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G = D7×SD16order 224 = 25·7

Direct product of D7 and SD16

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

Aliases: D7×SD16, C85D14, Q81D14, C565C22, D4.2D14, D14.13D4, C28.4C23, Dic7.4D4, D28.2C22, Dic142C22, (D4×D7).C2, (C8×D7)⋊4C2, C7⋊C86C22, Q8⋊D71C2, (Q8×D7)⋊1C2, C72(C2×SD16), C56⋊C25C2, D4.D73C2, C2.18(D4×D7), (C7×SD16)⋊3C2, C14.30(C2×D4), (C7×Q8)⋊1C22, C4.4(C22×D7), (C4×D7).9C22, (C7×D4).2C22, SmallGroup(224,108)

Series: Derived Chief Lower central Upper central

C1C28 — D7×SD16
C1C7C14C28C4×D7D4×D7 — D7×SD16
C7C14C28 — D7×SD16
C1C2C4SD16

Generators and relations for D7×SD16
 G = < a,b,c,d | a7=b2=c8=d2=1, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd=c3 >

Subgroups: 342 in 68 conjugacy classes, 29 normal (27 characteristic)
C1, C2, C2 [×4], C4, C4 [×3], C22 [×5], C7, C8, C8, C2×C4 [×2], D4, D4 [×2], Q8, Q8 [×2], C23, D7 [×2], D7, C14, C14, C2×C8, SD16, SD16 [×3], C2×D4, C2×Q8, Dic7, Dic7, C28, C28, D14, D14 [×3], C2×C14, C2×SD16, C7⋊C8, C56, Dic14, Dic14, C4×D7, C4×D7, D28, C7⋊D4, C7×D4, C7×Q8, C22×D7, C8×D7, C56⋊C2, D4.D7, Q8⋊D7, C7×SD16, D4×D7, Q8×D7, D7×SD16
Quotients: C1, C2 [×7], C22 [×7], D4 [×2], C23, D7, SD16 [×2], C2×D4, D14 [×3], C2×SD16, C22×D7, D4×D7, D7×SD16

Smallest permutation representation of D7×SD16
On 56 points
Generators in S56
(1 49 39 12 23 45 27)(2 50 40 13 24 46 28)(3 51 33 14 17 47 29)(4 52 34 15 18 48 30)(5 53 35 16 19 41 31)(6 54 36 9 20 42 32)(7 55 37 10 21 43 25)(8 56 38 11 22 44 26)
(1 31)(2 32)(3 25)(4 26)(5 27)(6 28)(7 29)(8 30)(9 13)(10 14)(11 15)(12 16)(17 37)(18 38)(19 39)(20 40)(21 33)(22 34)(23 35)(24 36)(41 49)(42 50)(43 51)(44 52)(45 53)(46 54)(47 55)(48 56)
(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)
(2 4)(3 7)(6 8)(9 11)(10 14)(13 15)(17 21)(18 24)(20 22)(25 29)(26 32)(28 30)(33 37)(34 40)(36 38)(42 44)(43 47)(46 48)(50 52)(51 55)(54 56)

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

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

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

D7×SD16 is a maximal subgroup of
D28.29D4  D811D14  D86D14  C56.C23
D7×SD16 is a maximal quotient of
Dic76SD16  Dic7.SD16  D4⋊Dic14  Dic142D4  D4.6D28  D14.SD16  D14⋊SD16  Dic77SD16  Q8⋊Dic14  Dic7.1Q16  D14.1SD16  Q82D28  D142SD16  Dic7⋊SD16  Dic78SD16  Dic14⋊Q8  C565Q8  D14.2SD16  D14.4SD16  C88D28  D28⋊Q8  Dic73SD16  Dic75SD16  D146SD16  Dic147D4  C5614D4  C5615D4

35 conjugacy classes

class 1 2A2B2C2D2E4A4B4C4D7A7B7C8A8B8C8D14A14B14C14D14E14F28A28B28C28D28E28F56A···56F
order1222224444777888814141414141428282828282856···56
size11477282414282222214142228884448884···4

35 irreducible representations

dim11111111222222244
type+++++++++++++++
imageC1C2C2C2C2C2C2C2D4D4D7SD16D14D14D14D4×D7D7×SD16
kernelD7×SD16C8×D7C56⋊C2D4.D7Q8⋊D7C7×SD16D4×D7Q8×D7Dic7D14SD16D7C8D4Q8C2C1
# reps11111111113433336

Matrix representation of D7×SD16 in GL4(𝔽113) generated by

0100
112900
0010
0001
,
0100
1000
001120
000112
,
112000
011200
002622
00360
,
1000
0100
0010
0081112
G:=sub<GL(4,GF(113))| [0,112,0,0,1,9,0,0,0,0,1,0,0,0,0,1],[0,1,0,0,1,0,0,0,0,0,112,0,0,0,0,112],[112,0,0,0,0,112,0,0,0,0,26,36,0,0,22,0],[1,0,0,0,0,1,0,0,0,0,1,81,0,0,0,112] >;

D7×SD16 in GAP, Magma, Sage, TeX

D_7\times {\rm SD}_{16}
% in TeX

G:=Group("D7xSD16");
// GroupNames label

G:=SmallGroup(224,108);
// by ID

G=gap.SmallGroup(224,108);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-7,116,86,297,159,69,6917]);
// Polycyclic

G:=Group<a,b,c,d|a^7=b^2=c^8=d^2=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^3>;
// generators/relations

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