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G = C23.24D4order 64 = 26

3rd non-split extension by C23 of D4 acting via D4/C4=C2

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

Aliases: C23.24D4, C4○D41C4, (C22×C8)⋊4C2, D4.5(C2×C4), C4.49(C2×D4), Q8.5(C2×C4), C4(D4⋊C4), C4(Q8⋊C4), C2.1(C4○D8), (C2×C4).145D4, C4.3(C22×C4), D4⋊C420C2, C42⋊C22C2, Q8⋊C420C2, C4⋊C4.43C22, (C2×C8).59C22, (C2×C4).61C23, C22.43(C2×D4), C4.33(C22⋊C4), (C2×D4).47C22, (C2×Q8).41C22, C22.3(C22⋊C4), (C22×C4).109C22, (C2×C4).45(C2×C4), (C2×C4○D4).4C2, (C2×C4)(D4⋊C4), (C2×C4)(Q8⋊C4), C2.19(C2×C22⋊C4), SmallGroup(64,97)

Series: Derived Chief Lower central Upper central Jennings

C1C4 — C23.24D4
C1C2C22C2×C4C22×C4C2×C4○D4 — C23.24D4
C1C2C4 — C23.24D4
C1C2×C4C22×C4 — C23.24D4
C1C2C2C2×C4 — C23.24D4

Generators and relations for C23.24D4
 G = < a,b,c,d,e | a2=b2=c2=1, d4=c, e2=b, ab=ba, eae-1=ac=ca, ad=da, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede-1=bcd3 >

Subgroups: 129 in 79 conjugacy classes, 41 normal (15 characteristic)
C1, C2, C2 [×2], C2 [×4], C4 [×2], C4 [×2], C4 [×4], C22, C22 [×2], C22 [×6], C8 [×2], C2×C4 [×2], C2×C4 [×4], C2×C4 [×7], D4 [×2], D4 [×5], Q8 [×2], Q8, C23, C23, C42, C22⋊C4, C4⋊C4 [×2], C2×C8 [×2], C2×C8 [×2], C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4 [×4], C4○D4 [×2], D4⋊C4 [×2], Q8⋊C4 [×2], C42⋊C2, C22×C8, C2×C4○D4, C23.24D4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×4], C23, C22⋊C4 [×4], C22×C4, C2×D4 [×2], C2×C22⋊C4, C4○D8 [×2], C23.24D4

Character table of C23.24D4

 class 12A2B2C2D2E2F2G4A4B4C4D4E4F4G4H4I4J4K4L8A8B8C8D8E8F8G8H
 size 1111224411112244444422222222
ρ11111111111111111111111111111    trivial
ρ21111-1-1-1-1-1-1-1-111111-11-11-1-1111-1-1    linear of order 2
ρ31111-1-1-1-1-1-1-1-111-1111-11-111-1-1-111    linear of order 2
ρ411111111111111-111-1-1-1-1-1-1-1-1-1-1-1    linear of order 2
ρ5111111-1-1111111-1-1-1-1-1-111111111    linear of order 2
ρ61111-1-111-1-1-1-111-1-1-11-111-1-1111-1-1    linear of order 2
ρ71111-1-111-1-1-1-1111-1-1-11-1-111-1-1-111    linear of order 2
ρ8111111-1-11111111-1-1111-1-1-1-1-1-1-1-1    linear of order 2
ρ91-1-111-11-11-11-11-1-i1-1-iiiii-i-i-iii-i    linear of order 4
ρ101-1-111-1-111-11-11-1i-11i-i-iii-i-i-iii-i    linear of order 4
ρ111-1-11-111-1-11-111-1-i-11ii-i-ii-iii-ii-i    linear of order 4
ρ121-1-11-11-11-11-111-1i1-1-i-ii-ii-iii-ii-i    linear of order 4
ρ131-1-11-11-11-11-111-1-i1-1ii-ii-ii-i-ii-ii    linear of order 4
ρ141-1-11-111-1-11-111-1i-11-i-iii-ii-i-ii-ii    linear of order 4
ρ151-1-111-1-111-11-11-1-i-11-iii-i-iiii-i-ii    linear of order 4
ρ161-1-111-11-11-11-11-1i1-1i-i-i-i-iiii-i-ii    linear of order 4
ρ1722222200-2-2-2-2-2-200000000000000    orthogonal lifted from D4
ρ182-2-22-22002-22-2-2200000000000000    orthogonal lifted from D4
ρ192-2-222-200-22-22-2200000000000000    orthogonal lifted from D4
ρ202222-2-2002222-2-200000000000000    orthogonal lifted from D4
ρ2122-2-200002i-2i-2i2i00000000-2--2--22-22-2-2    complex lifted from C4○D8
ρ222-22-200002i2i-2i-2i00000000--22-2--2-2-2-22    complex lifted from C4○D8
ρ2322-2-20000-2i2i2i-2i000000002--2--2-22-2-2-2    complex lifted from C4○D8
ρ242-22-20000-2i-2i2i2i00000000--2-22--2-2-22-2    complex lifted from C4○D8
ρ2522-2-200002i-2i-2i2i000000002-2-2-22-2--2--2    complex lifted from C4○D8
ρ262-22-200002i2i-2i-2i00000000-2-22-2--2--22-2    complex lifted from C4○D8
ρ2722-2-20000-2i2i2i-2i00000000-2-2-22-22--2--2    complex lifted from C4○D8
ρ282-22-20000-2i-2i2i2i00000000-22-2-2--2--2-22    complex lifted from C4○D8

Smallest permutation representation of C23.24D4
On 32 points
Generators in S32
(1 10)(2 11)(3 12)(4 13)(5 14)(6 15)(7 16)(8 9)(17 27)(18 28)(19 29)(20 30)(21 31)(22 32)(23 25)(24 26)
(1 27)(2 28)(3 29)(4 30)(5 31)(6 32)(7 25)(8 26)(9 24)(10 17)(11 18)(12 19)(13 20)(14 21)(15 22)(16 23)
(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 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 26 27 8)(2 7 28 25)(3 32 29 6)(4 5 30 31)(9 14 24 21)(10 20 17 13)(11 12 18 19)(15 16 22 23)

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

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

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

C23.24D4 is a maximal subgroup of
Q8.C42  D4.3C42  2+ 1+44C4  M4(2).42D4  M4(2).44D4  M4(2).24D4  C42.428D4  C42.107D4  C22⋊C4.7D4  C42.9D4  C24.98D4  2+ 1+45C4  2- 1+44C4  C4×C4○D8  C42.275C23  C42.276C23  C24.103D4  C4○D4⋊D4  D4.(C2×D4)  (C2×Q8)⋊16D4  Q8.(C2×D4)  C42.443D4  C42.14C23  C42.15C23  C42.16C23  C42.17C23  C42.447D4  C42.22C23  C42.23C23  C24.115D4  (C2×D4).303D4  (C2×D4).304D4  C42.355D4  C42.239D4  C42.366C23  C42.367C23  C42.461C23  C42.462C23  C42.465C23  C42.466C23  C42.467C23  C42.468C23  C42.469C23  C42.470C23  C42.42C23  C42.44C23  C42.46C23  C42.48C23  C42.50C23  C42.52C23  C42.54C23  C42.56C23  C4.A4⋊C4
 C4○D4p⋊C4: C42.383D4  C4.(C2×D12)  C23.28D12  C4○D2010C4  C23.23D20  C4○D20⋊C4  C4.(C2×D28)  C23.23D28 ...
 C4p.(C2×D4): C24.144D4  C24.110D4  M4(2)⋊16D4  M4(2)⋊17D4  D42S3⋊C4  C4⋊C4.150D6  C4○D44Dic3  D42D5⋊C4 ...
C23.24D4 is a maximal quotient of
C42.455D4  C42.373D4  C42.374D4  C42.305D4  C42.375D4  C24.53D4  C24.59D4  C42.63D4  C42.410D4  C42.411D4  C42.412D4  C42.80D4  C42.81D4  C42.417D4  C42.418D4  C24.132D4  C4×D4⋊C4  C4×Q8⋊C4  C24.65D4  C42.100D4  C42.101D4  C24.69D4  C24.74D4  C42.123D4  C42.437D4  C4○D20⋊C4
 C23.D4p: C23.23D8  C23.28D12  C23.23D20  C23.23D28 ...
 C4.(C2×D4p): C42.409D4  C4.(C2×D12)  C4○D2010C4  C4.(C2×D28) ...
 (C2×C4p).D4: C24.135D4  C42.433D4  C4○D44Dic3  C20.(C2×D4)  C28.(C2×D4) ...
 C4⋊C4.D2p: C24.73D4  C42.119D4  D42S3⋊C4  C4⋊C4.150D6  D42D5⋊C4  Q82D5⋊C4  D42D7⋊C4  Q82D7⋊C4 ...

Matrix representation of C23.24D4 in GL4(𝔽17) generated by

16000
01600
0004
00130
,
16000
01600
00160
00016
,
1000
0100
00160
00016
,
0100
1000
00125
001212
,
0100
16000
00125
0055
G:=sub<GL(4,GF(17))| [16,0,0,0,0,16,0,0,0,0,0,13,0,0,4,0],[16,0,0,0,0,16,0,0,0,0,16,0,0,0,0,16],[1,0,0,0,0,1,0,0,0,0,16,0,0,0,0,16],[0,1,0,0,1,0,0,0,0,0,12,12,0,0,5,12],[0,16,0,0,1,0,0,0,0,0,12,5,0,0,5,5] >;

C23.24D4 in GAP, Magma, Sage, TeX

C_2^3._{24}D_4
% in TeX

G:=Group("C2^3.24D4");
// GroupNames label

G:=SmallGroup(64,97);
// by ID

G=gap.SmallGroup(64,97);
# by ID

G:=PCGroup([6,-2,2,2,-2,2,-2,96,121,158,963,489,117]);
// Polycyclic

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

Export

Character table of C23.24D4 in TeX

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