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G = C42.5C23order 128 = 27

5th non-split extension by C42 of C23 acting faithfully

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

Aliases: C42.5C23, C4⋊Q83C22, (C2×D4).30D4, C8.2D41C2, (C2×Q8).30D4, C8⋊C428C22, C24⋊C22.C2, C2.27(D44D4), C4.4D4.9C22, C2.20(D4.9D4), C22.186C22≀C2, C42.C223C2, C42.28C2226C2, (C2×C4).218(C2×D4), SmallGroup(128,391)

Series: Derived Chief Lower central Upper central Jennings

C1C42 — C42.5C23
C1C2C22C2×C4C42C4.4D4C24⋊C22 — C42.5C23
C1C22C42 — C42.5C23
C1C22C42 — C42.5C23
C1C22C22C42 — C42.5C23

Generators and relations for C42.5C23
 G = < a,b,c,d,e | a4=b4=d2=e2=1, c2=a2, ab=ba, cac-1=dad=a-1, eae=a-1b2, cbc-1=ebe=b-1, dbd=a2b-1, dcd=ac, ece=bc, de=ed >

Subgroups: 336 in 114 conjugacy classes, 30 normal (14 characteristic)
C1, C2, C2 [×2], C2 [×3], C4 [×7], C22, C22 [×13], C8 [×4], C2×C4, C2×C4 [×2], C2×C4 [×4], D4 [×6], Q8 [×5], C23 [×6], C42, C42, C22⋊C4 [×9], C4⋊C4 [×2], C2×C8 [×3], SD16 [×2], Q16 [×2], C2×D4, C2×D4 [×2], C2×D4 [×3], C2×Q8, C2×Q8 [×2], C2×Q8, C24, C8⋊C4, C8⋊C4 [×2], D4⋊C4 [×2], Q8⋊C4 [×2], C22≀C2 [×3], C4.4D4, C4.4D4 [×2], C4.4D4 [×3], C4⋊Q8, C2×SD16, C2×Q16, C42.C22, C42.C22 [×2], C42.28C22 [×2], C8.2D4, C24⋊C22, C42.5C23
Quotients: C1, C2 [×7], C22 [×7], D4 [×6], C23, C2×D4 [×3], C22≀C2, D44D4, D4.9D4 [×2], C42.5C23

Character table of C42.5C23

 class 12A2B2C2D2E2F4A4B4C4D4E4F4G8A8B8C8D8E8F
 size 111188844488816888888
ρ111111111111111111111    trivial
ρ21111111111111-1-1-1-1-1-1-1    linear of order 2
ρ31111-11-1111-11-111-1-1-11-1    linear of order 2
ρ41111-11-1111-11-1-1-1111-11    linear of order 2
ρ51111-1-11111-1-11-11-1111-1    linear of order 2
ρ61111-1-11111-1-111-11-1-1-11    linear of order 2
ρ711111-1-11111-1-1-111-1-111    linear of order 2
ρ811111-1-11111-1-11-1-111-1-1    linear of order 2
ρ92222200-22-2-2000000000    orthogonal lifted from D4
ρ1022220022-2-200-20000000    orthogonal lifted from D4
ρ1122220-20-2-220200000000    orthogonal lifted from D4
ρ12222200-22-2-20020000000    orthogonal lifted from D4
ρ132222-200-22-22000000000    orthogonal lifted from D4
ρ142222020-2-220-200000000    orthogonal lifted from D4
ρ154-4-4400000000002000-20    orthogonal lifted from D44D4
ρ164-4-440000000000-200020    orthogonal lifted from D44D4
ρ174-44-4000000000002i000-2i    complex lifted from D4.9D4
ρ184-44-400000000000-2i0002i    complex lifted from D4.9D4
ρ1944-4-40000000000002i-2i00    complex lifted from D4.9D4
ρ2044-4-4000000000000-2i2i00    complex lifted from D4.9D4

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

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

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

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

Matrix representation of C42.5C23 in GL8(𝔽17)

00100000
00010000
10000000
01000000
00000010
00000001
000016000
000001600
,
01000000
160000000
00010000
001600000
00000100
000016000
00000001
000000160
,
89880000
99890000
88890000
89990000
0000152215
0000221515
0000215215
000015151515
,
160000000
01000000
001600000
00010000
00001000
00000100
000000160
000000016
,
10000000
016000000
001600000
00010000
00001000
000001600
00000010
000000016

G:=sub<GL(8,GF(17))| [0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0],[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],[8,9,8,8,0,0,0,0,9,9,8,9,0,0,0,0,8,8,8,9,0,0,0,0,8,9,9,9,0,0,0,0,0,0,0,0,15,2,2,15,0,0,0,0,2,2,15,15,0,0,0,0,2,15,2,15,0,0,0,0,15,15,15,15],[16,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16],[1,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,16] >;

C42.5C23 in GAP, Magma, Sage, TeX

C_4^2._5C_2^3
% in TeX

G:=Group("C4^2.5C2^3");
// GroupNames label

G:=SmallGroup(128,391);
// by ID

G=gap.SmallGroup(128,391);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,2,224,141,422,1123,570,521,136,3924,1411,998,242]);
// Polycyclic

G:=Group<a,b,c,d,e|a^4=b^4=d^2=e^2=1,c^2=a^2,a*b=b*a,c*a*c^-1=d*a*d=a^-1,e*a*e=a^-1*b^2,c*b*c^-1=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 C42.5C23 in TeX

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