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G = C23.3C42order 128 = 27

3rd non-split extension by C23 of C42 acting via C42/C4=C4

p-group, metabelian, nilpotent (class 4), monomial

Aliases: C23.3C42, (C2×D4).2Q8, (C22×C8).9C4, (C2×Q8).39D4, C23.1(C4⋊C4), C4.D4.3C4, (C22×C4).37D4, C4.26(C23⋊C4), (C2×M4(2)).9C4, C2.14(C23.9D4), C22.3(C2.C42), M4(2).8C22.3C2, (C2×C4).8(C4⋊C4), (C2×D4).45(C2×C4), (C2×C4).2(C22⋊C4), (C22×C4).65(C2×C4), (C2×C4○D4).2C22, (C22×C8)⋊C2.11C2, SmallGroup(128,124)

Series: Derived Chief Lower central Upper central Jennings

C1C23 — C23.3C42
C1C2C22C2×C4C22×C4C2×C4○D4(C22×C8)⋊C2 — C23.3C42
C1C2C22C23 — C23.3C42
C1C4C2×C4C2×C4○D4 — C23.3C42
C1C2C22C2×C4○D4 — C23.3C42

Generators and relations for C23.3C42
 G = < a,b,c,d,e | a2=b2=c2=1, d4=e4=c, ab=ba, ac=ca, dad-1=abc, ae=ea, dbd-1=bc=cb, be=eb, cd=dc, ce=ec, ede-1=abcd >

Subgroups: 176 in 79 conjugacy classes, 30 normal (16 characteristic)
C1, C2, C2 [×4], C4 [×2], C4 [×3], C22, C22 [×6], C8 [×6], C2×C4 [×2], C2×C4 [×2], C2×C4 [×6], D4 [×3], Q8, C23, C23 [×2], C2×C8 [×5], M4(2) [×7], C22×C4, C22×C4 [×2], C2×D4, C2×D4 [×2], C2×Q8, C4○D4 [×2], C22⋊C8 [×2], C4.D4 [×4], C4.10D4 [×2], C22×C8, C2×M4(2), C2×M4(2) [×2], C2×C4○D4, (C22×C8)⋊C2, M4(2).8C22 [×2], C23.3C42
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], C23.9D4, C23.3C42

Character table of C23.3C42

 class 12A2B2C2D2E4A4B4C4D4E4F8A8B8C8D8E8F8G8H8I8J8K8L8M8N
 size 11244411244444448888888888
ρ111111111111111111111111111    trivial
ρ2111111111111-1-1-1-1-1111-11-1-1-1-1    linear of order 2
ρ31111111111111111-1-1-1-1-1-111-1-1    linear of order 2
ρ4111111111111-1-1-1-11-1-1-11-1-1-111    linear of order 2
ρ5111-111-1-1-1-1-11-ii-iii11-1i-1-ii-i-i    linear of order 4
ρ6111-111-1-1-1-1-11i-ii-i-i11-1-i-1i-iii    linear of order 4
ρ711111-1-1-1-11-1-1-ii-ii1i-ii-1-ii-i1-1    linear of order 4
ρ8111-111-1-1-1-1-11-ii-ii-i-1-11-i1-iiii    linear of order 4
ρ9111-11-1111-11-1-1-1-1-1ii-i-i-ii11-ii    linear of order 4
ρ10111-111-1-1-1-1-11i-ii-ii-1-11i1i-i-i-i    linear of order 4
ρ11111-11-1111-11-11111-ii-i-iii-1-1i-i    linear of order 4
ρ1211111-1-1-1-11-1-1i-ii-i-1i-ii1-i-ii-11    linear of order 4
ρ13111-11-1111-11-1-1-1-1-1-i-iiii-i11i-i    linear of order 4
ρ1411111-1-1-1-11-1-1-ii-ii-1-ii-i1ii-i-11    linear of order 4
ρ1511111-1-1-1-11-1-1i-ii-i1-ii-i-1i-ii1-1    linear of order 4
ρ16111-11-1111-11-11111i-iii-i-i-1-1-ii    linear of order 4
ρ172222-2-2222-2-2200000000000000    orthogonal lifted from D4
ρ18222-2-222222-2-200000000000000    orthogonal lifted from D4
ρ19222-2-2-2-2-2-222200000000000000    orthogonal lifted from D4
ρ202222-22-2-2-2-22-200000000000000    symplectic lifted from Q8, Schur index 2
ρ2144-400044-400000000000000000    orthogonal lifted from C23⋊C4
ρ2244-4000-4-4400000000000000000    orthogonal lifted from C23⋊C4
ρ234-40000-4i4i000085878830000000000    complex faithful
ρ244-40000-4i4i000088385870000000000    complex faithful
ρ254-400004i-4i000087858380000000000    complex faithful
ρ264-400004i-4i000083887850000000000    complex faithful

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

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

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

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

Matrix representation of C23.3C42 in GL4(𝔽17) generated by

11500
01600
00115
00016
,
1000
0100
00160
00016
,
16000
01600
00160
00016
,
0010
0001
13000
13400
,
2000
0200
00213
00015
G:=sub<GL(4,GF(17))| [1,0,0,0,15,16,0,0,0,0,1,0,0,0,15,16],[1,0,0,0,0,1,0,0,0,0,16,0,0,0,0,16],[16,0,0,0,0,16,0,0,0,0,16,0,0,0,0,16],[0,0,13,13,0,0,0,4,1,0,0,0,0,1,0,0],[2,0,0,0,0,2,0,0,0,0,2,0,0,0,13,15] >;

C23.3C42 in GAP, Magma, Sage, TeX

C_2^3._3C_4^2
% in TeX

G:=Group("C2^3.3C4^2");
// GroupNames label

G:=SmallGroup(128,124);
// by ID

G=gap.SmallGroup(128,124);
# by ID

G:=PCGroup([7,-2,2,-2,2,2,-2,-2,56,85,120,758,570,521,248,172,4037]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^2=c^2=1,d^4=e^4=c,a*b=b*a,a*c=c*a,d*a*d^-1=a*b*c,a*e=e*a,d*b*d^-1=b*c=c*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 C23.3C42 in TeX

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