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G = (C22×C8)⋊C4order 128 = 27

4th semidirect product of C22×C8 and C4 acting faithfully

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

Aliases: (C22×C8)⋊4C4, (C2×D4).4Q8, (C2×D4).42D4, C23⋊C4.1C4, (C2×C4).3C42, C4.10D46C4, C23.3(C4⋊C4), (C2×M4(2))⋊4C4, (C22×C4).39D4, C4.43(C23⋊C4), C23.4(C22⋊C4), C2.17(C23.9D4), C23.C23.2C2, C22.6(C2.C42), M4(2).8C22.4C2, (C2×C4).4(C4⋊C4), (C2×D4).47(C2×C4), (C2×Q8).42(C2×C4), (C2×C4).4(C22⋊C4), (C22×C4).68(C2×C4), (C2×C4○D4).3C22, (C22×C8)⋊C2.12C2, SmallGroup(128,127)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — (C22×C8)⋊C4
C1C2C22C2×C4C22×C4C2×C4○D4(C22×C8)⋊C2 — (C22×C8)⋊C4
C1C2C22C2×C4 — (C22×C8)⋊C4
C1C4C2×C4C2×C4○D4 — (C22×C8)⋊C4
C1C2C22C2×C4○D4 — (C22×C8)⋊C4

Generators and relations for (C22×C8)⋊C4
 G = < a,b,c,d | a2=b2=c8=d4=1, dad-1=ab=ba, ac=ca, bc=cb, dbd-1=bc4, dcd-1=abc3 >

Subgroups: 192 in 81 conjugacy classes, 30 normal (26 characteristic)
C1, C2, C2 [×4], C4 [×2], C4 [×5], C22, C22 [×6], C8 [×4], C2×C4 [×4], C2×C4 [×8], D4 [×3], Q8, C23 [×3], C42, C22⋊C4 [×2], C4⋊C4, C2×C8 [×4], M4(2) [×4], C22×C4 [×3], C2×D4 [×3], C2×Q8, C4○D4 [×2], C22⋊C8 [×2], C23⋊C4 [×2], C23⋊C4, C4.D4, C4.10D4 [×2], C42⋊C2, C22×C8, C2×M4(2), C2×M4(2), C2×C4○D4, (C22×C8)⋊C2, C23.C23, M4(2).8C22, (C22×C8)⋊C4
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, (C22×C8)⋊C4

Character table of (C22×C8)⋊C4

 class 12A2B2C2D2E4A4B4C4D4E4F4G4H4I4J8A8B8C8D8E8F8G8H8I8J
 size 11244411244488884444888888
ρ111111111111111111111111111    trivial
ρ21111111111111111-1-1-1-1-1-1-1-1-1-1    linear of order 2
ρ3111111111111-1-1-1-1-1-1-1-1-11111-1    linear of order 2
ρ4111111111111-1-1-1-111111-1-1-1-11    linear of order 2
ρ51111-11-1-1-1-11-11-11-1-iii-i-i-iii-ii    linear of order 4
ρ61111-11-1-1-1-11-1-11-11i-i-iii-iii-i-i    linear of order 4
ρ7111-11-1111-11-1ii-i-i-1-1-1-11-i-iii1    linear of order 4
ρ8111-11-1111-11-1ii-i-i1111-1ii-i-i-1    linear of order 4
ρ9111-1-1-1-1-1-1111i-i-iii-i-ii-i-11-11i    linear of order 4
ρ10111-1-1-1-1-1-1111-iii-i-iii-ii-11-11-i    linear of order 4
ρ11111-1-1-1-1-1-1111-iii-ii-i-ii-i1-11-1i    linear of order 4
ρ12111-1-1-1-1-1-1111i-i-ii-iii-ii1-11-1-i    linear of order 4
ρ13111-11-1111-11-1-i-iii1111-1-i-iii-1    linear of order 4
ρ14111-11-1111-11-1-i-iii-1-1-1-11ii-i-i1    linear of order 4
ρ151111-11-1-1-1-11-1-11-11-iii-i-ii-i-iii    linear of order 4
ρ161111-11-1-1-1-11-11-11-1i-i-iiii-i-ii-i    linear of order 4
ρ17222-222-2-2-22-2-200000000000000    orthogonal lifted from D4
ρ182222-2-22222-2-200000000000000    orthogonal lifted from D4
ρ19222-2-22222-2-2200000000000000    orthogonal lifted from D4
ρ2022222-2-2-2-2-2-2200000000000000    symplectic lifted from Q8, Schur index 2
ρ2144-400044-400000000000000000    orthogonal lifted from C23⋊C4
ρ2244-4000-4-4400000000000000000    orthogonal lifted from C23⋊C4
ρ234-40000-4i4i000000008838785000000    complex faithful
ρ244-400004i-4i000000008388587000000    complex faithful
ρ254-40000-4i4i000000008587838000000    complex faithful
ρ264-400004i-4i000000008785883000000    complex faithful

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

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

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

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

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

00138
0004
4900
01300
,
11500
01600
00115
00016
,
0898
0800
8908
0008
,
1000
11600
00162
00161
G:=sub<GL(4,GF(17))| [0,0,4,0,0,0,9,13,13,0,0,0,8,4,0,0],[1,0,0,0,15,16,0,0,0,0,1,0,0,0,15,16],[0,0,8,0,8,8,9,0,9,0,0,0,8,0,8,8],[1,1,0,0,0,16,0,0,0,0,16,16,0,0,2,1] >;

(C22×C8)⋊C4 in GAP, Magma, Sage, TeX

(C_2^2\times C_8)\rtimes C_4
% in TeX

G:=Group("(C2^2xC8):C4");
// GroupNames label

G:=SmallGroup(128,127);
// by ID

G=gap.SmallGroup(128,127);
# by ID

G:=PCGroup([7,-2,2,-2,2,2,-2,-2,56,85,120,758,352,1466,521,2804,172,4037]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^2=c^8=d^4=1,d*a*d^-1=a*b=b*a,a*c=c*a,b*c=c*b,d*b*d^-1=b*c^4,d*c*d^-1=a*b*c^3>;
// generators/relations

Export

Character table of (C22×C8)⋊C4 in TeX

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