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G = C2×C4○D28order 224 = 25·7

Direct product of C2 and C4○D28

direct product, metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C2×C4○D28, C14.4C24, D2812C22, C28.43C23, D14.1C23, C23.26D14, Dic7.2C23, Dic1411C22, (C2×C4)⋊10D14, (C22×C4)⋊6D7, (C2×D28)⋊14C2, C141(C4○D4), (C22×C28)⋊8C2, (C4×D7)⋊6C22, C7⋊D46C22, C2.5(C23×D7), (C2×C28)⋊13C22, C4.43(C22×D7), (C2×Dic14)⋊15C2, (C2×C14).65C23, C22.5(C22×D7), (C22×C14).46C22, (C2×Dic7).43C22, (C22×D7).28C22, C71(C2×C4○D4), (C2×C4×D7)⋊15C2, (C2×C7⋊D4)⋊12C2, SmallGroup(224,177)

Series: Derived Chief Lower central Upper central

C1C14 — C2×C4○D28
C1C7C14D14C22×D7C2×C4×D7 — C2×C4○D28
C7C14 — C2×C4○D28
C1C2×C4C22×C4

Generators and relations for C2×C4○D28
 G = < a,b,c,d | a2=b4=d2=1, c14=b2, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=b2c13 >

Subgroups: 590 in 164 conjugacy classes, 89 normal (17 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C7, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, D7, C14, C14, C14, C22×C4, C22×C4, C2×D4, C2×Q8, C4○D4, Dic7, C28, D14, D14, C2×C14, C2×C14, C2×C14, C2×C4○D4, Dic14, C4×D7, D28, C2×Dic7, C7⋊D4, C2×C28, C2×C28, C22×D7, C22×C14, C2×Dic14, C2×C4×D7, C2×D28, C4○D28, C2×C7⋊D4, C22×C28, C2×C4○D28
Quotients: C1, C2, C22, C23, D7, C4○D4, C24, D14, C2×C4○D4, C22×D7, C4○D28, C23×D7, C2×C4○D28

Smallest permutation representation of C2×C4○D28
On 112 points
Generators in S112
(1 47)(2 48)(3 49)(4 50)(5 51)(6 52)(7 53)(8 54)(9 55)(10 56)(11 29)(12 30)(13 31)(14 32)(15 33)(16 34)(17 35)(18 36)(19 37)(20 38)(21 39)(22 40)(23 41)(24 42)(25 43)(26 44)(27 45)(28 46)(57 93)(58 94)(59 95)(60 96)(61 97)(62 98)(63 99)(64 100)(65 101)(66 102)(67 103)(68 104)(69 105)(70 106)(71 107)(72 108)(73 109)(74 110)(75 111)(76 112)(77 85)(78 86)(79 87)(80 88)(81 89)(82 90)(83 91)(84 92)
(1 97 15 111)(2 98 16 112)(3 99 17 85)(4 100 18 86)(5 101 19 87)(6 102 20 88)(7 103 21 89)(8 104 22 90)(9 105 23 91)(10 106 24 92)(11 107 25 93)(12 108 26 94)(13 109 27 95)(14 110 28 96)(29 71 43 57)(30 72 44 58)(31 73 45 59)(32 74 46 60)(33 75 47 61)(34 76 48 62)(35 77 49 63)(36 78 50 64)(37 79 51 65)(38 80 52 66)(39 81 53 67)(40 82 54 68)(41 83 55 69)(42 84 56 70)
(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 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112)
(1 46)(2 45)(3 44)(4 43)(5 42)(6 41)(7 40)(8 39)(9 38)(10 37)(11 36)(12 35)(13 34)(14 33)(15 32)(16 31)(17 30)(18 29)(19 56)(20 55)(21 54)(22 53)(23 52)(24 51)(25 50)(26 49)(27 48)(28 47)(57 100)(58 99)(59 98)(60 97)(61 96)(62 95)(63 94)(64 93)(65 92)(66 91)(67 90)(68 89)(69 88)(70 87)(71 86)(72 85)(73 112)(74 111)(75 110)(76 109)(77 108)(78 107)(79 106)(80 105)(81 104)(82 103)(83 102)(84 101)

G:=sub<Sym(112)| (1,47)(2,48)(3,49)(4,50)(5,51)(6,52)(7,53)(8,54)(9,55)(10,56)(11,29)(12,30)(13,31)(14,32)(15,33)(16,34)(17,35)(18,36)(19,37)(20,38)(21,39)(22,40)(23,41)(24,42)(25,43)(26,44)(27,45)(28,46)(57,93)(58,94)(59,95)(60,96)(61,97)(62,98)(63,99)(64,100)(65,101)(66,102)(67,103)(68,104)(69,105)(70,106)(71,107)(72,108)(73,109)(74,110)(75,111)(76,112)(77,85)(78,86)(79,87)(80,88)(81,89)(82,90)(83,91)(84,92), (1,97,15,111)(2,98,16,112)(3,99,17,85)(4,100,18,86)(5,101,19,87)(6,102,20,88)(7,103,21,89)(8,104,22,90)(9,105,23,91)(10,106,24,92)(11,107,25,93)(12,108,26,94)(13,109,27,95)(14,110,28,96)(29,71,43,57)(30,72,44,58)(31,73,45,59)(32,74,46,60)(33,75,47,61)(34,76,48,62)(35,77,49,63)(36,78,50,64)(37,79,51,65)(38,80,52,66)(39,81,53,67)(40,82,54,68)(41,83,55,69)(42,84,56,70), (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)(85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112), (1,46)(2,45)(3,44)(4,43)(5,42)(6,41)(7,40)(8,39)(9,38)(10,37)(11,36)(12,35)(13,34)(14,33)(15,32)(16,31)(17,30)(18,29)(19,56)(20,55)(21,54)(22,53)(23,52)(24,51)(25,50)(26,49)(27,48)(28,47)(57,100)(58,99)(59,98)(60,97)(61,96)(62,95)(63,94)(64,93)(65,92)(66,91)(67,90)(68,89)(69,88)(70,87)(71,86)(72,85)(73,112)(74,111)(75,110)(76,109)(77,108)(78,107)(79,106)(80,105)(81,104)(82,103)(83,102)(84,101)>;

G:=Group( (1,47)(2,48)(3,49)(4,50)(5,51)(6,52)(7,53)(8,54)(9,55)(10,56)(11,29)(12,30)(13,31)(14,32)(15,33)(16,34)(17,35)(18,36)(19,37)(20,38)(21,39)(22,40)(23,41)(24,42)(25,43)(26,44)(27,45)(28,46)(57,93)(58,94)(59,95)(60,96)(61,97)(62,98)(63,99)(64,100)(65,101)(66,102)(67,103)(68,104)(69,105)(70,106)(71,107)(72,108)(73,109)(74,110)(75,111)(76,112)(77,85)(78,86)(79,87)(80,88)(81,89)(82,90)(83,91)(84,92), (1,97,15,111)(2,98,16,112)(3,99,17,85)(4,100,18,86)(5,101,19,87)(6,102,20,88)(7,103,21,89)(8,104,22,90)(9,105,23,91)(10,106,24,92)(11,107,25,93)(12,108,26,94)(13,109,27,95)(14,110,28,96)(29,71,43,57)(30,72,44,58)(31,73,45,59)(32,74,46,60)(33,75,47,61)(34,76,48,62)(35,77,49,63)(36,78,50,64)(37,79,51,65)(38,80,52,66)(39,81,53,67)(40,82,54,68)(41,83,55,69)(42,84,56,70), (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)(85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112), (1,46)(2,45)(3,44)(4,43)(5,42)(6,41)(7,40)(8,39)(9,38)(10,37)(11,36)(12,35)(13,34)(14,33)(15,32)(16,31)(17,30)(18,29)(19,56)(20,55)(21,54)(22,53)(23,52)(24,51)(25,50)(26,49)(27,48)(28,47)(57,100)(58,99)(59,98)(60,97)(61,96)(62,95)(63,94)(64,93)(65,92)(66,91)(67,90)(68,89)(69,88)(70,87)(71,86)(72,85)(73,112)(74,111)(75,110)(76,109)(77,108)(78,107)(79,106)(80,105)(81,104)(82,103)(83,102)(84,101) );

G=PermutationGroup([[(1,47),(2,48),(3,49),(4,50),(5,51),(6,52),(7,53),(8,54),(9,55),(10,56),(11,29),(12,30),(13,31),(14,32),(15,33),(16,34),(17,35),(18,36),(19,37),(20,38),(21,39),(22,40),(23,41),(24,42),(25,43),(26,44),(27,45),(28,46),(57,93),(58,94),(59,95),(60,96),(61,97),(62,98),(63,99),(64,100),(65,101),(66,102),(67,103),(68,104),(69,105),(70,106),(71,107),(72,108),(73,109),(74,110),(75,111),(76,112),(77,85),(78,86),(79,87),(80,88),(81,89),(82,90),(83,91),(84,92)], [(1,97,15,111),(2,98,16,112),(3,99,17,85),(4,100,18,86),(5,101,19,87),(6,102,20,88),(7,103,21,89),(8,104,22,90),(9,105,23,91),(10,106,24,92),(11,107,25,93),(12,108,26,94),(13,109,27,95),(14,110,28,96),(29,71,43,57),(30,72,44,58),(31,73,45,59),(32,74,46,60),(33,75,47,61),(34,76,48,62),(35,77,49,63),(36,78,50,64),(37,79,51,65),(38,80,52,66),(39,81,53,67),(40,82,54,68),(41,83,55,69),(42,84,56,70)], [(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),(85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112)], [(1,46),(2,45),(3,44),(4,43),(5,42),(6,41),(7,40),(8,39),(9,38),(10,37),(11,36),(12,35),(13,34),(14,33),(15,32),(16,31),(17,30),(18,29),(19,56),(20,55),(21,54),(22,53),(23,52),(24,51),(25,50),(26,49),(27,48),(28,47),(57,100),(58,99),(59,98),(60,97),(61,96),(62,95),(63,94),(64,93),(65,92),(66,91),(67,90),(68,89),(69,88),(70,87),(71,86),(72,85),(73,112),(74,111),(75,110),(76,109),(77,108),(78,107),(79,106),(80,105),(81,104),(82,103),(83,102),(84,101)]])

C2×C4○D28 is a maximal subgroup of
D14⋊C8⋊C2  D28.32D4  D2814D4  C4○D28⋊C4  C4.(C2×D28)  C424D14  (C2×D28)⋊13C4  D2817D4  D28.37D4  (C22×C8)⋊D7  C23.23D28  (C2×D28).14C4  M4(2).31D14  C23.49D28  C23.20D28  C42.276D14  C24.27D14  C14.82+ 1+4  C14.2- 1+4  C14.2+ 1+4  C42.188D14  C42.91D14  C428D14  C429D14  C42.92D14  C4212D14  C42.228D14  D2823D4  D2824D4  Dic1423D4  Dic1424D4  Dic1420D4  C14.382+ 1+4  C14.722- 1+4  D2820D4  C14.162- 1+4  C14.172- 1+4  D2822D4  Dic1422D4  C14.1212+ 1+4  C14.822- 1+4  C28.70C24  C56.9C23  C24.72D14  C24.41D14  C14.442- 1+4  C28.C24  (C2×C28)⋊17D4  C14.1082- 1+4  C2×D7×C4○D4  C14.C25
C2×C4○D28 is a maximal quotient of
C2×C4×Dic14  C42.274D14  C2×C4×D28  C42.276D14  C42.277D14  C24.27D14  C24.30D14  C24.31D14  C14.2- 1+4  C14.102+ 1+4  C14.52- 1+4  C14.112+ 1+4  C14.62- 1+4  C42.89D14  C4210D14  C42.93D14  C42.94D14  C42.95D14  C42.96D14  C42.97D14  C42.98D14  C42.99D14  C42.100D14  C42.102D14  C42.104D14  C42.105D14  C42.106D14  C4212D14  C42.228D14  D2823D4  D2824D4  Dic1423D4  Dic1424D4  C4216D14  C42.229D14  C42.113D14  C42.114D14  C4217D14  C42.115D14  C42.116D14  C42.117D14  C42.118D14  C42.119D14  Dic1410Q8  C42.122D14  C42.232D14  D2810Q8  C42.131D14  C42.132D14  C42.133D14  C42.134D14  C42.135D14  C42.136D14  C2×C4×C7⋊D4  C24.72D14

68 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I4A4B4C4D4E4F4G4H4I4J7A7B7C14A···14U28A···28X
order1222222222444444444477714···1428···28
size11112214141414111122141414142222···22···2

68 irreducible representations

dim111111122222
type++++++++++
imageC1C2C2C2C2C2C2D7C4○D4D14D14C4○D28
kernelC2×C4○D28C2×Dic14C2×C4×D7C2×D28C4○D28C2×C7⋊D4C22×C28C22×C4C14C2×C4C23C2
# reps11218213418324

Matrix representation of C2×C4○D28 in GL3(𝔽29) generated by

2800
0280
0028
,
100
0120
0012
,
100
0255
01221
,
100
02716
0272
G:=sub<GL(3,GF(29))| [28,0,0,0,28,0,0,0,28],[1,0,0,0,12,0,0,0,12],[1,0,0,0,25,12,0,5,21],[1,0,0,0,27,27,0,16,2] >;

C2×C4○D28 in GAP, Magma, Sage, TeX

C_2\times C_4\circ D_{28}
% in TeX

G:=Group("C2xC4oD28");
// GroupNames label

G:=SmallGroup(224,177);
// by ID

G=gap.SmallGroup(224,177);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-7,86,579,6917]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^4=d^2=1,c^14=b^2,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=b^2*c^13>;
// generators/relations

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