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G = C23.2C42order 128 = 27

2nd non-split extension by C23 of C42 acting via C42/C4=C4

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

Aliases: C23.2C42, C23⋊C42C4, (C22×C8)⋊3C4, (C2×Q8).1Q8, (C2×D4).40D4, (C2×M4(2))⋊3C4, (C22×C4).36D4, C4.25(C23⋊C4), C23.2(C22⋊C4), C2.13(C23.9D4), C23.C23.1C2, C22.2(C2.C42), (C2×C4).2(C4⋊C4), (C2×D4).44(C2×C4), (C22×C4).64(C2×C4), (C2×C4○D4).1C22, (C2×C4).349(C22⋊C4), (C22×C8)⋊C2.10C2, SmallGroup(128,123)

Series: Derived Chief Lower central Upper central Jennings

C1C23 — C23.2C42
C1C2C22C2×C4C22×C4C2×C4○D4(C22×C8)⋊C2 — C23.2C42
C1C2C22C23 — C23.2C42
C1C4C2×C4C2×C4○D4 — C23.2C42
C1C2C22C2×C4○D4 — C23.2C42

Generators and relations for C23.2C42
 G = < a,b,c,d,e | a2=b2=c2=d4=1, 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: 208 in 83 conjugacy classes, 30 normal (16 characteristic)
C1, C2, C2 [×4], C4 [×2], C4 [×7], C22, C22 [×6], C8 [×2], C2×C4 [×2], C2×C4 [×2], C2×C4 [×10], D4 [×3], Q8, C23, C23 [×2], C42 [×2], C22⋊C4 [×4], C4⋊C4 [×2], C2×C8 [×3], M4(2), C22×C4, C22×C4 [×2], C2×D4, C2×D4 [×2], C2×Q8, C4○D4 [×2], C22⋊C8 [×2], C23⋊C4 [×4], C23⋊C4 [×2], C42⋊C2 [×2], C22×C8, C2×M4(2), C2×C4○D4, (C22×C8)⋊C2, C23.C23 [×2], C23.2C42
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, C23.2C42

Character table of C23.2C42

 class 12A2B2C2D2E4A4B4C4D4E4F4G4H4I4J4K4L4M4N8A8B8C8D8E8F
 size 11244411244488888888444488
ρ111111111111111111111111111    trivial
ρ21111111111111-1-1-1-1111-1-1-1-1-1-1    linear of order 2
ρ3111111111111-11111-1-1-1-1-1-1-1-1-1    linear of order 2
ρ4111111111111-1-1-1-1-1-1-1-1111111    linear of order 2
ρ5111-11-1111-1-11i-iii-i-i-ii1111-1-1    linear of order 4
ρ611111-1-1-1-1-11-1-i11-1-1i-iii-ii-ii-i    linear of order 4
ρ7111-111-1-1-11-1-1-1-ii-ii-111-ii-iii-i    linear of order 4
ρ811111-1-1-1-1-11-1i11-1-1-ii-i-ii-ii-ii    linear of order 4
ρ9111-11-1111-1-11-i-iii-iii-i-1-1-1-111    linear of order 4
ρ1011111-1-1-1-1-11-1-i-1-111i-ii-ii-ii-ii    linear of order 4
ρ11111-11-1111-1-11ii-i-ii-i-ii-1-1-1-111    linear of order 4
ρ12111-11-1111-1-11-ii-i-iiii-i1111-1-1    linear of order 4
ρ1311111-1-1-1-1-11-1i-1-111-ii-ii-ii-ii-i    linear of order 4
ρ14111-111-1-1-11-1-1-1i-ii-i-111i-ii-i-ii    linear of order 4
ρ15111-111-1-1-11-1-11-ii-ii1-1-1i-ii-i-ii    linear of order 4
ρ16111-111-1-1-11-1-11i-ii-i1-1-1-ii-iii-i    linear of order 4
ρ17222-2-22222-22-200000000000000    orthogonal lifted from D4
ρ182222-2-22222-2-200000000000000    orthogonal lifted from D4
ρ192222-22-2-2-2-2-2200000000000000    orthogonal lifted from D4
ρ20222-2-2-2-2-2-222200000000000000    symplectic lifted from Q8, Schur index 2
ρ2144-400044-400000000000000000    orthogonal lifted from C23⋊C4
ρ2244-4000-4-4400000000000000000    orthogonal lifted from C23⋊C4
ρ234-400004i-4i000000000000858788300    complex faithful
ρ244-400004i-4i000000000000883858700    complex faithful
ρ254-40000-4i4i000000000000878583800    complex faithful
ρ264-40000-4i4i000000000000838878500    complex faithful

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

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

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

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

Matrix representation of C23.2C42 in GL4(𝔽17) generated by

0010
0001
1000
0100
,
0100
1000
0001
0010
,
16000
01600
00160
00016
,
4000
01300
00013
0040
,
1313134
1313413
1341313
4131313
G:=sub<GL(4,GF(17))| [0,0,1,0,0,0,0,1,1,0,0,0,0,1,0,0],[0,1,0,0,1,0,0,0,0,0,0,1,0,0,1,0],[16,0,0,0,0,16,0,0,0,0,16,0,0,0,0,16],[4,0,0,0,0,13,0,0,0,0,0,4,0,0,13,0],[13,13,13,4,13,13,4,13,13,4,13,13,4,13,13,13] >;

C23.2C42 in GAP, Magma, Sage, TeX

C_2^3._2C_4^2
% in TeX

G:=Group("C2^3.2C4^2");
// GroupNames label

G:=SmallGroup(128,123);
// by ID

G=gap.SmallGroup(128,123);
# by ID

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

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

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