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G = D24.C4order 192 = 26·3

4th non-split extension by D24 of C4 acting via C4/C2=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C24.7D4, D24.4C4, C8.17D12, C12.48D8, C8.2(C4×S3), (C2×C8).43D6, C8.C41S3, C24.22(C2×C4), C4.4(D6⋊C4), (C2×C12).95D4, C12.C85C2, (C2×C6).4SD16, C4.21(D4⋊S3), (C2×D24).12C2, C31(M5(2)⋊C2), C6.7(D4⋊C4), C12.4(C22⋊C4), C2.9(C6.D8), (C2×C24).100C22, C22.3(Q82S3), (C3×C8.C4)⋊9C2, (C2×C4).18(C3⋊D4), SmallGroup(192,54)

Series: Derived Chief Lower central Upper central

C1C24 — D24.C4
C1C3C6C12C2×C12C2×C24C2×D24 — D24.C4
C3C6C12C24 — D24.C4
C1C2C2×C4C2×C8C8.C4

Generators and relations for D24.C4
 G = < a,b,c | a24=b2=1, c4=a12, bab=a-1, cac-1=a19, cbc-1=a15b >

Subgroups: 288 in 62 conjugacy classes, 25 normal (23 characteristic)
C1, C2, C2 [×3], C3, C4 [×2], C22, C22 [×4], S3 [×2], C6, C6, C8 [×2], C8, C2×C4, D4 [×3], C23, C12 [×2], D6 [×4], C2×C6, C16, C2×C8, M4(2), D8 [×3], C2×D4, C24 [×2], C24, D12 [×3], C2×C12, C22×S3, C8.C4, M5(2), C2×D8, C3⋊C16, D24 [×2], D24, C2×C24, C3×M4(2), C2×D12, M5(2)⋊C2, C12.C8, C3×C8.C4, C2×D24, D24.C4
Quotients: C1, C2 [×3], C4 [×2], C22, S3, C2×C4, D4 [×2], D6, C22⋊C4, D8, SD16, C4×S3, D12, C3⋊D4, D4⋊C4, D6⋊C4, D4⋊S3, Q82S3, M5(2)⋊C2, C6.D8, D24.C4

Character table of D24.C4

 class 12A2B2C2D34A4B6A6B8A8B8C8D8E12A12B12C16A16B16C16D24A24B24C24D24E24F24G24H
 size 112242422224224882241212121244448888
ρ1111111111111111111111111111111    trivial
ρ2111-1-111111111-1-111111111111-1-1-1-1    linear of order 2
ρ3111-1-11111111111111-1-1-1-111111111    linear of order 2
ρ41111111111111-1-1111-1-1-1-11111-1-1-1-1    linear of order 2
ρ511-1-111-111-1-1-11i-i-1-11-ii-ii-11-11-iii-i    linear of order 4
ρ611-11-11-111-1-1-11-ii-1-11-ii-ii-11-11i-i-ii    linear of order 4
ρ711-11-11-111-1-1-11i-i-1-11i-ii-i-11-11-iii-i    linear of order 4
ρ811-1-111-111-1-1-11-ii-1-11i-ii-i-11-11i-i-ii    linear of order 4
ρ92220022222-2-2-2002220000-2-2-2-20000    orthogonal lifted from D4
ρ1022-2002-222-222-200-2-2200002-22-20000    orthogonal lifted from D4
ρ1122200-122-1-122222-1-1-10000-1-1-1-1-1-1-1-1    orthogonal lifted from S3
ρ1222200-122-1-1222-2-2-1-1-10000-1-1-1-11111    orthogonal lifted from D6
ρ1322-20022-22-20000022-222-2-200000000    orthogonal lifted from D8
ρ1422-20022-22-20000022-2-2-22200000000    orthogonal lifted from D8
ρ1522-200-1-22-1122-20011-10000-11-11-3-333    orthogonal lifted from D12
ρ1622-200-1-22-1122-20011-10000-11-1133-3-3    orthogonal lifted from D12
ρ1722-200-1-22-11-2-222i-2i11-100001-11-1i-i-ii    complex lifted from C4×S3
ρ1822-200-1-22-11-2-22-2i2i11-100001-11-1-iii-i    complex lifted from C4×S3
ρ19222002-2-22200000-2-2-2--2-2-2--200000000    complex lifted from SD16
ρ20222002-2-22200000-2-2-2-2--2--2-200000000    complex lifted from SD16
ρ2122200-122-1-1-2-2-200-1-1-100001111--3-3--3-3    complex lifted from C3⋊D4
ρ2222200-122-1-1-2-2-200-1-1-100001111-3--3-3--3    complex lifted from C3⋊D4
ρ2344400-2-4-4-2-200000222000000000000    orthogonal lifted from Q82S3
ρ2444-400-24-4-2200000-2-22000000000000    orthogonal lifted from D4⋊S3, Schur index 2
ρ254-4000400-4022-220000000000220-2200000    orthogonal lifted from M5(2)⋊C2
ρ264-4000400-40-22220000000000-2202200000    orthogonal lifted from M5(2)⋊C2
ρ274-4000-2002022-22000-232300000-262-60000    orthogonal faithful
ρ284-4000-20020-222200023-230000026-2-60000    orthogonal faithful
ρ294-4000-2002022-2200023-2300000-2-6260000    orthogonal faithful
ρ304-4000-20020-2222000-2323000002-6-260000    orthogonal faithful

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

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

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

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

Matrix representation of D24.C4 in GL4(𝔽97) generated by

181600
81200
120162
12129518
,
799500
161800
60256858
25622929
,
56153333
82413133
53545682
96531541
G:=sub<GL(4,GF(97))| [18,81,12,12,16,2,0,12,0,0,16,95,0,0,2,18],[79,16,60,25,95,18,25,62,0,0,68,29,0,0,58,29],[56,82,53,96,15,41,54,53,33,31,56,15,33,33,82,41] >;

D24.C4 in GAP, Magma, Sage, TeX

D_{24}.C_4
% in TeX

G:=Group("D24.C4");
// GroupNames label

G:=SmallGroup(192,54);
// by ID

G=gap.SmallGroup(192,54);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,141,36,758,184,675,794,192,1684,851,102,6278]);
// Polycyclic

G:=Group<a,b,c|a^24=b^2=1,c^4=a^12,b*a*b=a^-1,c*a*c^-1=a^19,c*b*c^-1=a^15*b>;
// generators/relations

Export

Character table of D24.C4 in TeX

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