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G = C24.97D4order 192 = 26·3

20th non-split extension by C24 of D4 acting via D4/C22=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C24.97D4, C24.10Q8, C8.9Dic6, M5(2).2S3, C3:C8.1C8, C6.9(C4:C8), C4.13(S3xC8), C12.5(C2xC8), C3:2(C8.C8), (C2xC8).266D6, C12.41(C4:C4), C8.50(C3:D4), (C4xDic3).4C4, (C2xC6).3M4(2), C12.C8.7C2, C2.5(Dic3:C8), (C8xDic3).15C2, (C3xM5(2)).4C2, C4.28(Dic3:C4), (C2xC24).261C22, C22.5(C8:S3), (C2xC3:C8).6C4, (C2xC4).135(C4xS3), (C2xC12).50(C2xC4), SmallGroup(192,70)

Series: Derived Chief Lower central Upper central

C1C12 — C24.97D4
C1C3C6C12C24C2xC24C8xDic3 — C24.97D4
C3C6C12 — C24.97D4
C1C8C2xC8M5(2)

Generators and relations for C24.97D4
 G = < a,b,c | a24=1, b4=a12, c2=a9, bab-1=cac-1=a17, cbc-1=a12b3 >

Subgroups: 88 in 46 conjugacy classes, 27 normal (all characteristic)
Quotients: C1, C2, C4, C22, S3, C8, C2xC4, D4, Q8, D6, C4:C4, C2xC8, M4(2), Dic6, C4xS3, C3:D4, C4:C8, S3xC8, C8:S3, Dic3:C4, C8.C8, Dic3:C8, C24.97D4
2C2
6C4
6C4
2C6
3C8
3C8
6C2xC4
2Dic3
2Dic3
2C16
3C42
3C2xC8
6C16
2C2xDic3
3M5(2)
3C4xC8
2C3:C16
2C48
3C8.C8

Smallest permutation representation of C24.97D4
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 4 7 10 13 16 19 22)(2 21 8 3 14 9 20 15)(5 24 11 6 17 12 23 18)(25 46 43 40 37 34 31 28)(26 39 44 33 38 27 32 45)(29 42 47 36 41 30 35 48)
(1 46 10 31 19 40 4 25 13 34 22 43 7 28 16 37)(2 39 11 48 20 33 5 42 14 27 23 36 8 45 17 30)(3 32 12 41 21 26 6 35 15 44 24 29 9 38 18 47)

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

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

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

48 conjugacy classes

class 1 2A2B 3 4A4B4C4D4E4F4G6A6B8A8B8C8D8E8F8G8H8I8J12A12B12C16A16B16C16D16E16F16G16H24A24B24C24D24E24F48A···48H
order12234444444668888888888121212161616161616161624242424242448···48
size112211266662411112266662244444121212122222444···4

48 irreducible representations

dim1111111222222222224
type++++++-+-
imageC1C2C2C2C4C4C8S3D4Q8D6M4(2)Dic6C3:D4C4xS3S3xC8C8:S3C8.C8C24.97D4
kernelC24.97D4C12.C8C8xDic3C3xM5(2)C2xC3:C8C4xDic3C3:C8M5(2)C24C24C2xC8C2xC6C8C8C2xC4C4C22C3C1
# reps1111228111122224484

Matrix representation of C24.97D4 in GL4(F97) generated by

09600
19600
00330
00033
,
02200
22000
00470
004064
,
392900
685800
006466
008133
G:=sub<GL(4,GF(97))| [0,1,0,0,96,96,0,0,0,0,33,0,0,0,0,33],[0,22,0,0,22,0,0,0,0,0,47,40,0,0,0,64],[39,68,0,0,29,58,0,0,0,0,64,81,0,0,66,33] >;

C24.97D4 in GAP, Magma, Sage, TeX

C_{24}._{97}D_4
% in TeX

G:=Group("C24.97D4");
// GroupNames label

G:=SmallGroup(192,70);
// by ID

G=gap.SmallGroup(192,70);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,56,141,36,100,570,136,102,6278]);
// Polycyclic

G:=Group<a,b,c|a^24=1,b^4=a^12,c^2=a^9,b*a*b^-1=c*a*c^-1=a^17,c*b*c^-1=a^12*b^3>;
// generators/relations

Export

Subgroup lattice of C24.97D4 in TeX

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