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G = C22⋊C4.C8order 128 = 27

The non-split extension by C22⋊C4 of C8 acting via C8/C2=C4

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

Aliases: C22⋊C4.C8, (C2×C8).18D4, C23.2(C2×C8), (C22×C8).8C4, C23.C8.3C2, (C2×C4).3M4(2), C4.42(C23⋊C4), C42⋊C2.2C4, C4.14(C4.10D4), C22.21(C22⋊C8), C42.6C22.9C2, (C2×M4(2)).146C22, C2.9(C22.M4(2)), (C22×C4).63(C2×C4), (C2×C4).345(C22⋊C4), SmallGroup(128,60)

Series: Derived Chief Lower central Upper central Jennings

C1C23 — C22⋊C4.C8
C1C2C4C2×C4C2×C8C2×M4(2)C42.6C22 — C22⋊C4.C8
C1C2C22C23 — C22⋊C4.C8
C1C4C2×C4C2×M4(2) — C22⋊C4.C8
C1C2C2C2C2C4C2×C4C2×M4(2) — C22⋊C4.C8

Generators and relations for C22⋊C4.C8
 G = < a,b,c,d | a2=b2=c4=1, d8=b, cac-1=ab=ba, dad-1=abc2, bc=cb, bd=db, dcd-1=abc >

2C2
4C2
2C22
2C4
4C4
4C4
4C22
2C8
2C2×C4
2C2×C4
2C8
2C2×C4
2C2×C4
4C8
2C42
2C2×C8
2C4⋊C4
4C2×C8
4C16
4C16
4M4(2)
2C4⋊C8
2M5(2)
2M5(2)
2C4⋊C8

Character table of C22⋊C4.C8

 class 12A2B2C4A4B4C4D4E4F8A8B8C8D8E8F8G8H16A16B16C16D16E16F16G16H
 size 11241124884444444488888888
ρ111111111111111111111111111    trivial
ρ211111111-1-1-1111-11-1-11-1-11-1-111    linear of order 2
ρ3111111111111111111-1-1-1-1-1-1-1-1    linear of order 2
ρ411111111-1-1-1111-11-1-1-111-111-1-1    linear of order 2
ρ51111111111-1-1-1-1-1-1-1-1ii-i-i-ii-ii    linear of order 4
ρ611111111-1-11-1-1-11-111i-ii-ii-i-ii    linear of order 4
ρ71111111111-1-1-1-1-1-1-1-1-i-iiii-ii-i    linear of order 4
ρ811111111-1-11-1-1-11-111-ii-ii-iii-i    linear of order 4
ρ91111-1-1-1-11-1iii-i-i-ii-iζ8ζ8ζ87ζ83ζ83ζ85ζ87ζ85    linear of order 8
ρ101111-1-1-1-1-11-iii-ii-i-iiζ8ζ85ζ83ζ83ζ87ζ8ζ87ζ85    linear of order 8
ρ111111-1-1-1-11-1iii-i-i-ii-iζ85ζ85ζ83ζ87ζ87ζ8ζ83ζ8    linear of order 8
ρ121111-1-1-1-11-1-i-i-iiii-iiζ83ζ83ζ85ζ8ζ8ζ87ζ85ζ87    linear of order 8
ρ131111-1-1-1-1-11-iii-ii-i-iiζ85ζ8ζ87ζ87ζ83ζ85ζ83ζ8    linear of order 8
ρ141111-1-1-1-1-11i-i-ii-iii-iζ83ζ87ζ8ζ8ζ85ζ83ζ85ζ87    linear of order 8
ρ151111-1-1-1-1-11i-i-ii-iii-iζ87ζ83ζ85ζ85ζ8ζ87ζ8ζ83    linear of order 8
ρ161111-1-1-1-11-1-i-i-iiii-iiζ87ζ87ζ8ζ85ζ85ζ83ζ8ζ83    linear of order 8
ρ17222-2222-20002-220-20000000000    orthogonal lifted from D4
ρ18222-2222-2000-22-2020000000000    orthogonal lifted from D4
ρ19222-2-2-2-220002i-2i-2i02i0000000000    complex lifted from M4(2)
ρ20222-2-2-2-22000-2i2i2i0-2i0000000000    complex lifted from M4(2)
ρ2144-4044-40000000000000000000    orthogonal lifted from C23⋊C4
ρ2244-40-4-440000000000000000000    symplectic lifted from C4.10D4, Schur index 2
ρ234-400-4i4i00008700080838500000000    complex faithful
ρ244-4004i-4i00008500083088700000000    complex faithful
ρ254-400-4i4i00008300085087800000000    complex faithful
ρ264-4004i-4i00008000870858300000000    complex faithful

Smallest permutation representation of C22⋊C4.C8
On 32 points
Generators in S32
(1 17)(2 18)(3 27)(4 28)(5 21)(6 22)(7 31)(8 32)(9 25)(10 26)(11 19)(12 20)(13 29)(14 30)(15 23)(16 24)
(1 9)(2 10)(3 11)(4 12)(5 13)(6 14)(7 15)(8 16)(17 25)(18 26)(19 27)(20 28)(21 29)(22 30)(23 31)(24 32)
(1 5 9 13)(2 22)(3 7 11 15)(4 32)(6 26)(8 20)(10 30)(12 24)(14 18)(16 28)(17 29 25 21)(19 31 27 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)

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

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

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

Matrix representation of C22⋊C4.C8 in GL4(𝔽17) generated by

1200
01600
001615
0001
,
16000
01600
00160
00016
,
13000
4400
00139
0044
,
0010
0001
9000
8800
G:=sub<GL(4,GF(17))| [1,0,0,0,2,16,0,0,0,0,16,0,0,0,15,1],[16,0,0,0,0,16,0,0,0,0,16,0,0,0,0,16],[13,4,0,0,0,4,0,0,0,0,13,4,0,0,9,4],[0,0,9,8,0,0,0,8,1,0,0,0,0,1,0,0] >;

C22⋊C4.C8 in GAP, Magma, Sage, TeX

C_2^2\rtimes C_4.C_8
% in TeX

G:=Group("C2^2:C4.C8");
// GroupNames label

G:=SmallGroup(128,60);
// by ID

G=gap.SmallGroup(128,60);
# by ID

G:=PCGroup([7,-2,2,-2,2,-2,2,-2,56,85,120,422,723,346,521,136,2804,124]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^2=c^4=1,d^8=b,c*a*c^-1=a*b=b*a,d*a*d^-1=a*b*c^2,b*c=c*b,b*d=d*b,d*c*d^-1=a*b*c>;
// generators/relations

Export

Subgroup lattice of C22⋊C4.C8 in TeX
Character table of C22⋊C4.C8 in TeX

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