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G = C22⋊C4.Q8order 128 = 27

1st non-split extension by C22⋊C4 of Q8 acting via Q8/C2=C22

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

Aliases: (C2×C8).10D4, C22⋊C4.1Q8, C23.8(C2×Q8), (C22×C8).93C22, C4.25(C422C2), M4(2).C4.5C2, C4.10C42.2C2, C4.104(C4.4D4), M4(2)⋊4C4.7C2, C4.C42.11C2, (C22×C4).745C23, C22.41(C22⋊Q8), C42⋊C2.75C22, C42.6C22.2C2, C22.13(C42.C2), C4.123(C22.D4), (C2×M4(2)).241C22, C2.9(C23.83C23), (C2×C4).1387(C2×D4), (C2×C4).788(C4○D4), SmallGroup(128,835)

Series: Derived Chief Lower central Upper central Jennings

C1C22×C4 — C22⋊C4.Q8
C1C2C4C2×C4C22×C4C22×C8C4.C42 — C22⋊C4.Q8
C1C2C22×C4 — C22⋊C4.Q8
C1C4C22×C4 — C22⋊C4.Q8
C1C2C2C22×C4 — C22⋊C4.Q8

Generators and relations for C22⋊C4.Q8
 G = < a,b,c,d,e | a2=b2=c4=1, d4=b, e2=bc2d2, cac-1=dad-1=eae-1=ab=ba, bc=cb, bd=db, be=eb, dcd-1=c-1, ece-1=abc-1, ede-1=bd3 >

Subgroups: 128 in 74 conjugacy classes, 36 normal (16 characteristic)
C1, C2, C2 [×3], C4 [×2], C4 [×2], C4 [×2], C22, C22 [×2], C22, C8 [×7], C2×C4 [×2], C2×C4 [×4], C2×C4 [×2], C23, C42, C22⋊C4 [×2], C4⋊C4, C2×C8 [×2], C2×C8 [×7], M4(2) [×10], C22×C4, C4⋊C8 [×2], C8.C4 [×2], C42⋊C2, C22×C8, C2×M4(2), C2×M4(2) [×4], C4.10C42, C4.C42 [×2], M4(2)⋊4C4 [×2], C42.6C22, M4(2).C4, C22⋊C4.Q8
Quotients: C1, C2 [×7], C22 [×7], D4 [×2], Q8 [×2], C23, C2×D4, C2×Q8, C4○D4 [×5], C22⋊Q8, C22.D4, C4.4D4, C42.C2 [×2], C422C2 [×2], C23.83C23, C22⋊C4.Q8

Character table of C22⋊C4.Q8

 class 12A2B2C2D4A4B4C4D4E4F4G8A8B8C8D8E8F8G8H8I8J8K8L8M8N
 size 11222112228844448888888888
ρ111111111111111111111111111    trivial
ρ21111111111-1-1-1-1-1-1-1111-111-1-11    linear of order 2
ρ31111111111-1-1111111-11-1-1-11-1-1    linear of order 2
ρ4111111111111-1-1-1-1-11-111-1-1-11-1    linear of order 2
ρ5111111111111-1-1-1-11-1-1-1-11-11-11    linear of order 2
ρ61111111111-1-11111-1-1-1-111-1-111    linear of order 2
ρ71111111111-1-1-1-1-1-11-11-11-1111-1    linear of order 2
ρ81111111111111111-1-11-1-1-11-1-1-1    linear of order 2
ρ922-22-222-2-2200000000-20002000    orthogonal lifted from D4
ρ1022-22-222-2-22000000002000-2000    orthogonal lifted from D4
ρ1122222-2-2-2-2-2-2200000000000000    symplectic lifted from Q8, Schur index 2
ρ1222222-2-2-2-2-22-200000000000000    symplectic lifted from Q8, Schur index 2
ρ1322-22-2-2-222-2002i-2i2i-2i0000000000    complex lifted from C4○D4
ρ1422-22-2-2-222-200-2i2i-2i2i0000000000    complex lifted from C4○D4
ρ15222-2-2-2-22-22000000-2i0000002i00    complex lifted from C4○D4
ρ1622-2-22-2-2-22200000000002i000-2i0    complex lifted from C4○D4
ρ1722-2-22222-2-20000000-2i02i000000    complex lifted from C4○D4
ρ18222-2-222-22-2000000000002i000-2i    complex lifted from C4○D4
ρ1922-2-22-2-2-2220000000000-2i0002i0    complex lifted from C4○D4
ρ20222-2-222-22-200000000000-2i0002i    complex lifted from C4○D4
ρ2122-2-22222-2-200000002i0-2i000000    complex lifted from C4○D4
ρ22222-2-2-2-22-220000002i000000-2i00    complex lifted from C4○D4
ρ234-4000-4i4i0000085838870000000000    complex faithful
ρ244-4000-4i4i0000088785830000000000    complex faithful
ρ254-40004i-4i0000083858780000000000    complex faithful
ρ264-40004i-4i0000087883850000000000    complex faithful

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

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

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

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

Matrix representation of C22⋊C4.Q8 in GL4(𝔽17) generated by

16000
0100
6010
00016
,
16000
01600
00160
00016
,
12240
10001
141158
1615110
,
1116150
140015
13361
9830
,
0100
13000
1314013
31610
G:=sub<GL(4,GF(17))| [16,0,6,0,0,1,0,0,0,0,1,0,0,0,0,16],[16,0,0,0,0,16,0,0,0,0,16,0,0,0,0,16],[12,10,14,16,2,0,11,15,4,0,5,11,0,1,8,0],[11,14,13,9,16,0,3,8,15,0,6,3,0,15,1,0],[0,13,13,3,1,0,14,16,0,0,0,1,0,0,13,0] >;

C22⋊C4.Q8 in GAP, Magma, Sage, TeX

C_2^2\rtimes C_4.Q_8
% in TeX

G:=Group("C2^2:C4.Q8");
// GroupNames label

G:=SmallGroup(128,835);
// by ID

G=gap.SmallGroup(128,835);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,56,141,176,422,387,58,1018,248,1411,4037,2028,124]);
// Polycyclic

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

Export

Character table of C22⋊C4.Q8 in TeX

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