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G = (C22×Q8)⋊C4order 128 = 27

6th semidirect product of C22×Q8 and C4 acting faithfully

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

Aliases: C4○D4.4D4, (C2×Q8).68D4, (C22×Q8)⋊6C4, C4.59C22≀C2, (C2×D4).267D4, C23.124(C2×D4), D4.14(C22⋊C4), C22.13C22≀C2, Q8.14(C22⋊C4), (C22×C4).26C23, C2.21(C243C4), C42⋊C2210C2, (C22×Q8).8C22, C23.38D420C2, C23.32(C22⋊C4), C23.C233C2, C42⋊C2.1C22, (C2×2- 1+4).2C2, M4(2).8C229C2, (C2×M4(2)).150C22, (C2×C4○D4)⋊6C4, C4.9(C2×C22⋊C4), (C2×C4).228(C2×D4), (C2×D4).204(C2×C4), (C22×C4).16(C2×C4), (C2×Q8).187(C2×C4), (C2×C4○D4).9C22, (C2×C4).45(C22⋊C4), (C2×C4).177(C22×C4), C22.34(C2×C22⋊C4), SmallGroup(128,528)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — (C22×Q8)⋊C4
C1C2C22C23C22×C4C2×C4○D4C2×2- 1+4 — (C22×Q8)⋊C4
C1C2C2×C4 — (C22×Q8)⋊C4
C1C2C22×C4 — (C22×Q8)⋊C4
C1C2C2C22×C4 — (C22×Q8)⋊C4

Generators and relations for (C22×Q8)⋊C4
 G = < a,b,c,d,e | a2=b2=c4=e4=1, d2=c2, ab=ba, ac=ca, ad=da, eae-1=bc2, ede-1=bc=cb, bd=db, ebe-1=a, dcd-1=c-1, ece-1=ac2d >

Subgroups: 508 in 268 conjugacy classes, 66 normal (20 characteristic)
C1, C2, C2 [×7], C4 [×2], C4 [×2], C4 [×10], C22, C22 [×2], C22 [×9], C8 [×2], C2×C4 [×2], C2×C4 [×6], C2×C4 [×33], D4 [×2], D4 [×19], Q8 [×6], Q8 [×17], C23, C23 [×2], C23, C42 [×2], C22⋊C4 [×2], C4⋊C4 [×2], C2×C8 [×2], M4(2) [×4], C22×C4, C22×C4 [×2], C22×C4 [×6], C2×D4 [×2], C2×D4 [×2], C2×D4 [×3], C2×Q8 [×2], C2×Q8 [×6], C2×Q8 [×21], C4○D4 [×4], C4○D4 [×38], C23⋊C4 [×2], C4.D4, C4.10D4, Q8⋊C4 [×4], C4≀C2 [×4], C42⋊C2 [×2], C2×M4(2) [×2], C22×Q8, C22×Q8 [×2], C22×Q8, C2×C4○D4 [×2], C2×C4○D4 [×2], C2×C4○D4 [×3], 2- 1+4 [×8], C23.C23, M4(2).8C22, C23.38D4 [×2], C42⋊C22 [×2], C2×2- 1+4, (C22×Q8)⋊C4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×12], C23, C22⋊C4 [×12], C22×C4, C2×D4 [×6], C2×C22⋊C4 [×3], C22≀C2 [×4], C243C4, (C22×Q8)⋊C4

Character table of (C22×Q8)⋊C4

 class 12A2B2C2D2E2F2G2H4A4B4C4D4E4F4G4H4I4J4K4L4M4N4O4P8A8B8C8D
 size 11222444422224444444488888888
ρ111111111111111111111111111111    trivial
ρ211111-1-1111111-1-1-1-1-11-11-11-111-11-1    linear of order 2
ρ311111-1-1111111-1-1-1-1-11-111-11-1-11-11    linear of order 2
ρ4111111111111111111111-1-1-1-1-1-1-1-1    linear of order 2
ρ511111-1-1-1-11111-11111-1-1-1-11-11-11-11    linear of order 2
ρ61111111-1-111111-1-1-1-1-11-11111-1-1-1-1    linear of order 2
ρ71111111-1-111111-1-1-1-1-11-1-1-1-1-11111    linear of order 2
ρ811111-1-1-1-11111-11111-1-1-11-11-11-11-1    linear of order 2
ρ911-1-11-1111-11-111-11-11-1-1-1-i-iii-i-iii    linear of order 4
ρ1011-1-111-111-11-11-11-11-1-11-1i-i-ii-iii-i    linear of order 4
ρ1111-1-11-1111-11-111-11-11-1-1-1ii-i-iii-i-i    linear of order 4
ρ1211-1-111-111-11-11-11-11-1-11-1-iii-ii-i-ii    linear of order 4
ρ1311-1-111-1-1-1-11-11-1-11-11111i-i-iii-i-ii    linear of order 4
ρ1411-1-11-11-1-1-11-1111-11-11-11-i-iiiii-i-i    linear of order 4
ρ1511-1-111-1-1-1-11-11-1-11-11111-iii-i-iii-i    linear of order 4
ρ1611-1-11-11-1-1-11-1111-11-11-11ii-i-i-i-iii    linear of order 4
ρ1722-22-200-222-2-220000020-200000000    orthogonal lifted from D4
ρ1822-22-20000-222-202-2-2200000000000    orthogonal lifted from D4
ρ1922-22-20000-222-20-222-200000000000    orthogonal lifted from D4
ρ2022222-2200-2-2-2-2-2000002000000000    orthogonal lifted from D4
ρ2122-2-22-2-2002-22-22000002000000000    orthogonal lifted from D4
ρ2222-22-2002-22-2-2200000-20200000000    orthogonal lifted from D4
ρ23222-2-2000022-2-20-2-22200000000000    orthogonal lifted from D4
ρ24222-2-2002-2-2-2220000020-200000000    orthogonal lifted from D4
ρ25222222-200-2-2-2-2200000-2000000000    orthogonal lifted from D4
ρ2622-2-2222002-22-2-200000-2000000000    orthogonal lifted from D4
ρ27222-2-2000022-2-2022-2-200000000000    orthogonal lifted from D4
ρ28222-2-200-22-2-22200000-20200000000    orthogonal lifted from D4
ρ298-8000000000000000000000000000    symplectic faithful, Schur index 2

Smallest permutation representation of (C22×Q8)⋊C4
On 32 points
Generators in S32
(1 19)(2 20)(3 17)(4 18)(5 16)(6 13)(7 14)(8 15)(9 32)(10 29)(11 30)(12 31)(21 27)(22 28)(23 25)(24 26)
(1 31)(2 32)(3 29)(4 30)(5 26)(6 27)(7 28)(8 25)(9 20)(10 17)(11 18)(12 19)(13 21)(14 22)(15 23)(16 24)
(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 7 3 5)(2 6 4 8)(9 21 11 23)(10 24 12 22)(13 18 15 20)(14 17 16 19)(25 32 27 30)(26 31 28 29)
(2 16)(4 14)(5 32 7 30)(6 23 8 21)(9 28)(10 12)(11 26)(13 15)(17 29 19 31)(18 22 20 24)

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

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

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

Matrix representation of (C22×Q8)⋊C4 in GL8(𝔽17)

000000160
0000116162
00001000
000011601
00100000
1611150000
160000000
1610160000
,
0000161115
00000010
000001600
000011601
1161620000
001600000
01000000
1610160000
,
040130000
40040000
001340000
00040000
000001304
0000130013
000000413
000000013
,
000001304
0000130013
000000413
000000013
013040000
1300130000
004130000
000130000
,
10000000
016000000
001600000
0161610000
000001600
00001000
0000161115
000001116

G:=sub<GL(8,GF(17))| [0,0,0,0,0,16,16,16,0,0,0,0,0,1,0,1,0,0,0,0,1,1,0,0,0,0,0,0,0,15,0,16,0,1,1,1,0,0,0,0,0,16,0,16,0,0,0,0,16,16,0,0,0,0,0,0,0,2,0,1,0,0,0,0],[0,0,0,0,1,0,0,16,0,0,0,0,16,0,1,1,0,0,0,0,16,16,0,0,0,0,0,0,2,0,0,16,16,0,0,1,0,0,0,0,1,0,16,16,0,0,0,0,1,1,0,0,0,0,0,0,15,0,0,1,0,0,0,0],[0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,13,0,0,0,0,0,13,4,4,4,0,0,0,0,0,0,0,0,0,13,0,0,0,0,0,0,13,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,4,13,13,13],[0,0,0,0,0,13,0,0,0,0,0,0,13,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,4,13,13,13,0,13,0,0,0,0,0,0,13,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,4,13,13,13,0,0,0,0],[1,0,0,0,0,0,0,0,0,16,0,16,0,0,0,0,0,0,16,16,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,16,0,0,0,0,0,16,0,1,1,0,0,0,0,0,0,1,1,0,0,0,0,0,0,15,16] >;

(C22×Q8)⋊C4 in GAP, Magma, Sage, TeX

(C_2^2\times Q_8)\rtimes C_4
% in TeX

G:=Group("(C2^2xQ8):C4");
// GroupNames label

G:=SmallGroup(128,528);
// by ID

G=gap.SmallGroup(128,528);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,456,422,2019,1018,248,2804,1027]);
// Polycyclic

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

Export

Character table of (C22×Q8)⋊C4 in TeX

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