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## G = C2×Q8⋊5D4order 128 = 27

### Direct product of C2 and Q8⋊5D4

direct product, p-group, metabelian, nilpotent (class 2), monomial

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C22 — C2×Q8⋊5D4
 Chief series C1 — C2 — C22 — C23 — C22×C4 — C23×C4 — Q8×C23 — C2×Q8⋊5D4
 Lower central C1 — C22 — C2×Q8⋊5D4
 Upper central C1 — C23 — C2×Q8⋊5D4
 Jennings C1 — C22 — C2×Q8⋊5D4

Generators and relations for C2×Q85D4
G = < a,b,c,d,e | a2=b4=d4=e2=1, c2=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=b-1, bd=db, be=eb, dcd-1=ece=b2c, ede=d-1 >

Subgroups: 1212 in 810 conjugacy classes, 444 normal (15 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C2×C4, C2×C4, D4, Q8, Q8, C23, C23, C23, C42, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C2×Q8, C4○D4, C24, C24, C2×C42, C2×C22⋊C4, C2×C22⋊C4, C2×C4⋊C4, C4×D4, C4×Q8, C4⋊D4, C22⋊Q8, C4.4D4, C23×C4, C22×D4, C22×Q8, C22×Q8, C22×Q8, C2×C4○D4, C2×C4○D4, C2×C4×D4, C2×C4×Q8, C2×C4⋊D4, C2×C22⋊Q8, C2×C4.4D4, Q85D4, Q8×C23, C22×C4○D4, C2×Q85D4
Quotients: C1, C2, C22, D4, C23, C2×D4, C4○D4, C24, C22×D4, C2×C4○D4, 2- 1+4, C25, Q85D4, D4×C23, C22×C4○D4, C2×2- 1+4, C2×Q85D4

Smallest permutation representation of C2×Q85D4
On 64 points
Generators in S64
(1 7)(2 8)(3 5)(4 6)(9 24)(10 21)(11 22)(12 23)(13 49)(14 50)(15 51)(16 52)(17 38)(18 39)(19 40)(20 37)(25 32)(26 29)(27 30)(28 31)(33 54)(34 55)(35 56)(36 53)(41 46)(42 47)(43 48)(44 45)(57 62)(58 63)(59 64)(60 61)
(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 33 3 35)(2 36 4 34)(5 56 7 54)(6 55 8 53)(9 42 11 44)(10 41 12 43)(13 17 15 19)(14 20 16 18)(21 46 23 48)(22 45 24 47)(25 64 27 62)(26 63 28 61)(29 58 31 60)(30 57 32 59)(37 52 39 50)(38 51 40 49)
(1 52 24 62)(2 49 21 63)(3 50 22 64)(4 51 23 61)(5 14 11 59)(6 15 12 60)(7 16 9 57)(8 13 10 58)(17 43 31 55)(18 44 32 56)(19 41 29 53)(20 42 30 54)(25 35 39 45)(26 36 40 46)(27 33 37 47)(28 34 38 48)
(1 43)(2 44)(3 41)(4 42)(5 46)(6 47)(7 48)(8 45)(9 34)(10 35)(11 36)(12 33)(13 39)(14 40)(15 37)(16 38)(17 52)(18 49)(19 50)(20 51)(21 56)(22 53)(23 54)(24 55)(25 58)(26 59)(27 60)(28 57)(29 64)(30 61)(31 62)(32 63)

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

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

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

50 conjugacy classes

 class 1 2A ··· 2G 2H 2I 2J 2K 2L ··· 2Q 4A ··· 4T 4U ··· 4AF order 1 2 ··· 2 2 2 2 2 2 ··· 2 4 ··· 4 4 ··· 4 size 1 1 ··· 1 2 2 2 2 4 ··· 4 2 ··· 2 4 ··· 4

50 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 2 4 type + + + + + + + + + + - image C1 C2 C2 C2 C2 C2 C2 C2 C2 D4 C4○D4 2- 1+4 kernel C2×Q8⋊5D4 C2×C4×D4 C2×C4×Q8 C2×C4⋊D4 C2×C22⋊Q8 C2×C4.4D4 Q8⋊5D4 Q8×C23 C22×C4○D4 C2×Q8 C23 C22 # reps 1 3 1 3 3 3 16 1 1 8 8 2

Matrix representation of C2×Q85D4 in GL5(𝔽5)

 4 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 4 0 0 0 0 0 4
,
 1 0 0 0 0 0 4 0 0 0 0 0 4 0 0 0 0 0 4 4 0 0 0 2 1
,
 4 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 3 3 0 0 0 0 2
,
 4 0 0 0 0 0 0 1 0 0 0 4 0 0 0 0 0 0 2 2 0 0 0 1 3
,
 4 0 0 0 0 0 1 0 0 0 0 0 4 0 0 0 0 0 3 3 0 0 0 4 2

G:=sub<GL(5,GF(5))| [4,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,4,0,0,0,0,0,4],[1,0,0,0,0,0,4,0,0,0,0,0,4,0,0,0,0,0,4,2,0,0,0,4,1],[4,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,3,0,0,0,0,3,2],[4,0,0,0,0,0,0,4,0,0,0,1,0,0,0,0,0,0,2,1,0,0,0,2,3],[4,0,0,0,0,0,1,0,0,0,0,0,4,0,0,0,0,0,3,4,0,0,0,3,2] >;

C2×Q85D4 in GAP, Magma, Sage, TeX

C_2\times Q_8\rtimes_5D_4
% in TeX

G:=Group("C2xQ8:5D4");
// GroupNames label

G:=SmallGroup(128,2197);
// by ID

G=gap.SmallGroup(128,2197);
# by ID

G:=PCGroup([7,-2,2,2,2,2,-2,2,477,232,1430,570,136]);
// Polycyclic

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

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