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G = C24.D4order 192 = 26·3

52nd non-split extension by C24 of D4 acting via D4/C2=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C24.52D4, C12.25M4(2), (C2×C12).1C8, (C2×C24).2C4, (C2×C8).153D6, C32(C23.C8), C23.2(C3⋊C8), (C22×C6).2C8, (C2×C8).2Dic3, C8.33(C3⋊D4), (C22×C12).7C4, C12.C811C2, C6.18(C22⋊C8), (C2×M4(2)).4S3, (C6×M4(2)).6C2, C4.7(C4.Dic3), (C22×C4).7Dic3, C12.93(C22⋊C4), (C2×C24).265C22, C4.26(C6.D4), C2.7(C12.55D4), (C2×C4).(C3⋊C8), C22.4(C2×C3⋊C8), (C2×C6).32(C2×C8), (C2×C12).303(C2×C4), (C2×C4).75(C2×Dic3), SmallGroup(192,112)

Series: Derived Chief Lower central Upper central

C1C2×C6 — C24.D4
C1C3C6C12C24C2×C24C12.C8 — C24.D4
C3C6C2×C6 — C24.D4
C1C4C2×C8C2×M4(2)

Generators and relations for C24.D4
 G = < a,b,c | a24=1, b4=a18, c2=a9, bab-1=a5, cac-1=a17, cbc-1=a15b3 >

Subgroups: 104 in 58 conjugacy classes, 31 normal (25 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C6, C6, C8, C8, C2×C4, C2×C4, C23, C12, C12, C2×C6, C2×C6, C16, C2×C8, M4(2), C22×C4, C24, C24, C2×C12, C2×C12, C22×C6, M5(2), C2×M4(2), C3⋊C16, C2×C24, C3×M4(2), C22×C12, C23.C8, C12.C8, C6×M4(2), C24.D4
Quotients: C1, C2, C4, C22, S3, C8, C2×C4, D4, Dic3, D6, C22⋊C4, C2×C8, M4(2), C3⋊C8, C2×Dic3, C3⋊D4, C22⋊C8, C2×C3⋊C8, C4.Dic3, C6.D4, C23.C8, C12.55D4, C24.D4

Smallest permutation representation of C24.D4
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 46 10 43 19 40 4 37 13 34 22 31 7 28 16 25)(2 27 11 48 20 45 5 42 14 39 23 36 8 33 17 30)(3 32 12 29 21 26 6 47 15 44 24 41 9 38 18 35)
(1 34 10 43 19 28 4 37 13 46 22 31 7 40 16 25)(2 27 11 36 20 45 5 30 14 39 23 48 8 33 17 42)(3 44 12 29 21 38 6 47 15 32 24 41 9 26 18 35)

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

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

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

42 conjugacy classes

class 1 2A2B2C 3 4A4B4C4D6A6B6C6D6E8A8B8C8D8E8F12A12B12C12D12E12F16A···16H24A···24H
order1222344446666688888812121212121216···1624···24
size1124211242224422224422224412···124···4

42 irreducible representations

dim1111111222222222244
type+++++-+-
imageC1C2C2C4C4C8C8S3D4Dic3D6Dic3M4(2)C3⋊D4C3⋊C8C3⋊C8C4.Dic3C23.C8C24.D4
kernelC24.D4C12.C8C6×M4(2)C2×C24C22×C12C2×C12C22×C6C2×M4(2)C24C2×C8C2×C8C22×C4C12C8C2×C4C23C4C3C1
# reps1212244121112422424

Matrix representation of C24.D4 in GL4(𝔽97) generated by

611000
653600
003528
008562
,
00122
00096
963700
53100
,
0010
0001
15900
449600
G:=sub<GL(4,GF(97))| [61,65,0,0,10,36,0,0,0,0,35,85,0,0,28,62],[0,0,96,53,0,0,37,1,1,0,0,0,22,96,0,0],[0,0,1,44,0,0,59,96,1,0,0,0,0,1,0,0] >;

C24.D4 in GAP, Magma, Sage, TeX

C_{24}.D_4
% in TeX

G:=Group("C24.D4");
// GroupNames label

G:=SmallGroup(192,112);
// by ID

G=gap.SmallGroup(192,112);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,28,141,387,100,1123,102,6278]);
// Polycyclic

G:=Group<a,b,c|a^24=1,b^4=a^18,c^2=a^9,b*a*b^-1=a^5,c*a*c^-1=a^17,c*b*c^-1=a^15*b^3>;
// generators/relations

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