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G = C23.C42order 128 = 27

2nd non-split extension by C23 of C42 acting via C42/C22=C22

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

Aliases: C23.7C42, C22⋊C82C4, (C2×M4(2))⋊2C4, C24.38(C2×C4), (C22×C4).26Q8, C4.24(C23⋊C4), C23.17(C4⋊C4), (C22×C4).183D4, C24.4C4.8C2, C23.7Q8.3C2, (C23×C4).195C22, C22.5(C4.D4), C2.9(C22.C42), C23.145(C22⋊C4), C2.10(C23.9D4), C22.5(C4.10D4), C2.12(M4(2)⋊4C4), C22.55(C2.C42), (C2×C4).23(C4⋊C4), (C2×C22⋊C4).1C4, (C22×C4).159(C2×C4), (C2×C4).304(C22⋊C4), SmallGroup(128,37)

Series: Derived Chief Lower central Upper central Jennings

C1C23 — C23.C42
C1C2C22C23C22×C4C23×C4C23.7Q8 — C23.C42
C1C22C23 — C23.C42
C1C22C23×C4 — C23.C42
C1C2C22C23×C4 — C23.C42

Generators and relations for C23.C42
 G = < a,b,c,d,e | a2=b2=c2=e4=1, d4=b, dad-1=ab=ba, eae-1=ac=ca, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede-1=abcd >

Subgroups: 240 in 100 conjugacy classes, 34 normal (26 characteristic)
C1, C2 [×3], C2 [×4], C4 [×2], C4 [×5], C22 [×3], C22 [×12], C8 [×4], C2×C4 [×4], C2×C4 [×15], C23 [×3], C23 [×4], C22⋊C4 [×2], C4⋊C4 [×2], C2×C8 [×4], M4(2) [×4], C22×C4 [×6], C22×C4 [×4], C24, C2.C42, C22⋊C8 [×2], C22⋊C8 [×3], C2×C22⋊C4 [×2], C2×C4⋊C4, C2×M4(2) [×2], C2×M4(2), C23×C4, C23.7Q8, C24.4C4 [×2], C23.C42
Quotients: C1, C2 [×3], C4 [×6], C22, C2×C4 [×3], D4 [×3], Q8, C42, C22⋊C4 [×3], C4⋊C4 [×3], C2.C42, C23⋊C4 [×2], C4.D4, C4.10D4, C23.9D4, C22.C42, M4(2)⋊4C4, C23.C42

Character table of C23.C42

 class 12A2B2C2D2E2F2G4A4B4C4D4E4F4G4H4I4J8A8B8C8D8E8F8G8H
 size 11112244222244888888888888
ρ111111111111111111111111111    trivial
ρ211111111111111-1-1-1-111-11-1-1-11    linear of order 2
ρ311111111111111-1-1-1-1-1-11-1111-1    linear of order 2
ρ4111111111111111111-1-1-1-1-1-1-1-1    linear of order 2
ρ51111-1-1-1111-1-1-11-iii-i-11i1i-i-i-1    linear of order 4
ρ61111-1-1-11-1-1111-1ii-i-ii-i-1i1-11-i    linear of order 4
ρ71111-1-1-1111-1-1-11-iii-i1-1-i-1-iii1    linear of order 4
ρ81111-1-1-11-1-1111-1ii-i-i-ii1-i-11-1i    linear of order 4
ρ911111111-1-1-1-1-1-11-11-1iii-i-i-ii-i    linear of order 4
ρ1011111111-1-1-1-1-1-1-11-11ii-i-iii-i-i    linear of order 4
ρ111111-1-1-1111-1-1-11i-i-ii-11-i1-iii-1    linear of order 4
ρ121111-1-1-11-1-1111-1-i-iiii-i1i-11-1-i    linear of order 4
ρ1311111111-1-1-1-1-1-11-11-1-i-i-iiii-ii    linear of order 4
ρ1411111111-1-1-1-1-1-1-11-11-i-iii-i-iii    linear of order 4
ρ151111-1-1-11-1-1111-1-i-iii-ii-1-i1-11i    linear of order 4
ρ161111-1-1-1111-1-1-11i-i-ii1-1i-1i-i-i1    linear of order 4
ρ172222-2-22-22222-2-2000000000000    orthogonal lifted from D4
ρ18222222-2-222-2-22-2000000000000    orthogonal lifted from D4
ρ19222222-2-2-2-222-22000000000000    orthogonal lifted from D4
ρ202222-2-22-2-2-2-2-222000000000000    symplectic lifted from Q8, Schur index 2
ρ214-44-40000004-400000000000000    orthogonal lifted from C23⋊C4
ρ224-44-4000000-4400000000000000    orthogonal lifted from C23⋊C4
ρ2344-4-44-400000000000000000000    orthogonal lifted from C4.D4
ρ2444-4-4-4400000000000000000000    symplectic lifted from C4.10D4, Schur index 2
ρ254-4-440000-4i4i0000000000000000    complex lifted from M4(2)⋊4C4
ρ264-4-4400004i-4i0000000000000000    complex lifted from M4(2)⋊4C4

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

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

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

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

Matrix representation of C23.C42 in GL8(𝔽17)

013000000
40000000
441390000
013440000
000013200
00001400
00001611132
000016114
,
160000000
016000000
001600000
000160000
00001000
00000100
00000010
00000001
,
160000000
016000000
001600000
000160000
000016000
000001600
000000160
000000016
,
00100000
1116150000
01000000
100160000
0000120150
000011015
000012050
00001501616
,
10000000
016000000
441390000
76040000
00001000
000041600
000001415
00001310013

G:=sub<GL(8,GF(17))| [0,4,4,0,0,0,0,0,13,0,4,13,0,0,0,0,0,0,13,4,0,0,0,0,0,0,9,4,0,0,0,0,0,0,0,0,13,1,16,16,0,0,0,0,2,4,11,1,0,0,0,0,0,0,13,1,0,0,0,0,0,0,2,4],[16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16],[0,1,0,1,0,0,0,0,0,1,1,0,0,0,0,0,1,16,0,0,0,0,0,0,0,15,0,16,0,0,0,0,0,0,0,0,12,1,12,15,0,0,0,0,0,1,0,0,0,0,0,0,15,0,5,16,0,0,0,0,0,15,0,16],[1,0,4,7,0,0,0,0,0,16,4,6,0,0,0,0,0,0,13,0,0,0,0,0,0,0,9,4,0,0,0,0,0,0,0,0,1,4,0,13,0,0,0,0,0,16,1,10,0,0,0,0,0,0,4,0,0,0,0,0,0,0,15,13] >;

C23.C42 in GAP, Magma, Sage, TeX

C_2^3.C_4^2
% in TeX

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

G:=SmallGroup(128,37);
// by ID

G=gap.SmallGroup(128,37);
# by ID

G:=PCGroup([7,-2,2,-2,2,2,-2,2,56,85,120,758,723,570,136,2804]);
// Polycyclic

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

Export

Character table of C23.C42 in TeX

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