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## G = Q8×C22⋊C4order 128 = 27

### Direct product of Q8 and C22⋊C4

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

Series: Derived Chief Lower central Upper central Jennings

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

Generators and relations for Q8×C22⋊C4
G = < a,b,c,d,e | a4=c2=d2=e4=1, b2=a2, bab-1=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, ece-1=cd=dc, de=ed >

Subgroups: 636 in 410 conjugacy classes, 196 normal (13 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C2×C4, C2×C4, Q8, Q8, C23, C23, C23, C42, C22⋊C4, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C2×Q8, C2×Q8, C24, C2.C42, C2×C42, C2×C22⋊C4, C2×C22⋊C4, C2×C4⋊C4, C4×Q8, C23×C4, C22×Q8, C22×Q8, C22×Q8, C4×C22⋊C4, C23.7Q8, C23.67C23, C2×C4×Q8, Q8×C23, Q8×C22⋊C4
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C23, C22⋊C4, C22×C4, C2×D4, C2×Q8, C4○D4, C24, C2×C22⋊C4, C4×Q8, C23×C4, C22×D4, C22×Q8, C2×C4○D4, 2- 1+4, C22×C22⋊C4, C2×C4×Q8, C23.32C23, Q85D4, D4×Q8, Q8×C22⋊C4

Smallest permutation representation of Q8×C22⋊C4
On 64 points
Generators in S64
(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 34 3 36)(2 33 4 35)(5 58 7 60)(6 57 8 59)(9 52 11 50)(10 51 12 49)(13 56 15 54)(14 55 16 53)(17 43 19 41)(18 42 20 44)(21 31 23 29)(22 30 24 32)(25 37 27 39)(26 40 28 38)(45 63 47 61)(46 62 48 64)
(1 3)(2 4)(5 13)(6 14)(7 15)(8 16)(9 62)(10 63)(11 64)(12 61)(17 19)(18 20)(21 23)(22 24)(25 27)(26 28)(29 31)(30 32)(33 35)(34 36)(37 39)(38 40)(41 43)(42 44)(45 49)(46 50)(47 51)(48 52)(53 59)(54 60)(55 57)(56 58)
(1 24)(2 21)(3 22)(4 23)(5 15)(6 16)(7 13)(8 14)(9 64)(10 61)(11 62)(12 63)(17 27)(18 28)(19 25)(20 26)(29 35)(30 36)(31 33)(32 34)(37 41)(38 42)(39 43)(40 44)(45 51)(46 52)(47 49)(48 50)(53 57)(54 58)(55 59)(56 60)
(1 47 18 53)(2 48 19 54)(3 45 20 55)(4 46 17 56)(5 29 11 39)(6 30 12 40)(7 31 9 37)(8 32 10 38)(13 33 64 41)(14 34 61 42)(15 35 62 43)(16 36 63 44)(21 50 25 58)(22 51 26 59)(23 52 27 60)(24 49 28 57)

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

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

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

50 conjugacy classes

 class 1 2A ··· 2G 2H 2I 2J 2K 4A ··· 4T 4U ··· 4AL order 1 2 ··· 2 2 2 2 2 4 ··· 4 4 ··· 4 size 1 1 ··· 1 2 2 2 2 2 ··· 2 4 ··· 4

50 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 4 type + + + + + + - + - image C1 C2 C2 C2 C2 C2 C4 Q8 D4 C4○D4 2- 1+4 kernel Q8×C22⋊C4 C4×C22⋊C4 C23.7Q8 C23.67C23 C2×C4×Q8 Q8×C23 C22×Q8 C22⋊C4 C2×Q8 C23 C22 # reps 1 3 3 6 2 1 16 4 8 4 2

Matrix representation of Q8×C22⋊C4 in GL5(𝔽5)

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

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

Q8×C22⋊C4 in GAP, Magma, Sage, TeX

Q_8\times C_2^2\rtimes C_4
% in TeX

G:=Group("Q8xC2^2:C4");
// GroupNames label

G:=SmallGroup(128,1072);
// by ID

G=gap.SmallGroup(128,1072);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,448,253,120,758,346,80]);
// Polycyclic

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

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