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G = C22.107C25order 128 = 27

88th central stem extension by C22 of C25

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

Aliases: C23.54C24, C42.98C23, C22.107C25, C4.1102- 1+4, Q8210C2, (D4×Q8)⋊23C2, Q85D422C2, (C2×C4).97C24, Q82(C4.4D4), C4⋊C4.500C23, Q8.42(C4○D4), C4⋊Q8.348C22, (C2×D4).479C23, (C4×D4).241C22, C22⋊C4.30C23, (C4×Q8).228C22, (C2×Q8).457C23, C4⋊D4.117C22, (C2×C42).956C22, C4.4D4.99C22, C22⋊Q8.232C22, C2.31(C2×2- 1+4), C2.38(C2.C25), C422C2.20C22, (C22×C4).1211C23, C42.C2.170C22, (C22×Q8).364C22, C23.37C2345C2, C42⋊C2.236C22, C22.50C2427C2, C22.36C2419C2, C23.36C2338C2, C23.32C2317C2, C22.49C2416C2, C22.D4.32C22, (C4×C4○D4)⋊36C2, C4.280(C2×C4○D4), (C2×Q8)(C4.4D4), C2.63(C22×C4○D4), (C2×C4○D4).332C22, SmallGroup(128,2250)

Series: Derived Chief Lower central Upper central Jennings

C1C22 — C22.107C25
C1C2C22C2×C4C42C2×C42C4×C4○D4 — C22.107C25
C1C22 — C22.107C25
C1C22 — C22.107C25
C1C22 — C22.107C25

Generators and relations for C22.107C25
 G = < a,b,c,d,e,f,g | a2=b2=1, c2=d2=e2=f2=a, g2=b, ab=ba, dcd-1=gcg-1=ac=ca, fdf-1=ad=da, ae=ea, af=fa, ag=ga, ece-1=bc=cb, bd=db, be=eb, bf=fb, bg=gb, cf=fc, de=ed, dg=gd, ef=fe, eg=ge, fg=gf >

Subgroups: 660 in 505 conjugacy classes, 390 normal (20 characteristic)
C1, C2 [×3], C2 [×5], C4 [×8], C4 [×22], C22, C22 [×15], C2×C4 [×3], C2×C4 [×23], C2×C4 [×27], D4 [×14], Q8 [×4], Q8 [×20], C23 [×5], C42, C42 [×27], C22⋊C4 [×40], C4⋊C4 [×48], C22×C4 [×15], C2×D4, C2×D4 [×9], C2×Q8 [×2], C2×Q8 [×12], C2×Q8 [×8], C4○D4 [×4], C2×C42 [×3], C42⋊C2 [×21], C4×D4 [×15], C4×Q8 [×2], C4×Q8 [×23], C4⋊D4 [×9], C22⋊Q8 [×21], C22.D4 [×6], C4.4D4, C4.4D4 [×18], C42.C2 [×3], C422C2 [×18], C4⋊Q8 [×12], C22×Q8 [×2], C2×C4○D4, C4×C4○D4, C23.32C23 [×2], C23.36C23 [×3], C23.37C23 [×3], C22.36C24 [×6], Q85D4 [×2], D4×Q8, C22.49C24 [×3], C22.50C24 [×9], Q82, C22.107C25
Quotients: C1, C2 [×31], C22 [×155], C23 [×155], C4○D4 [×4], C24 [×31], C2×C4○D4 [×6], 2- 1+4 [×2], C25, C22×C4○D4, C2×2- 1+4, C2.C25, C22.107C25

Smallest permutation representation of C22.107C25
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 3 17)(2 18 4 20)(5 39 7 37)(6 38 8 40)(9 13 11 15)(10 16 12 14)(21 36 23 34)(22 35 24 33)(25 29 27 31)(26 32 28 30)(41 45 43 47)(42 48 44 46)(49 61 51 63)(50 64 52 62)(53 57 55 59)(54 60 56 58)
(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 9 3 11)(2 10 4 12)(5 45 7 47)(6 46 8 48)(13 19 15 17)(14 20 16 18)(21 27 23 25)(22 28 24 26)(29 34 31 36)(30 35 32 33)(37 43 39 41)(38 44 40 42)(49 55 51 53)(50 56 52 54)(57 63 59 61)(58 64 60 62)
(1 23 51 39)(2 22 52 38)(3 21 49 37)(4 24 50 40)(5 17 36 61)(6 20 33 64)(7 19 34 63)(8 18 35 62)(9 25 53 41)(10 28 54 44)(11 27 55 43)(12 26 56 42)(13 29 57 45)(14 32 58 48)(15 31 59 47)(16 30 60 46)

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

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

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

44 conjugacy classes

class 1 2A2B2C2D···2H4A···4R4S···4AI
order12222···24···44···4
size11114···42···24···4

44 irreducible representations

dim11111111111244
type+++++++++++-
imageC1C2C2C2C2C2C2C2C2C2C2C4○D42- 1+4C2.C25
kernelC22.107C25C4×C4○D4C23.32C23C23.36C23C23.37C23C22.36C24Q85D4D4×Q8C22.49C24C22.50C24Q82Q8C4C2
# reps11233621391822

Matrix representation of C22.107C25 in GL6(𝔽5)

100000
010000
004000
000400
000040
000004
,
400000
040000
004000
000400
000040
000004
,
340000
320000
002100
000300
000021
000003
,
400000
040000
000013
000014
001300
001400
,
100000
140000
004200
004100
000042
000041
,
400000
040000
000010
000001
004000
000400
,
200000
020000
001300
001400
000013
000014

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

C22.107C25 in GAP, Magma, Sage, TeX

C_2^2._{107}C_2^5
% in TeX

G:=Group("C2^2.107C2^5");
// GroupNames label

G:=SmallGroup(128,2250);
// by ID

G=gap.SmallGroup(128,2250);
# by ID

G:=PCGroup([7,-2,2,2,2,2,-2,2,224,477,232,1430,184,570,136,1684,172]);
// Polycyclic

G:=Group<a,b,c,d,e,f,g|a^2=b^2=1,c^2=d^2=e^2=f^2=a,g^2=b,a*b=b*a,d*c*d^-1=g*c*g^-1=a*c=c*a,f*d*f^-1=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,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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