, C83D41C2, (C2×D4).27D4, (C2×Q8).27D4, C8⋊C41C22, C41D42C22, C2.24(D44D4), C24⋊C221C2, C4.4D4.6C22, C22.182C22≀C2, C42.C221C2, (C2×C4).214(C2×D4), 2-Sylow(PSL(3,4).C2), SmallGroup(128,387)

Series: Derived Chief Lower central Upper central Jennings

C1C42 — C42.C23
C1C2C22C2×C4C42C4.4D4C24⋊C22 — C42.C23
C1C22C42 — C42.C23
C1C22C42 — C42.C23
C1C22C22C42 — C42.C23

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

Subgroups: 416 in 131 conjugacy classes, 30 normal (6 characteristic)
C1, C2 [×3], C2 [×4], C4 [×6], C22, C22 [×16], C8 [×6], C2×C4 [×3], C2×C4 [×3], D4 [×12], Q8 [×3], C23 [×7], C42, C42, C22⋊C4 [×9], C2×C8 [×3], D8 [×6], SD16 [×6], C2×D4 [×3], C2×D4 [×6], C2×Q8 [×3], C24, C8⋊C4 [×3], C22≀C2 [×3], C4.4D4 [×3], C4.4D4 [×3], C41D4, C2×D8 [×3], C2×SD16 [×3], C42.C22 [×3], C83D4 [×3], C24⋊C22, C42.C23
Quotients: C1, C2 [×7], C22 [×7], D4 [×6], C23, C2×D4 [×3], C22≀C2, D44D4 [×3], C42.C23

Character table of C42.C23

 class 12A2B2C2D2E2F2G4A4B4C4D4E4F8A8B8C8D8E8F
 size 111188816444888888888
ρ111111111111111111111    trivial
ρ21111-1-11-1111-1-11-11-1111    linear of order 2
ρ31111-11-1-1111-11-11-111-11    linear of order 2
ρ411111-1-111111-1-1-1-1-11-11    linear of order 2
ρ51111-11-11111-11-1-11-1-11-1    linear of order 2
ρ611111-1-1-11111-1-1111-11-1    linear of order 2
ρ71111111-1111111-1-1-1-1-1-1    linear of order 2
ρ81111-1-111111-1-111-11-1-1-1    linear of order 2
ρ92222-2000-22-2200000000    orthogonal lifted from D4
ρ1022220200-2-220-20000000    orthogonal lifted from D4
ρ1122220-200-2-22020000000    orthogonal lifted from D4
ρ12222200-202-2-2002000000    orthogonal lifted from D4
ρ13222200202-2-200-2000000    orthogonal lifted from D4
ρ1422222000-22-2-200000000    orthogonal lifted from D4
ρ154-44-4000000000020-2000    orthogonal lifted from D44D4
ρ164-4-4400000000000200-20    orthogonal lifted from D44D4
ρ174-44-40000000000-202000    orthogonal lifted from D44D4
ρ1844-4-4000000000000020-2    orthogonal lifted from D44D4
ρ194-4-4400000000000-20020    orthogonal lifted from D44D4
ρ2044-4-40000000000000-202    orthogonal lifted from D44D4

Smallest permutation representation of C42.C23
On 32 points
Generators in S32
(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)
(1 19 13 7)(2 20 14 8)(3 17 15 5)(4 18 16 6)(9 28 21 30)(10 25 22 31)(11 26 23 32)(12 27 24 29)
(1 25)(2 28)(3 27)(4 26)(5 24)(6 23)(7 22)(8 21)(9 20)(10 19)(11 18)(12 17)(13 31)(14 30)(15 29)(16 32)
(1 15)(2 14)(3 13)(4 16)(6 8)(9 10)(11 12)(18 20)(21 22)(23 24)(25 32)(26 31)(27 30)(28 29)
(2 16)(4 14)(5 17)(6 8)(7 19)(9 32)(10 25)(11 30)(12 27)(18 20)(21 26)(22 31)(23 28)(24 29)

G:=sub<Sym(32)| (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), (1,19,13,7)(2,20,14,8)(3,17,15,5)(4,18,16,6)(9,28,21,30)(10,25,22,31)(11,26,23,32)(12,27,24,29), (1,25)(2,28)(3,27)(4,26)(5,24)(6,23)(7,22)(8,21)(9,20)(10,19)(11,18)(12,17)(13,31)(14,30)(15,29)(16,32), (1,15)(2,14)(3,13)(4,16)(6,8)(9,10)(11,12)(18,20)(21,22)(23,24)(25,32)(26,31)(27,30)(28,29), (2,16)(4,14)(5,17)(6,8)(7,19)(9,32)(10,25)(11,30)(12,27)(18,20)(21,26)(22,31)(23,28)(24,29)>;

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), (1,19,13,7)(2,20,14,8)(3,17,15,5)(4,18,16,6)(9,28,21,30)(10,25,22,31)(11,26,23,32)(12,27,24,29), (1,25)(2,28)(3,27)(4,26)(5,24)(6,23)(7,22)(8,21)(9,20)(10,19)(11,18)(12,17)(13,31)(14,30)(15,29)(16,32), (1,15)(2,14)(3,13)(4,16)(6,8)(9,10)(11,12)(18,20)(21,22)(23,24)(25,32)(26,31)(27,30)(28,29), (2,16)(4,14)(5,17)(6,8)(7,19)(9,32)(10,25)(11,30)(12,27)(18,20)(21,26)(22,31)(23,28)(24,29) );

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)], [(1,19,13,7),(2,20,14,8),(3,17,15,5),(4,18,16,6),(9,28,21,30),(10,25,22,31),(11,26,23,32),(12,27,24,29)], [(1,25),(2,28),(3,27),(4,26),(5,24),(6,23),(7,22),(8,21),(9,20),(10,19),(11,18),(12,17),(13,31),(14,30),(15,29),(16,32)], [(1,15),(2,14),(3,13),(4,16),(6,8),(9,10),(11,12),(18,20),(21,22),(23,24),(25,32),(26,31),(27,30),(28,29)], [(2,16),(4,14),(5,17),(6,8),(7,19),(9,32),(10,25),(11,30),(12,27),(18,20),(21,26),(22,31),(23,28),(24,29)])

Matrix representation of C42.C23 in GL8(𝔽17)

00100000
00010000
160000000
016000000
00000010
0000116115
00001000
00000001
,
01000000
10000000
00010000
00100000
00000100
000016000
0000116115
000010116
,
99890000
99980000
89880000
98880000
0000161161
000010016
0000016016
0000160161
,
160000000
01000000
00100000
000160000
000016000
00000100
000000160
0000160161
,
10000000
01000000
001600000
000160000
00001000
000001600
000000160
000001161

G:=sub<GL(8,GF(17))| [0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,16,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,15,0,1],[0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,16,1,1,0,0,0,0,1,0,16,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,15,16],[9,9,8,9,0,0,0,0,9,9,9,8,0,0,0,0,8,9,8,8,0,0,0,0,9,8,8,8,0,0,0,0,0,0,0,0,16,1,0,16,0,0,0,0,1,0,16,0,0,0,0,0,16,0,0,16,0,0,0,0,1,16,16,1],[16,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,16,0,0,0,0,0,1,0,0,0,0,0,0,0,0,16,16,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,1,0,0,0,0,0,0,16,16,0,0,0,0,0,0,0,1] >;

C42.C23 in GAP, Magma, Sage, TeX

C_4^2.C_2^3
% in TeX

G:=Group("C4^2.C2^3");
// GroupNames label

G:=SmallGroup(128,387);
// by ID

G=gap.SmallGroup(128,387);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,2,141,422,1123,570,521,136,3924,1411,998,242]);
// Polycyclic

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

Export

Character table of C42.C23 in TeX

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