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G = C2×C22⋊Q16order 128 = 27

Direct product of C2 and C22⋊Q16

direct product, p-group, metabelian, nilpotent (class 3), monomial

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C2×C4 — C2×C22⋊Q16
 Chief series C1 — C2 — C22 — C2×C4 — C22×C4 — C23×C4 — Q8×C23 — C2×C22⋊Q16
 Lower central C1 — C2 — C2×C4 — C2×C22⋊Q16
 Upper central C1 — C23 — C23×C4 — C2×C22⋊Q16
 Jennings C1 — C2 — C2 — C2×C4 — C2×C22⋊Q16

Generators and relations for C2×C22⋊Q16
G = < a,b,c,d,e | a2=b2=c2=d8=1, e2=d4, ab=ba, ac=ca, ad=da, ae=ea, dbd-1=bc=cb, be=eb, cd=dc, ce=ec, ede-1=d-1 >

Subgroups: 636 in 370 conjugacy classes, 124 normal (18 characteristic)
C1, C2 [×3], C2 [×4], C2 [×4], C4 [×4], C4 [×14], C22, C22 [×10], C22 [×12], C8 [×4], C2×C4 [×2], C2×C4 [×6], C2×C4 [×46], Q8 [×8], Q8 [×34], C23, C23 [×6], C23 [×4], C22⋊C4 [×4], C4⋊C4 [×2], C4⋊C4 [×5], C2×C8 [×4], C2×C8 [×4], Q16 [×16], C22×C4 [×2], C22×C4 [×4], C22×C4 [×20], C2×Q8 [×14], C2×Q8 [×55], C24, C22⋊C8 [×4], Q8⋊C4 [×8], C2×C22⋊C4, C2×C4⋊C4, C2×C4⋊C4, C22⋊Q8 [×4], C22⋊Q8 [×2], C22×C8 [×2], C2×Q16 [×8], C2×Q16 [×8], C23×C4, C23×C4, C22×Q8, C22×Q8 [×6], C22×Q8 [×11], C2×C22⋊C8, C2×Q8⋊C4 [×2], C22⋊Q16 [×8], C2×C22⋊Q8, C22×Q16 [×2], Q8×C23, C2×C22⋊Q16
Quotients: C1, C2 [×15], C22 [×35], D4 [×12], C23 [×15], Q16 [×4], C2×D4 [×18], C24, C22≀C2 [×4], C2×Q16 [×6], C8.C22 [×2], C22×D4 [×3], C22⋊Q16 [×4], C2×C22≀C2, C22×Q16, C2×C8.C22, C2×C22⋊Q16

Smallest permutation representation of C2×C22⋊Q16
On 64 points
Generators in S64
(1 52)(2 53)(3 54)(4 55)(5 56)(6 49)(7 50)(8 51)(9 30)(10 31)(11 32)(12 25)(13 26)(14 27)(15 28)(16 29)(17 60)(18 61)(19 62)(20 63)(21 64)(22 57)(23 58)(24 59)(33 48)(34 41)(35 42)(36 43)(37 44)(38 45)(39 46)(40 47)
(1 5)(2 64)(3 7)(4 58)(6 60)(8 62)(9 13)(10 35)(11 15)(12 37)(14 39)(16 33)(17 49)(18 22)(19 51)(20 24)(21 53)(23 55)(25 44)(26 30)(27 46)(28 32)(29 48)(31 42)(34 38)(36 40)(41 45)(43 47)(50 54)(52 56)(57 61)(59 63)
(1 59)(2 60)(3 61)(4 62)(5 63)(6 64)(7 57)(8 58)(9 38)(10 39)(11 40)(12 33)(13 34)(14 35)(15 36)(16 37)(17 53)(18 54)(19 55)(20 56)(21 49)(22 50)(23 51)(24 52)(25 48)(26 41)(27 42)(28 43)(29 44)(30 45)(31 46)(32 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)(57 58 59 60 61 62 63 64)
(1 36 5 40)(2 35 6 39)(3 34 7 38)(4 33 8 37)(9 61 13 57)(10 60 14 64)(11 59 15 63)(12 58 16 62)(17 27 21 31)(18 26 22 30)(19 25 23 29)(20 32 24 28)(41 50 45 54)(42 49 46 53)(43 56 47 52)(44 55 48 51)

G:=sub<Sym(64)| (1,52)(2,53)(3,54)(4,55)(5,56)(6,49)(7,50)(8,51)(9,30)(10,31)(11,32)(12,25)(13,26)(14,27)(15,28)(16,29)(17,60)(18,61)(19,62)(20,63)(21,64)(22,57)(23,58)(24,59)(33,48)(34,41)(35,42)(36,43)(37,44)(38,45)(39,46)(40,47), (1,5)(2,64)(3,7)(4,58)(6,60)(8,62)(9,13)(10,35)(11,15)(12,37)(14,39)(16,33)(17,49)(18,22)(19,51)(20,24)(21,53)(23,55)(25,44)(26,30)(27,46)(28,32)(29,48)(31,42)(34,38)(36,40)(41,45)(43,47)(50,54)(52,56)(57,61)(59,63), (1,59)(2,60)(3,61)(4,62)(5,63)(6,64)(7,57)(8,58)(9,38)(10,39)(11,40)(12,33)(13,34)(14,35)(15,36)(16,37)(17,53)(18,54)(19,55)(20,56)(21,49)(22,50)(23,51)(24,52)(25,48)(26,41)(27,42)(28,43)(29,44)(30,45)(31,46)(32,47), (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)(57,58,59,60,61,62,63,64), (1,36,5,40)(2,35,6,39)(3,34,7,38)(4,33,8,37)(9,61,13,57)(10,60,14,64)(11,59,15,63)(12,58,16,62)(17,27,21,31)(18,26,22,30)(19,25,23,29)(20,32,24,28)(41,50,45,54)(42,49,46,53)(43,56,47,52)(44,55,48,51)>;

G:=Group( (1,52)(2,53)(3,54)(4,55)(5,56)(6,49)(7,50)(8,51)(9,30)(10,31)(11,32)(12,25)(13,26)(14,27)(15,28)(16,29)(17,60)(18,61)(19,62)(20,63)(21,64)(22,57)(23,58)(24,59)(33,48)(34,41)(35,42)(36,43)(37,44)(38,45)(39,46)(40,47), (1,5)(2,64)(3,7)(4,58)(6,60)(8,62)(9,13)(10,35)(11,15)(12,37)(14,39)(16,33)(17,49)(18,22)(19,51)(20,24)(21,53)(23,55)(25,44)(26,30)(27,46)(28,32)(29,48)(31,42)(34,38)(36,40)(41,45)(43,47)(50,54)(52,56)(57,61)(59,63), (1,59)(2,60)(3,61)(4,62)(5,63)(6,64)(7,57)(8,58)(9,38)(10,39)(11,40)(12,33)(13,34)(14,35)(15,36)(16,37)(17,53)(18,54)(19,55)(20,56)(21,49)(22,50)(23,51)(24,52)(25,48)(26,41)(27,42)(28,43)(29,44)(30,45)(31,46)(32,47), (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)(57,58,59,60,61,62,63,64), (1,36,5,40)(2,35,6,39)(3,34,7,38)(4,33,8,37)(9,61,13,57)(10,60,14,64)(11,59,15,63)(12,58,16,62)(17,27,21,31)(18,26,22,30)(19,25,23,29)(20,32,24,28)(41,50,45,54)(42,49,46,53)(43,56,47,52)(44,55,48,51) );

G=PermutationGroup([(1,52),(2,53),(3,54),(4,55),(5,56),(6,49),(7,50),(8,51),(9,30),(10,31),(11,32),(12,25),(13,26),(14,27),(15,28),(16,29),(17,60),(18,61),(19,62),(20,63),(21,64),(22,57),(23,58),(24,59),(33,48),(34,41),(35,42),(36,43),(37,44),(38,45),(39,46),(40,47)], [(1,5),(2,64),(3,7),(4,58),(6,60),(8,62),(9,13),(10,35),(11,15),(12,37),(14,39),(16,33),(17,49),(18,22),(19,51),(20,24),(21,53),(23,55),(25,44),(26,30),(27,46),(28,32),(29,48),(31,42),(34,38),(36,40),(41,45),(43,47),(50,54),(52,56),(57,61),(59,63)], [(1,59),(2,60),(3,61),(4,62),(5,63),(6,64),(7,57),(8,58),(9,38),(10,39),(11,40),(12,33),(13,34),(14,35),(15,36),(16,37),(17,53),(18,54),(19,55),(20,56),(21,49),(22,50),(23,51),(24,52),(25,48),(26,41),(27,42),(28,43),(29,44),(30,45),(31,46),(32,47)], [(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),(57,58,59,60,61,62,63,64)], [(1,36,5,40),(2,35,6,39),(3,34,7,38),(4,33,8,37),(9,61,13,57),(10,60,14,64),(11,59,15,63),(12,58,16,62),(17,27,21,31),(18,26,22,30),(19,25,23,29),(20,32,24,28),(41,50,45,54),(42,49,46,53),(43,56,47,52),(44,55,48,51)])

38 conjugacy classes

 class 1 2A ··· 2G 2H 2I 2J 2K 4A 4B 4C 4D 4E ··· 4N 4O 4P 4Q 4R 8A ··· 8H order 1 2 ··· 2 2 2 2 2 4 4 4 4 4 ··· 4 4 4 4 4 8 ··· 8 size 1 1 ··· 1 2 2 2 2 2 2 2 2 4 ··· 4 8 8 8 8 4 ··· 4

38 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 4 type + + + + + + + + + + - - image C1 C2 C2 C2 C2 C2 C2 D4 D4 D4 Q16 C8.C22 kernel C2×C22⋊Q16 C2×C22⋊C8 C2×Q8⋊C4 C22⋊Q16 C2×C22⋊Q8 C22×Q16 Q8×C23 C22×C4 C2×Q8 C24 C23 C22 # reps 1 1 2 8 1 2 1 3 8 1 8 2

Matrix representation of C2×C22⋊Q16 in GL5(𝔽17)

 16 0 0 0 0 0 16 0 0 0 0 0 16 0 0 0 0 0 16 0 0 0 0 0 16
,
 16 0 0 0 0 0 16 0 0 0 0 0 16 0 0 0 0 0 1 0 0 0 0 16 16
,
 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 16 0 0 0 0 0 16
,
 1 0 0 0 0 0 2 0 0 0 0 0 9 0 0 0 0 0 16 15 0 0 0 0 1
,
 1 0 0 0 0 0 0 1 0 0 0 16 0 0 0 0 0 0 1 0 0 0 0 0 1

G:=sub<GL(5,GF(17))| [16,0,0,0,0,0,16,0,0,0,0,0,16,0,0,0,0,0,16,0,0,0,0,0,16],[16,0,0,0,0,0,16,0,0,0,0,0,16,0,0,0,0,0,1,16,0,0,0,0,16],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,16,0,0,0,0,0,16],[1,0,0,0,0,0,2,0,0,0,0,0,9,0,0,0,0,0,16,0,0,0,0,15,1],[1,0,0,0,0,0,0,16,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,1] >;

C2×C22⋊Q16 in GAP, Magma, Sage, TeX

C_2\times C_2^2\rtimes Q_{16}
% in TeX

G:=Group("C2xC2^2:Q16");
// GroupNames label

G:=SmallGroup(128,1731);
// by ID

G=gap.SmallGroup(128,1731);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,448,253,456,758,2804,1411,172]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^2=c^2=d^8=1,e^2=d^4,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,d*b*d^-1=b*c=c*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=d^-1>;
// generators/relations

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