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G = C242D4order 192 = 26·3

2nd semidirect product of C24 and D4 acting via D4/C2=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C242D4, C23.22D12, C36(C8⋊D4), C81(C3⋊D4), (C2×C8).77D6, C241C418C2, (C2×C4).51D12, C12.420(C2×D4), (C2×C12).297D4, C2.D2441C2, (C2×M4(2))⋊1S3, (C6×M4(2))⋊1C2, C127D4.17C2, C6.73(C4⋊D4), C2.22(C8⋊D6), C6.22(C8⋊C22), (C2×C24).63C22, C2.Dic1241C2, (C22×C4).158D6, (C22×C6).103D4, C12.230(C4○D4), C4.114(C4○D12), C12.48D441C2, C2.21(C127D4), (C2×C12).775C23, C2.22(C8.D6), (C2×D12).20C22, C22.134(C2×D12), C6.22(C8.C22), C4⋊Dic3.26C22, (C2×Dic6).19C22, (C22×C12).304C22, (C2×C24⋊C2)⋊2C2, (C2×C6).165(C2×D4), C4.113(C2×C3⋊D4), (C2×C4).724(C22×S3), SmallGroup(192,693)

Series: Derived Chief Lower central Upper central

C1C2×C12 — C242D4
C1C3C6C2×C6C2×C12C2×D12C2×C24⋊C2 — C242D4
C3C6C2×C12 — C242D4
C1C22C22×C4C2×M4(2)

Generators and relations for C242D4
 G = < a,b,c | a24=b4=c2=1, bab-1=a-1, cac=a11, cbc=b-1 >

Subgroups: 392 in 120 conjugacy classes, 43 normal (39 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C8, C8, C2×C4, C2×C4, D4, Q8, C23, C23, Dic3, C12, C12, D6, C2×C6, C2×C6, C22⋊C4, C4⋊C4, C2×C8, M4(2), SD16, C22×C4, C2×D4, C2×Q8, C24, C24, Dic6, D12, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C22×S3, C22×C6, D4⋊C4, Q8⋊C4, C2.D8, C4⋊D4, C22⋊Q8, C2×M4(2), C2×SD16, C24⋊C2, Dic3⋊C4, C4⋊Dic3, D6⋊C4, C6.D4, C2×C24, C3×M4(2), C2×Dic6, C2×D12, C2×C3⋊D4, C22×C12, C8⋊D4, C2.Dic12, C241C4, C2.D24, C2×C24⋊C2, C12.48D4, C127D4, C6×M4(2), C242D4
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C4○D4, D12, C3⋊D4, C22×S3, C4⋊D4, C8⋊C22, C8.C22, C2×D12, C4○D12, C2×C3⋊D4, C8⋊D4, C8⋊D6, C8.D6, C127D4, C242D4

Smallest permutation representation of C242D4
On 96 points
Generators in S96
(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)(49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72)(73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96)
(1 53 88 45)(2 52 89 44)(3 51 90 43)(4 50 91 42)(5 49 92 41)(6 72 93 40)(7 71 94 39)(8 70 95 38)(9 69 96 37)(10 68 73 36)(11 67 74 35)(12 66 75 34)(13 65 76 33)(14 64 77 32)(15 63 78 31)(16 62 79 30)(17 61 80 29)(18 60 81 28)(19 59 82 27)(20 58 83 26)(21 57 84 25)(22 56 85 48)(23 55 86 47)(24 54 87 46)
(2 12)(3 23)(4 10)(5 21)(6 8)(7 19)(9 17)(11 15)(14 24)(16 22)(18 20)(25 49)(26 60)(27 71)(28 58)(29 69)(30 56)(31 67)(32 54)(33 65)(34 52)(35 63)(36 50)(37 61)(38 72)(39 59)(40 70)(41 57)(42 68)(43 55)(44 66)(45 53)(46 64)(47 51)(48 62)(73 91)(74 78)(75 89)(77 87)(79 85)(80 96)(81 83)(82 94)(84 92)(86 90)(93 95)

G:=sub<Sym(96)| (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)(49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72)(73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96), (1,53,88,45)(2,52,89,44)(3,51,90,43)(4,50,91,42)(5,49,92,41)(6,72,93,40)(7,71,94,39)(8,70,95,38)(9,69,96,37)(10,68,73,36)(11,67,74,35)(12,66,75,34)(13,65,76,33)(14,64,77,32)(15,63,78,31)(16,62,79,30)(17,61,80,29)(18,60,81,28)(19,59,82,27)(20,58,83,26)(21,57,84,25)(22,56,85,48)(23,55,86,47)(24,54,87,46), (2,12)(3,23)(4,10)(5,21)(6,8)(7,19)(9,17)(11,15)(14,24)(16,22)(18,20)(25,49)(26,60)(27,71)(28,58)(29,69)(30,56)(31,67)(32,54)(33,65)(34,52)(35,63)(36,50)(37,61)(38,72)(39,59)(40,70)(41,57)(42,68)(43,55)(44,66)(45,53)(46,64)(47,51)(48,62)(73,91)(74,78)(75,89)(77,87)(79,85)(80,96)(81,83)(82,94)(84,92)(86,90)(93,95)>;

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)(49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72)(73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96), (1,53,88,45)(2,52,89,44)(3,51,90,43)(4,50,91,42)(5,49,92,41)(6,72,93,40)(7,71,94,39)(8,70,95,38)(9,69,96,37)(10,68,73,36)(11,67,74,35)(12,66,75,34)(13,65,76,33)(14,64,77,32)(15,63,78,31)(16,62,79,30)(17,61,80,29)(18,60,81,28)(19,59,82,27)(20,58,83,26)(21,57,84,25)(22,56,85,48)(23,55,86,47)(24,54,87,46), (2,12)(3,23)(4,10)(5,21)(6,8)(7,19)(9,17)(11,15)(14,24)(16,22)(18,20)(25,49)(26,60)(27,71)(28,58)(29,69)(30,56)(31,67)(32,54)(33,65)(34,52)(35,63)(36,50)(37,61)(38,72)(39,59)(40,70)(41,57)(42,68)(43,55)(44,66)(45,53)(46,64)(47,51)(48,62)(73,91)(74,78)(75,89)(77,87)(79,85)(80,96)(81,83)(82,94)(84,92)(86,90)(93,95) );

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),(49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72),(73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96)], [(1,53,88,45),(2,52,89,44),(3,51,90,43),(4,50,91,42),(5,49,92,41),(6,72,93,40),(7,71,94,39),(8,70,95,38),(9,69,96,37),(10,68,73,36),(11,67,74,35),(12,66,75,34),(13,65,76,33),(14,64,77,32),(15,63,78,31),(16,62,79,30),(17,61,80,29),(18,60,81,28),(19,59,82,27),(20,58,83,26),(21,57,84,25),(22,56,85,48),(23,55,86,47),(24,54,87,46)], [(2,12),(3,23),(4,10),(5,21),(6,8),(7,19),(9,17),(11,15),(14,24),(16,22),(18,20),(25,49),(26,60),(27,71),(28,58),(29,69),(30,56),(31,67),(32,54),(33,65),(34,52),(35,63),(36,50),(37,61),(38,72),(39,59),(40,70),(41,57),(42,68),(43,55),(44,66),(45,53),(46,64),(47,51),(48,62),(73,91),(74,78),(75,89),(77,87),(79,85),(80,96),(81,83),(82,94),(84,92),(86,90),(93,95)]])

36 conjugacy classes

class 1 2A2B2C2D2E 3 4A4B4C4D4E4F6A6B6C6D6E8A8B8C8D12A12B12C12D12E12F24A···24H
order122222344444466666888812121212121224···24
size111142422242424242224444442222444···4

36 irreducible representations

dim11111111222222222224444
type+++++++++++++++++-+-
imageC1C2C2C2C2C2C2C2S3D4D4D4D6D6C4○D4C3⋊D4D12D12C4○D12C8⋊C22C8.C22C8⋊D6C8.D6
kernelC242D4C2.Dic12C241C4C2.D24C2×C24⋊C2C12.48D4C127D4C6×M4(2)C2×M4(2)C24C2×C12C22×C6C2×C8C22×C4C12C8C2×C4C23C4C6C6C2C2
# reps11111111121121242241122

Matrix representation of C242D4 in GL8(𝔽73)

720000000
072000000
00010000
007210000
000056366
00004153180
000052525535
000062112655
,
12000000
7272000000
000720000
007200000
000058020
0000281733
0000330150
000070257156
,
10000000
7272000000
00010000
00100000
00001000
000037200
0000150720
0000566001

G:=sub<GL(8,GF(73))| [72,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,56,41,52,62,0,0,0,0,3,53,52,11,0,0,0,0,6,18,55,26,0,0,0,0,6,0,35,55],[1,72,0,0,0,0,0,0,2,72,0,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,0,58,28,33,70,0,0,0,0,0,17,0,25,0,0,0,0,2,3,15,71,0,0,0,0,0,3,0,56],[1,72,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,3,15,56,0,0,0,0,0,72,0,60,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,1] >;

C242D4 in GAP, Magma, Sage, TeX

C_{24}\rtimes_2D_4
% in TeX

G:=Group("C24:2D4");
// GroupNames label

G:=SmallGroup(192,693);
// by ID

G=gap.SmallGroup(192,693);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,253,120,254,387,1684,102,6278]);
// Polycyclic

G:=Group<a,b,c|a^24=b^4=c^2=1,b*a*b^-1=a^-1,c*a*c=a^11,c*b*c=b^-1>;
// generators/relations

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