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G = C22:D28order 224 = 25·7

The semidirect product of C22 and D28 acting via D28/D14=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D14:4D4, C22:2D28, C23.15D14, C7:1C22wrC2, (C2xC4):1D14, (C2xC14):1D4, C2.7(D4xD7), D14:C4:4C2, (C2xD28):2C2, C22:C4:2D7, C14.5(C2xD4), C2.7(C2xD28), (C2xC28):1C22, (C23xD7):1C2, (C2xC14).23C23, (C2xDic7):1C22, (C22xD7):1C22, C22.41(C22xD7), (C22xC14).12C22, (C2xC7:D4):1C2, (C7xC22:C4):3C2, SmallGroup(224,77)

Series: Derived Chief Lower central Upper central

C1C2xC14 — C22:D28
C1C7C14C2xC14C22xD7C23xD7 — C22:D28
C7C2xC14 — C22:D28
C1C22C22:C4

Generators and relations for C22:D28
 G = < a,b,c,d | a2=b2=c28=d2=1, cac-1=dad=ab=ba, bc=cb, bd=db, dcd=c-1 >

Subgroups: 710 in 130 conjugacy classes, 37 normal (15 characteristic)
C1, C2, C2, C2, C4, C22, C22, C22, C7, C2xC4, C2xC4, D4, C23, C23, D7, C14, C14, C14, C22:C4, C22:C4, C2xD4, C24, Dic7, C28, D14, D14, C2xC14, C2xC14, C2xC14, C22wrC2, D28, C2xDic7, C7:D4, C2xC28, C22xD7, C22xD7, C22xD7, C22xC14, D14:C4, C7xC22:C4, C2xD28, C2xC7:D4, C23xD7, C22:D28
Quotients: C1, C2, C22, D4, C23, D7, C2xD4, D14, C22wrC2, D28, C22xD7, C2xD28, D4xD7, C22:D28

Smallest permutation representation of C22:D28
On 56 points
Generators in S56
(2 49)(4 51)(6 53)(8 55)(10 29)(12 31)(14 33)(16 35)(18 37)(20 39)(22 41)(24 43)(26 45)(28 47)
(1 48)(2 49)(3 50)(4 51)(5 52)(6 53)(7 54)(8 55)(9 56)(10 29)(11 30)(12 31)(13 32)(14 33)(15 34)(16 35)(17 36)(18 37)(19 38)(20 39)(21 40)(22 41)(23 42)(24 43)(25 44)(26 45)(27 46)(28 47)
(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)
(1 47)(2 46)(3 45)(4 44)(5 43)(6 42)(7 41)(8 40)(9 39)(10 38)(11 37)(12 36)(13 35)(14 34)(15 33)(16 32)(17 31)(18 30)(19 29)(20 56)(21 55)(22 54)(23 53)(24 52)(25 51)(26 50)(27 49)(28 48)

G:=sub<Sym(56)| (2,49)(4,51)(6,53)(8,55)(10,29)(12,31)(14,33)(16,35)(18,37)(20,39)(22,41)(24,43)(26,45)(28,47), (1,48)(2,49)(3,50)(4,51)(5,52)(6,53)(7,54)(8,55)(9,56)(10,29)(11,30)(12,31)(13,32)(14,33)(15,34)(16,35)(17,36)(18,37)(19,38)(20,39)(21,40)(22,41)(23,42)(24,43)(25,44)(26,45)(27,46)(28,47), (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), (1,47)(2,46)(3,45)(4,44)(5,43)(6,42)(7,41)(8,40)(9,39)(10,38)(11,37)(12,36)(13,35)(14,34)(15,33)(16,32)(17,31)(18,30)(19,29)(20,56)(21,55)(22,54)(23,53)(24,52)(25,51)(26,50)(27,49)(28,48)>;

G:=Group( (2,49)(4,51)(6,53)(8,55)(10,29)(12,31)(14,33)(16,35)(18,37)(20,39)(22,41)(24,43)(26,45)(28,47), (1,48)(2,49)(3,50)(4,51)(5,52)(6,53)(7,54)(8,55)(9,56)(10,29)(11,30)(12,31)(13,32)(14,33)(15,34)(16,35)(17,36)(18,37)(19,38)(20,39)(21,40)(22,41)(23,42)(24,43)(25,44)(26,45)(27,46)(28,47), (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), (1,47)(2,46)(3,45)(4,44)(5,43)(6,42)(7,41)(8,40)(9,39)(10,38)(11,37)(12,36)(13,35)(14,34)(15,33)(16,32)(17,31)(18,30)(19,29)(20,56)(21,55)(22,54)(23,53)(24,52)(25,51)(26,50)(27,49)(28,48) );

G=PermutationGroup([[(2,49),(4,51),(6,53),(8,55),(10,29),(12,31),(14,33),(16,35),(18,37),(20,39),(22,41),(24,43),(26,45),(28,47)], [(1,48),(2,49),(3,50),(4,51),(5,52),(6,53),(7,54),(8,55),(9,56),(10,29),(11,30),(12,31),(13,32),(14,33),(15,34),(16,35),(17,36),(18,37),(19,38),(20,39),(21,40),(22,41),(23,42),(24,43),(25,44),(26,45),(27,46),(28,47)], [(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)], [(1,47),(2,46),(3,45),(4,44),(5,43),(6,42),(7,41),(8,40),(9,39),(10,38),(11,37),(12,36),(13,35),(14,34),(15,33),(16,32),(17,31),(18,30),(19,29),(20,56),(21,55),(22,54),(23,53),(24,52),(25,51),(26,50),(27,49),(28,48)]])

C22:D28 is a maximal subgroup of
C23:D28  C24.27D14  C23:3D28  C42:8D14  C42:9D14  C42:10D14  C42:12D14  D4xD28  D4:5D28  C42:17D14  D7xC22wrC2  C24:3D14  C24.34D14  C14.372+ 1+4  C14.382+ 1+4  D28:19D4  C14.482+ 1+4  C4:C4:26D14  D28:21D4  C14.532+ 1+4  C14.562+ 1+4  C14.1202+ 1+4  C14.1212+ 1+4  C4:C4:28D14  C14.612+ 1+4  C14.682+ 1+4  C42:18D14  D28:10D4  C42:20D14  C42:22D14  C42:23D14  C42:24D14  C42:25D14
C22:D28 is a maximal quotient of
(C2xDic7):Q8  (C2xC4):9D28  D14:C4:C4  (C2xC28):5D4  (C2xDic7):3D4  (C2xC4).20D28  D28.31D4  D28:13D4  D28.32D4  D28:14D4  Dic14:14D4  C22:Dic28  C23:D28  C23.5D28  D28.1D4  D28:1D4  D28.4D4  D28.5D4  D4:D28  D4.6D28  D4:3D28  D4.D28  Q8:2D28  D14:4Q16  Q8.D28  D28:4D4  D4:4D28  M4(2):D14  D4.9D28  D4.10D28  C24.47D14  C23.44D28  C23.45D28  C23:2D28  C23.16D28

44 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I2J4A4B4C7A7B7C14A···14I14J···14O28A···28L
order1222222222244477714···1414···1428···28
size111122141414142844282222···24···44···4

44 irreducible representations

dim1111112222224
type+++++++++++++
imageC1C2C2C2C2C2D4D4D7D14D14D28D4xD7
kernelC22:D28D14:C4C7xC22:C4C2xD28C2xC7:D4C23xD7D14C2xC14C22:C4C2xC4C23C22C2
# reps12121142363126

Matrix representation of C22:D28 in GL4(F29) generated by

1000
0100
0010
00028
,
1000
0100
00280
00028
,
222400
252600
0001
0010
,
241700
2500
00028
00280
G:=sub<GL(4,GF(29))| [1,0,0,0,0,1,0,0,0,0,1,0,0,0,0,28],[1,0,0,0,0,1,0,0,0,0,28,0,0,0,0,28],[22,25,0,0,24,26,0,0,0,0,0,1,0,0,1,0],[24,2,0,0,17,5,0,0,0,0,0,28,0,0,28,0] >;

C22:D28 in GAP, Magma, Sage, TeX

C_2^2\rtimes D_{28}
% in TeX

G:=Group("C2^2:D28");
// GroupNames label

G:=SmallGroup(224,77);
// by ID

G=gap.SmallGroup(224,77);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-7,218,188,50,6917]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^2=c^28=d^2=1,c*a*c^-1=d*a*d=a*b=b*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations

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x
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Z
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