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G = C4○D24order 96 = 25·3

Central product of C4 and D24

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C4D24, D247C2, C4Dic12, C8.17D6, C4.20D12, C12.35D4, Dic127C2, C22.1D12, C12.30C23, C24.17C22, D12.7C22, Dic6.6C22, (C2×C8)⋊4S3, (C2×C24)⋊6C2, C31(C4○D8), C4(C24⋊C2), C24⋊C27C2, C4○D121C2, C6.11(C2×D4), (C2×C6).18D4, (C2×C4).81D6, C2.13(C2×D12), C4.28(C22×S3), (C2×C12).99C22, SmallGroup(96,111)

Series: Derived Chief Lower central Upper central

C1C12 — C4○D24
C1C3C6C12D12C4○D12 — C4○D24
C3C6C12 — C4○D24
C1C4C2×C4C2×C8

Generators and relations for C4○D24
 G = < a,b,c | a4=c2=1, b12=a2, ab=ba, ac=ca, cbc=a2b11 >

Subgroups: 162 in 62 conjugacy classes, 29 normal (21 characteristic)
C1, C2, C2 [×3], C3, C4 [×2], C4 [×2], C22, C22 [×2], S3 [×2], C6, C6, C8 [×2], C2×C4, C2×C4 [×2], D4 [×4], Q8 [×2], Dic3 [×2], C12 [×2], D6 [×2], C2×C6, C2×C8, D8, SD16 [×2], Q16, C4○D4 [×2], C24 [×2], Dic6 [×2], C4×S3 [×2], D12 [×2], C3⋊D4 [×2], C2×C12, C4○D8, C24⋊C2 [×2], D24, Dic12, C2×C24, C4○D12 [×2], C4○D24
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], C23, D6 [×3], C2×D4, D12 [×2], C22×S3, C4○D8, C2×D12, C4○D24

Character table of C4○D24

 class 12A2B2C2D34A4B4C4D4E6A6B6C8A8B8C8D12A12B12C12D24A24B24C24D24E24F24G24H
 size 1121212211212122222222222222222222
ρ1111111111111111111111111111111    trivial
ρ211-1111-1-11-1-11-1-1-111-111-1-11-111-1-11-1    linear of order 2
ρ311-1-111-1-11-111-1-11-1-1111-1-1-11-1-111-11    linear of order 2
ρ4111-1111111-1111-1-1-1-11111-1-1-1-1-1-1-1-1    linear of order 2
ρ511-11-11-1-111-11-1-11-1-1111-1-1-11-1-111-11    linear of order 2
ρ61111-11111-11111-1-1-1-11111-1-1-1-1-1-1-1-1    linear of order 2
ρ7111-1-11111-1-11111111111111111111    linear of order 2
ρ811-1-1-11-1-11111-1-1-111-111-1-11-111-1-11-1    linear of order 2
ρ9222002-2-2-2002220000-2-2-2-200000000    orthogonal lifted from D4
ρ1022200-122200-1-1-1-2-2-2-2-1-1-1-111111111    orthogonal lifted from D6
ρ1122-200-1-2-2200-1112-2-22-1-1111-111-1-11-1    orthogonal lifted from D6
ρ1222-200-1-2-2200-111-222-2-1-111-11-1-111-11    orthogonal lifted from D6
ρ1322-200222-2002-2-20000-2-22200000000    orthogonal lifted from D4
ρ1422200-122200-1-1-12222-1-1-1-1-1-1-1-1-1-1-1-1    orthogonal lifted from S3
ρ1522-200-122-200-111000011-1-1-333-3-333-3    orthogonal lifted from D12
ρ1622200-1-2-2-200-1-1-100001111-3-33-33-333    orthogonal lifted from D12
ρ1722200-1-2-2-200-1-1-10000111133-33-33-3-3    orthogonal lifted from D12
ρ1822-200-122-200-111000011-1-13-3-333-3-33    orthogonal lifted from D12
ρ192-20002-2i2i000-200-2-22--2002i-2i2--2-2-2--2-22-2    complex lifted from C4○D8
ρ202-200022i-2i000-200-22-2--200-2i2i-2--222--2-2-2-2    complex lifted from C4○D8
ρ212-200022i-2i000-200--2-22-200-2i2i2-2-2-2-2--22--2    complex lifted from C4○D8
ρ222-20002-2i2i000-200--22-2-2002i-2i-2-222-2--2-2--2    complex lifted from C4○D8
ρ232-2000-1-2i2i0001--3-3--22-2-23-3-iiζ83ζ38ζ38ζ87ζ328785ζ32ζ87ζ3285ζ3285ζ83ζ32838ζ32ζ87ζ38785ζ3ζ83ζ32838ζ32ζ87ζ38785ζ3ζ83ζ3838ζ3    complex faithful
ρ242-2000-1-2i2i0001--3-3-2-22--23-3-iiζ83ζ32838ζ32ζ83ζ32838ζ32ζ87ζ38785ζ3ζ83ζ38ζ38ζ83ζ3838ζ3ζ87ζ328785ζ32ζ87ζ3285ζ3285ζ87ζ38785ζ3    complex faithful
ρ252-2000-1-2i2i0001-3--3-2-22--2-33-iiζ87ζ3285ζ3285ζ83ζ3838ζ3ζ83ζ38ζ38ζ87ζ38785ζ3ζ83ζ32838ζ32ζ87ζ38785ζ3ζ83ζ32838ζ32ζ87ζ328785ζ32    complex faithful
ρ262-2000-12i-2i0001--3-3-22-2--2-33i-iζ87ζ38785ζ3ζ83ζ3838ζ3ζ83ζ32838ζ32ζ87ζ3285ζ3285ζ83ζ32838ζ32ζ87ζ38785ζ3ζ83ζ38ζ38ζ87ζ328785ζ32    complex faithful
ρ272-2000-12i-2i0001-3--3-22-2--23-3i-iζ83ζ38ζ38ζ83ζ32838ζ32ζ87ζ3285ζ3285ζ83ζ32838ζ32ζ83ζ3838ζ3ζ87ζ328785ζ32ζ87ζ38785ζ3ζ87ζ38785ζ3    complex faithful
ρ282-2000-12i-2i0001-3--3--2-22-23-3i-iζ83ζ32838ζ32ζ87ζ328785ζ32ζ87ζ38785ζ3ζ83ζ38ζ38ζ87ζ38785ζ3ζ83ζ32838ζ32ζ87ζ3285ζ3285ζ83ζ3838ζ3    complex faithful
ρ292-2000-12i-2i0001--3-3--2-22-2-33i-iζ87ζ3285ζ3285ζ87ζ38785ζ3ζ83ζ38ζ38ζ87ζ38785ζ3ζ87ζ328785ζ32ζ83ζ3838ζ3ζ83ζ32838ζ32ζ83ζ32838ζ32    complex faithful
ρ302-2000-1-2i2i0001-3--3--22-2-2-33-iiζ87ζ38785ζ3ζ87ζ38785ζ3ζ83ζ32838ζ32ζ87ζ3285ζ3285ζ87ζ328785ζ32ζ83ζ3838ζ3ζ83ζ38ζ38ζ83ζ32838ζ32    complex faithful

Smallest permutation representation of C4○D24
On 48 points
Generators in S48
(1 44 13 32)(2 45 14 33)(3 46 15 34)(4 47 16 35)(5 48 17 36)(6 25 18 37)(7 26 19 38)(8 27 20 39)(9 28 21 40)(10 29 22 41)(11 30 23 42)(12 31 24 43)
(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 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48)
(1 15)(2 14)(3 13)(4 12)(5 11)(6 10)(7 9)(16 24)(17 23)(18 22)(19 21)(25 29)(26 28)(30 48)(31 47)(32 46)(33 45)(34 44)(35 43)(36 42)(37 41)(38 40)

G:=sub<Sym(48)| (1,44,13,32)(2,45,14,33)(3,46,15,34)(4,47,16,35)(5,48,17,36)(6,25,18,37)(7,26,19,38)(8,27,20,39)(9,28,21,40)(10,29,22,41)(11,30,23,42)(12,31,24,43), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48), (1,15)(2,14)(3,13)(4,12)(5,11)(6,10)(7,9)(16,24)(17,23)(18,22)(19,21)(25,29)(26,28)(30,48)(31,47)(32,46)(33,45)(34,44)(35,43)(36,42)(37,41)(38,40)>;

G:=Group( (1,44,13,32)(2,45,14,33)(3,46,15,34)(4,47,16,35)(5,48,17,36)(6,25,18,37)(7,26,19,38)(8,27,20,39)(9,28,21,40)(10,29,22,41)(11,30,23,42)(12,31,24,43), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48), (1,15)(2,14)(3,13)(4,12)(5,11)(6,10)(7,9)(16,24)(17,23)(18,22)(19,21)(25,29)(26,28)(30,48)(31,47)(32,46)(33,45)(34,44)(35,43)(36,42)(37,41)(38,40) );

G=PermutationGroup([(1,44,13,32),(2,45,14,33),(3,46,15,34),(4,47,16,35),(5,48,17,36),(6,25,18,37),(7,26,19,38),(8,27,20,39),(9,28,21,40),(10,29,22,41),(11,30,23,42),(12,31,24,43)], [(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24),(25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48)], [(1,15),(2,14),(3,13),(4,12),(5,11),(6,10),(7,9),(16,24),(17,23),(18,22),(19,21),(25,29),(26,28),(30,48),(31,47),(32,46),(33,45),(34,44),(35,43),(36,42),(37,41),(38,40)])

C4○D24 is a maximal subgroup of
D248C4  Dic12.C4  D24.1C4  D242C4  D2411C4  D244C4  D2410C4  D247C4  D487C2  C16⋊D6  C16.D6  D8.D6  C24.27C23  Q16.D6  C24.9C23  D4.11D12  D4.12D12  D4.13D12  D813D6  SD1613D6  D12.30D4  S3×C4○D8  SD16⋊D6  D727C2  D6.1D12  D247S3  D6.3D12  D12.27D6  C24.78D6  C40.31D6  D247D5  D120⋊C2  C20.60D12  C40.69D6
C4○D24 is a maximal quotient of
C24.13Q8  C4×C24⋊C2  C4×D24  C8.8D12  C42.264D6  C4×Dic12  C23.15D12  D12.32D4  D1214D4  C23.18D12  Dic6.3Q8  D12.19D4  C42.36D6  D12.3Q8  C23.27D12  C23.28D12  C2430D4  C2429D4  C24.82D4  D727C2  D6.1D12  D247S3  D6.3D12  D12.27D6  C24.78D6  C40.31D6  D247D5  D120⋊C2  C20.60D12  C40.69D6

Matrix representation of C4○D24 in GL2(𝔽73) generated by

270
027
,
5550
235
,
667
147
G:=sub<GL(2,GF(73))| [27,0,0,27],[55,23,50,5],[66,14,7,7] >;

C4○D24 in GAP, Magma, Sage, TeX

C_4\circ D_{24}
% in TeX

G:=Group("C4oD24");
// GroupNames label

G:=SmallGroup(96,111);
// by ID

G=gap.SmallGroup(96,111);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-3,103,218,50,579,69,2309]);
// Polycyclic

G:=Group<a,b,c|a^4=c^2=1,b^12=a^2,a*b=b*a,a*c=c*a,c*b*c=a^2*b^11>;
// generators/relations

Export

Character table of C4○D24 in TeX

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