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## G = C4×C22⋊Q8order 128 = 27

### Direct product of C4 and C22⋊Q8

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C22 — C4×C22⋊Q8
 Chief series C1 — C2 — C22 — C23 — C22×C4 — C23×C4 — C22×C42 — C4×C22⋊Q8
 Lower central C1 — C22 — C4×C22⋊Q8
 Upper central C1 — C22×C4 — C4×C22⋊Q8
 Jennings C1 — C23 — C4×C22⋊Q8

Generators and relations for C4×C22⋊Q8
G = < a,b,c,d,e | a4=b2=c2=d4=1, e2=d2, ab=ba, ac=ca, ad=da, ae=ea, ebe-1=bc=cb, bd=db, cd=dc, ce=ec, ede-1=d-1 >

Subgroups: 492 in 328 conjugacy classes, 172 normal (42 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C2×C4, C2×C4, Q8, C23, C23, C23, C42, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C22×C4, C2×Q8, C2×Q8, C24, C2.C42, C2×C42, C2×C42, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C2×C4⋊C4, C4×Q8, C22⋊Q8, C23×C4, C22×Q8, C4×C22⋊C4, C4×C4⋊C4, C4×C4⋊C4, C23.7Q8, C23.8Q8, C23.63C23, C23.65C23, C23.67C23, C22×C42, C2×C4×Q8, C2×C22⋊Q8, C4×C22⋊Q8
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C23, C22×C4, C2×D4, C2×Q8, C4○D4, C24, C4×D4, C4×Q8, C22⋊Q8, C23×C4, C22×D4, C22×Q8, C2×C4○D4, C2×C4×D4, C2×C4×Q8, C4×C4○D4, C2×C22⋊Q8, C22.19C24, C23.36C23, C23.37C23, C4×C22⋊Q8

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

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

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

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

56 conjugacy classes

 class 1 2A ··· 2G 2H 2I 2J 2K 4A ··· 4H 4I ··· 4AB 4AC ··· 4AR order 1 2 ··· 2 2 2 2 2 4 ··· 4 4 ··· 4 4 ··· 4 size 1 1 ··· 1 2 2 2 2 1 ··· 1 2 ··· 2 4 ··· 4

56 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 type + + + + + + + + + + + + - image C1 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 C4 D4 Q8 C4○D4 C4○D4 kernel C4×C22⋊Q8 C4×C22⋊C4 C4×C4⋊C4 C23.7Q8 C23.8Q8 C23.63C23 C23.65C23 C23.67C23 C22×C42 C2×C4×Q8 C2×C22⋊Q8 C22⋊Q8 C42 C22×C4 C2×C4 C23 # reps 1 2 3 1 2 2 1 1 1 1 1 16 4 4 12 4

Matrix representation of C4×C22⋊Q8 in GL6(𝔽5)

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

G:=sub<GL(6,GF(5))| [3,0,0,0,0,0,0,3,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[2,0,0,0,0,0,0,3,0,0,0,0,0,0,3,0,0,0,0,0,0,2,0,0,0,0,0,0,3,0,0,0,0,0,0,2],[0,3,0,0,0,0,3,0,0,0,0,0,0,0,0,4,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,1,0] >;

C4×C22⋊Q8 in GAP, Magma, Sage, TeX

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

G:=Group("C4xC2^2:Q8");
// GroupNames label

G:=SmallGroup(128,1034);
// by ID

G=gap.SmallGroup(128,1034);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,448,253,120,758,304]);
// Polycyclic

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

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