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G = D24⋊C22  order 192 = 26·3

3rd semidirect product of D24 and C22 acting faithfully

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: Q16⋊4D6, SD16⋊6D6, D24⋊3C22, C24.5C23, M4(2)⋊12D6, C12.24C24, D12.17C23, Dic6.17C23, C8⋊D6⋊3C2, Q8⋊3D6⋊3C2, (C2×Q8)⋊26D6, (S3×C8)⋊5C22, D4⋊S3⋊7C22, C4○D4.46D6, D6.55(C2×D4), (C4×S3).45D4, C4.192(S3×D4), Q16⋊S3⋊2C2, C8.C22⋊6S3, C3⋊C8.12C23, C8.5(C22×S3), D4⋊D6⋊10C2, D24⋊C2⋊1C2, C8⋊S3⋊6C22, C24⋊C2⋊6C22, Q8.7D6⋊3C2, C12.245(C2×D4), (S3×D4)⋊10C22, (S3×M4(2))⋊4C2, C4.24(S3×C23), (C3×Q16)⋊2C22, (C6×Q8)⋊21C22, (S3×Q8)⋊12C22, C3⋊Q16⋊5C22, C22.49(S3×D4), (C2×D12)⋊37C22, C3⋊4(D8⋊C22), (C4×S3).31C23, Dic3.62(C2×D4), Q8⋊2S3⋊6C22, (C3×SD16)⋊6C22, (C3×D4).17C23, D4.17(C22×S3), (C22×S3).44D4, C6.125(C22×D4), (C3×Q8).17C23, Q8.27(C22×S3), Q8.11D6⋊10C2, D4⋊2S3⋊11C22, (C2×C12).115C23, (C2×Dic3).196D4, Q8⋊3S3⋊11C22, C4○D12.31C22, (C3×M4(2))⋊6C22, C4.Dic3⋊15C22, C2.98(C2×S3×D4), (S3×C4○D4)⋊5C2, (C2×C6).70(C2×D4), (C3×C8.C22)⋊2C2, (C2×Q8⋊3S3)⋊17C2, (S3×C2×C4).163C22, (C2×C4).99(C22×S3), (C3×C4○D4).26C22, SmallGroup(192,1336)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C12 — D24⋊C22
Chief series
C1 — C3 — C6 — C12 — C4×S3 — S3×C2×C4 — S3×C4○D4 — D24⋊C22
Lower central
C3 — C6 — C12 — D24⋊C22
Upper central
C1 — C2 — C2×C4 — C8.C22

Generators and relations for D24⋊C22
 G = < a,b,c,d | a24=b2=c2=d2=1, bab=a-1, cac=a13, dad=a5, cbc=a12b, dbd=a16b, cd=dc >

Subgroups: 720 in 262 conjugacy classes, 99 normal (51 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C8, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, Q8, C23, Dic3, Dic3, C12, C12, D6, D6, C2×C6, C2×C6, C2×C8, M4(2), M4(2), D8, SD16, SD16, Q16, Q16, C22×C4, C2×D4, C2×Q8, C2×Q8, C4○D4, C4○D4, C3⋊C8, C24, Dic6, Dic6, C4×S3, C4×S3, D12, D12, D12, C2×Dic3, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C3×D4, C3×D4, C3×Q8, C3×Q8, C3×Q8, C22×S3, C22×S3, C2×M4(2), C4○D8, C8⋊C22, C8.C22, C8.C22, C2×C4○D4, S3×C8, C8⋊S3, C24⋊C2, D24, C4.Dic3, D4⋊S3, Q8⋊2S3, C3⋊Q16, C3×M4(2), C3×SD16, C3×Q16, S3×C2×C4, S3×C2×C4, C2×D12, C2×D12, C4○D12, C4○D12, S3×D4, S3×D4, D4⋊2S3, D4⋊2S3, S3×Q8, Q8⋊3S3, Q8⋊3S3, Q8⋊3S3, C6×Q8, C3×C4○D4, D8⋊C22, S3×M4(2), C8⋊D6, Q8⋊3D6, Q8.7D6, Q16⋊S3, D24⋊C2, Q8.11D6, D4⋊D6, C3×C8.C22, C2×Q8⋊3S3, S3×C4○D4, D24⋊C22
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C24, C22×S3, C22×D4, S3×D4, S3×C23, D8⋊C22, C2×S3×D4, D24⋊C22

Smallest permutation representation of D24⋊C22
►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 24)(2 23)(3 22)(4 21)(5 20)(6 19)(7 18)(8 17)(9 16)(10 15)(11 14)(12 13)(25 34)(26 33)(27 32)(28 31)(29 30)(35 48)(36 47)(37 46)(38 45)(39 44)(40 43)(41 42)

(1 13)(3 15)(5 17)(7 19)(9 21)(11 23)(26 38)(28 40)(30 42)(32 44)(34 46)(36 48)

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

33 conjugacy classes

class 1 2A2B2C2D2E2F2G2H 3 4A4B4C4D4E4F4G4H4I6A6B6C8A8B8C8D12A12B12C12D12E24A24B
order1222222223444444444666888812121212122424
size112466121212222334446122484412124488888

33 irreducible representations

dim1111111111112222222224448
type++++++++++++++++++++++++
imageC1C2C2C2C2C2C2C2C2C2C2C2S3D4D4D4D6D6D6D6D6S3×D4S3×D4D8⋊C22D24⋊C22
kernelD24⋊C22S3×M4(2)C8⋊D6Q8⋊3D6Q8.7D6Q16⋊S3D24⋊C2Q8.11D6D4⋊D6C3×C8.C22C2×Q8⋊3S3S3×C4○D4C8.C22C4×S3C2×Dic3C22×S3M4(2)SD16Q16C2×Q8C4○D4C4C22C3C1
# reps1112222111111211122111121

Matrix representation of D24⋊C22 ►in GL8(𝔽73)

464660600000
271913470000
131327270000
602646540000
0000002754
000000046
000046000
0000272700
,
271913470000
464660600000
602646540000
131327270000
0000002754
0000004646
0000275400
0000464600
,
10000000
01000000
00100000
00010000
000072000
000007200
00000010
00000001
,
00010000
00100000
01000000
10000000
0000275400
0000464600
0000002754
0000004646

G:=sub<GL(8,GF(73))| [46,27,13,60,0,0,0,0,46,19,13,26,0,0,0,0,60,13,27,46,0,0,0,0,60,47,27,54,0,0,0,0,0,0,0,0,0,0,46,27,0,0,0,0,0,0,0,27,0,0,0,0,27,0,0,0,0,0,0,0,54,46,0,0],[27,46,60,13,0,0,0,0,19,46,26,13,0,0,0,0,13,60,46,27,0,0,0,0,47,60,54,27,0,0,0,0,0,0,0,0,0,0,27,46,0,0,0,0,0,0,54,46,0,0,0,0,27,46,0,0,0,0,0,0,54,46,0,0],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,27,46,0,0,0,0,0,0,54,46,0,0,0,0,0,0,0,0,27,46,0,0,0,0,0,0,54,46] >;
 

D24⋊C22 in GAP, Magma, Sage, TeX 

D_{24}\rtimes C_2^2
 
% in TeX
 
G:=Group("D24:C2^2");
 
// GroupNames label
 
G:=SmallGroup(192,1336);
 
// by ID
 
G=gap.SmallGroup(192,1336);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,387,1123,185,136,438,235,102,6278]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^24=b^2=c^2=d^2=1,b*a*b=a^-1,c*a*c=a^13,d*a*d=a^5,c*b*c=a^12*b,d*b*d=a^16*b,c*d=d*c>;
 
// generators/relations
 

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