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## G = C22×C22⋊Q8order 128 = 27

### Direct product of C22 and C22⋊Q8

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C22 — C22×C22⋊Q8
 Chief series C1 — C2 — C22 — C23 — C24 — C25 — C24×C4 — C22×C22⋊Q8
 Lower central C1 — C22 — C22×C22⋊Q8
 Upper central C1 — C24 — C22×C22⋊Q8
 Jennings C1 — C22 — C22×C22⋊Q8

Generators and relations for C22×C22⋊Q8
G = < a,b,c,d,e,f | a2=b2=c2=d2=e4=1, f2=e2, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, fcf-1=cd=dc, ce=ec, de=ed, df=fd, fef-1=e-1 >

Subgroups: 1324 in 920 conjugacy classes, 516 normal (13 characteristic)
C1, C2 [×3], C2 [×12], C2 [×8], C4 [×8], C4 [×20], C22, C22 [×42], C22 [×56], C2×C4 [×48], C2×C4 [×92], Q8 [×32], C23 [×43], C23 [×56], C22⋊C4 [×32], C4⋊C4 [×48], C22×C4 [×60], C22×C4 [×68], C2×Q8 [×16], C2×Q8 [×48], C24, C24 [×14], C24 [×8], C2×C22⋊C4 [×24], C2×C4⋊C4 [×36], C22⋊Q8 [×64], C23×C4 [×2], C23×C4 [×16], C23×C4 [×8], C22×Q8 [×12], C22×Q8 [×8], C25, C22×C22⋊C4 [×2], C22×C4⋊C4, C22×C4⋊C4 [×2], C2×C22⋊Q8 [×24], C24×C4, Q8×C23, C22×C22⋊Q8
Quotients: C1, C2 [×31], C22 [×155], D4 [×8], Q8 [×8], C23 [×155], C2×D4 [×28], C2×Q8 [×28], C4○D4 [×4], C24 [×31], C22⋊Q8 [×16], C22×D4 [×14], C22×Q8 [×14], C2×C4○D4 [×6], C25, C2×C22⋊Q8 [×12], D4×C23, Q8×C23, C22×C4○D4, C22×C22⋊Q8

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

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

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

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

56 conjugacy classes

 class 1 2A ··· 2O 2P ··· 2W 4A ··· 4P 4Q ··· 4AF order 1 2 ··· 2 2 ··· 2 4 ··· 4 4 ··· 4 size 1 1 ··· 1 2 ··· 2 2 ··· 2 4 ··· 4

56 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 type + + + + + + + - image C1 C2 C2 C2 C2 C2 D4 Q8 C4○D4 kernel C22×C22⋊Q8 C22×C22⋊C4 C22×C4⋊C4 C2×C22⋊Q8 C24×C4 Q8×C23 C22×C4 C24 C23 # reps 1 2 3 24 1 1 8 8 8

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

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

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

C22×C22⋊Q8 in GAP, Magma, Sage, TeX

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

G:=Group("C2^2xC2^2:Q8");
// GroupNames label

G:=SmallGroup(128,2165);
// by ID

G=gap.SmallGroup(128,2165);
# by ID

G:=PCGroup([7,-2,2,2,2,2,-2,2,224,477,232,1430]);
// Polycyclic

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

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