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G = C22.71C25order 128 = 27

52nd central stem extension by C22 of C25

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

Aliases: C23.31C24, C22.71C25, C42.568C23, C4.1322- 1+4, Q83(C4×D4), D43(C4×Q8), C43(D4×Q8), (D4×Q8)⋊35C2, D412(C4○D4), C43(D46D4), D46D448C2, C43(Q85D4), C43(Q83Q8), Q811(C4○D4), Q85D438C2, Q83Q834C2, C4⋊C4.480C23, (C2×C4).167C24, C4⋊Q8.337C22, (C2×D4).504C23, (C4×D4).356C22, C22⋊C4.93C23, (C2×Q8).438C23, (C4×Q8).324C22, C4⋊D4.221C22, (C2×C42).938C22, C22⋊Q8.227C22, C2.17(C2×2- 1+4), C422C2.15C22, (C22×C4).1204C23, C4.4D4.172C22, C42.C2.150C22, (C22×Q8).498C22, C42(C22.46C24), C43(C22.50C24), C23.36C2324C2, C22.50C2446C2, C42⋊C2.222C22, C22.46C2445C2, C23.37C2333C2, C22.D4.28C22, (C2×C4×Q8)⋊58C2, (C4×C4○D4)⋊24C2, C4.273(C2×C4○D4), C22.40(C2×C4○D4), C2.42(C22×C4○D4), (C2×C4⋊C4).962C22, (C2×C4○D4).326C22, SmallGroup(128,2214)

Series: Derived Chief Lower central Upper central Jennings

C1C22 — C22.71C25
C1C2C22C2×C4C22×C4C2×C42C4×C4○D4 — C22.71C25
C1C22 — C22.71C25
C1C2×C4 — C22.71C25
C1C22 — C22.71C25

Generators and relations for C22.71C25
 G = < a,b,c,d,e,f,g | a2=b2=d2=f2=1, c2=e2=a, g2=ba=ab, dcd=ac=ca, fdf=ad=da, ae=ea, af=fa, ag=ga, ece-1=bc=cb, bd=db, be=eb, bf=fb, bg=gb, cf=fc, cg=gc, de=ed, dg=gd, ef=fe, eg=ge, fg=gf >

Subgroups: 700 in 547 conjugacy classes, 400 normal (20 characteristic)
C1, C2 [×3], C2 [×7], C4 [×10], C4 [×21], C22, C22 [×4], C22 [×13], C2×C4 [×3], C2×C4 [×23], C2×C4 [×46], D4 [×4], D4 [×18], Q8 [×4], Q8 [×18], C23 [×5], C42, C42 [×27], C22⋊C4 [×32], C4⋊C4, C4⋊C4 [×51], C22×C4 [×23], C2×D4, C2×D4 [×9], C2×Q8, C2×Q8 [×9], C2×Q8 [×8], C4○D4 [×16], C2×C42 [×9], C2×C4⋊C4 [×6], C42⋊C2 [×18], C4×D4, C4×D4 [×21], C4×Q8 [×2], C4×Q8 [×20], C4⋊D4 [×6], C22⋊Q8 [×18], C22.D4 [×12], C4.4D4 [×6], C42.C2 [×12], C422C2 [×12], C4⋊Q8 [×6], C22×Q8 [×2], C2×C4○D4, C2×C4○D4 [×3], C2×C4×Q8 [×2], C4×C4○D4, C4×C4○D4 [×3], C23.36C23 [×6], C23.37C23 [×3], D46D4 [×3], Q85D4 [×2], D4×Q8, C22.46C24 [×6], C22.50C24 [×3], Q83Q8, C22.71C25
Quotients: C1, C2 [×31], C22 [×155], C23 [×155], C4○D4 [×8], C24 [×31], C2×C4○D4 [×12], 2- 1+4 [×2], C25, C22×C4○D4 [×2], C2×2- 1+4, C22.71C25

Smallest permutation representation of C22.71C25
On 64 points
Generators in S64
(1 3)(2 4)(5 7)(6 8)(9 11)(10 12)(13 15)(14 16)(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 47)(46 48)(49 51)(50 52)(53 55)(54 56)(57 59)(58 60)(61 63)(62 64)
(1 51)(2 52)(3 49)(4 50)(5 36)(6 33)(7 34)(8 35)(9 53)(10 54)(11 55)(12 56)(13 57)(14 58)(15 59)(16 60)(17 61)(18 62)(19 63)(20 64)(21 37)(22 38)(23 39)(24 40)(25 41)(26 42)(27 43)(28 44)(29 45)(30 46)(31 47)(32 48)
(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 37)(6 40)(7 39)(8 38)(9 15)(10 14)(11 13)(12 16)(21 36)(22 35)(23 34)(24 33)(25 31)(26 30)(27 29)(28 32)(41 47)(42 46)(43 45)(44 48)(49 61)(50 64)(51 63)(52 62)(53 59)(54 58)(55 57)(56 60)
(1 13 3 15)(2 58 4 60)(5 41 7 43)(6 26 8 28)(9 19 11 17)(10 64 12 62)(14 50 16 52)(18 54 20 56)(21 31 23 29)(22 48 24 46)(25 34 27 36)(30 38 32 40)(33 42 35 44)(37 47 39 45)(49 59 51 57)(53 63 55 61)
(1 11)(2 12)(3 9)(4 10)(5 45)(6 46)(7 47)(8 48)(13 17)(14 18)(15 19)(16 20)(21 25)(22 26)(23 27)(24 28)(29 36)(30 33)(31 34)(32 35)(37 41)(38 42)(39 43)(40 44)(49 53)(50 54)(51 55)(52 56)(57 61)(58 62)(59 63)(60 64)
(1 21 49 39)(2 22 50 40)(3 23 51 37)(4 24 52 38)(5 17 34 63)(6 18 35 64)(7 19 36 61)(8 20 33 62)(9 27 55 41)(10 28 56 42)(11 25 53 43)(12 26 54 44)(13 31 59 45)(14 32 60 46)(15 29 57 47)(16 30 58 48)

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

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

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

50 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I2J4A4B4C4D4E···4X4Y···4AM
order1222222222244444···44···4
size1111222244411112···24···4

50 irreducible representations

dim11111111111224
type+++++++++++-
imageC1C2C2C2C2C2C2C2C2C2C2C4○D4C4○D42- 1+4
kernelC22.71C25C2×C4×Q8C4×C4○D4C23.36C23C23.37C23D46D4Q85D4D4×Q8C22.46C24C22.50C24Q83Q8D4Q8C4
# reps12463321631882

Matrix representation of C22.71C25 in GL4(𝔽5) generated by

1000
0100
0040
0004
,
4000
0400
0010
0001
,
0100
1000
0024
0003
,
4000
0400
0024
0033
,
4000
0100
0020
0002
,
1000
0100
0012
0004
,
2000
0200
0020
0002
G:=sub<GL(4,GF(5))| [1,0,0,0,0,1,0,0,0,0,4,0,0,0,0,4],[4,0,0,0,0,4,0,0,0,0,1,0,0,0,0,1],[0,1,0,0,1,0,0,0,0,0,2,0,0,0,4,3],[4,0,0,0,0,4,0,0,0,0,2,3,0,0,4,3],[4,0,0,0,0,1,0,0,0,0,2,0,0,0,0,2],[1,0,0,0,0,1,0,0,0,0,1,0,0,0,2,4],[2,0,0,0,0,2,0,0,0,0,2,0,0,0,0,2] >;

C22.71C25 in GAP, Magma, Sage, TeX

C_2^2._{71}C_2^5
% in TeX

G:=Group("C2^2.71C2^5");
// GroupNames label

G:=SmallGroup(128,2214);
// by ID

G=gap.SmallGroup(128,2214);
# by ID

G:=PCGroup([7,-2,2,2,2,2,-2,2,224,477,1430,184,570,242]);
// Polycyclic

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

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