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## G = C22×2- 1+4order 128 = 27

### Direct product of C22 and 2- 1+4

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

Aliases: C22×2- 1+4, C2.5C26, C4.16C25, D4.11C24, Q8.11C24, C22.19C25, C24.621C23, C23.283C24, D4(C22×Q8), Q8(C22×D4), C4○D48C23, (Q8×C23)⋊15C2, (C2×Q8)⋊23C23, (C2×C4).618C24, (C2×D4).509C23, (C22×Q8)⋊71C22, (C23×C4).625C22, (C22×C4).1594C23, (C22×D4).617C22, (C2×D4)2(C2×Q8), (C2×Q8)2(C22×D4), (C2×C4○D4)⋊80C22, (C22×C4○D4)⋊29C2, SmallGroup(128,2324)

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C2 — C22×2- 1+4
 Chief series C1 — C2 — C22 — C23 — C24 — C23×C4 — Q8×C23 — C22×2- 1+4
 Lower central C1 — C2 — C22×2- 1+4
 Upper central C1 — C23 — C22×2- 1+4
 Jennings C1 — C2 — C22×2- 1+4

Generators and relations for C22×2- 1+4
G = < a,b,c,d,e,f | a2=b2=c4=d2=1, e2=f2=c2, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, dcd=c-1, ce=ec, cf=fc, de=ed, df=fd, fef-1=c2e >

Subgroups: 3036 in 2936 conjugacy classes, 2836 normal (4 characteristic)
C1, C2, C2, C2, C4, C22, C22, C2×C4, D4, Q8, C23, C23, C23, C22×C4, C2×D4, C2×Q8, C4○D4, C24, C23×C4, C22×D4, C22×Q8, C2×C4○D4, 2- 1+4, Q8×C23, C22×C4○D4, C2×2- 1+4, C22×2- 1+4
Quotients: C1, C2, C22, C23, C24, 2- 1+4, C25, C2×2- 1+4, C26, C22×2- 1+4

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

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

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

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

68 conjugacy classes

 class 1 2A ··· 2G 2H ··· 2AA 4A ··· 4AN order 1 2 ··· 2 2 ··· 2 4 ··· 4 size 1 1 ··· 1 2 ··· 2 2 ··· 2

68 irreducible representations

 dim 1 1 1 1 4 type + + + + - image C1 C2 C2 C2 2- 1+4 kernel C22×2- 1+4 Q8×C23 C22×C4○D4 C2×2- 1+4 C22 # reps 1 5 10 48 4

Matrix representation of C22×2- 1+4 in GL6(𝔽5)

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

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

C22×2- 1+4 in GAP, Magma, Sage, TeX

C_2^2\times 2_-^{1+4}
% in TeX

G:=Group("C2^2xES-(2,2)");
// GroupNames label

G:=SmallGroup(128,2324);
// by ID

G=gap.SmallGroup(128,2324);
# by ID

G:=PCGroup([7,-2,2,2,2,2,2,-2,925,456,723,352,2019]);
// Polycyclic

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

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