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G = C23.422C24order 128 = 27

139th central stem extension by C23 of C24

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

Aliases: C23.422C24, C24.312C23, C22.2152+ 1+4, C22⋊C4.9Q8, C23.19(C2×Q8), C2.28(D43Q8), C23.4Q8.7C2, C22.95(C22×Q8), (C23×C4).386C22, (C22×C4).532C23, (C2×C42).537C22, C23.Q8.10C2, C23.7Q8.50C2, C23.8Q8.25C2, C23.65C2379C2, C23.63C2375C2, C23.81C2330C2, C24.C22.25C2, C2.38(C22.45C24), C2.C42.170C22, C2.46(C22.47C24), C2.24(C23.37C23), C2.21(C22.34C24), C2.65(C23.36C23), (C4×C4⋊C4)⋊81C2, (C2×C4).48(C2×Q8), (C4×C22⋊C4).57C2, (C2×C4).141(C4○D4), (C2×C4⋊C4).285C22, C22.299(C2×C4○D4), (C2×C22⋊C4).503C22, SmallGroup(128,1254)

Series: Derived Chief Lower central Upper central Jennings

C1C23 — C23.422C24
C1C2C22C23C22×C4C23×C4C4×C22⋊C4 — C23.422C24
C1C23 — C23.422C24
C1C23 — C23.422C24
C1C23 — C23.422C24

Generators and relations for C23.422C24
 G = < a,b,c,d,e,f,g | a2=b2=c2=f2=1, d2=ca=ac, e2=a, g2=ba=ab, ede-1=gdg-1=ad=da, ae=ea, af=fa, ag=ga, bc=cb, fdf=bd=db, be=eb, bf=fb, bg=gb, cd=dc, fef=ce=ec, cf=fc, cg=gc, eg=ge, fg=gf >

Subgroups: 388 in 214 conjugacy classes, 100 normal (82 characteristic)
C1, C2 [×7], C2 [×2], C4 [×20], C22 [×7], C22 [×10], C2×C4 [×12], C2×C4 [×40], C23, C23 [×2], C23 [×6], C42 [×5], C22⋊C4 [×4], C22⋊C4 [×7], C4⋊C4 [×19], C22×C4 [×14], C22×C4 [×5], C24, C2.C42 [×12], C2×C42 [×4], C2×C22⋊C4 [×6], C2×C4⋊C4 [×12], C23×C4, C4×C22⋊C4, C4×C4⋊C4, C23.7Q8, C23.8Q8, C23.63C23 [×3], C24.C22 [×2], C23.65C23 [×2], C23.Q8, C23.81C23 [×2], C23.4Q8, C23.422C24
Quotients: C1, C2 [×15], C22 [×35], Q8 [×4], C23 [×15], C2×Q8 [×6], C4○D4 [×8], C24, C22×Q8, C2×C4○D4 [×4], 2+ 1+4 [×2], C23.36C23, C23.37C23, C22.34C24, C22.45C24, C22.47C24, D43Q8 [×2], C23.422C24

Smallest permutation representation of C23.422C24
On 64 points
Generators in S64
(1 45)(2 46)(3 47)(4 48)(5 63)(6 64)(7 61)(8 62)(9 23)(10 24)(11 21)(12 22)(13 20)(14 17)(15 18)(16 19)(25 30)(26 31)(27 32)(28 29)(33 38)(34 39)(35 40)(36 37)(41 49)(42 50)(43 51)(44 52)(53 60)(54 57)(55 58)(56 59)
(1 58)(2 59)(3 60)(4 57)(5 52)(6 49)(7 50)(8 51)(9 34)(10 35)(11 36)(12 33)(13 31)(14 32)(15 29)(16 30)(17 27)(18 28)(19 25)(20 26)(21 37)(22 38)(23 39)(24 40)(41 64)(42 61)(43 62)(44 63)(45 55)(46 56)(47 53)(48 54)
(1 47)(2 48)(3 45)(4 46)(5 61)(6 62)(7 63)(8 64)(9 21)(10 22)(11 23)(12 24)(13 18)(14 19)(15 20)(16 17)(25 32)(26 29)(27 30)(28 31)(33 40)(34 37)(35 38)(36 39)(41 51)(42 52)(43 49)(44 50)(53 58)(54 59)(55 60)(56 57)
(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 44 45 52)(2 49 46 41)(3 42 47 50)(4 51 48 43)(5 58 63 55)(6 56 64 59)(7 60 61 53)(8 54 62 57)(9 18 23 15)(10 16 24 19)(11 20 21 13)(12 14 22 17)(25 35 30 40)(26 37 31 36)(27 33 32 38)(28 39 29 34)
(2 59)(4 57)(5 61)(6 43)(7 63)(8 41)(9 21)(10 38)(11 23)(12 40)(14 32)(16 30)(17 27)(19 25)(22 35)(24 33)(34 37)(36 39)(42 52)(44 50)(46 56)(48 54)(49 62)(51 64)
(1 26 55 13)(2 32 56 17)(3 28 53 15)(4 30 54 19)(5 11 44 37)(6 22 41 33)(7 9 42 39)(8 24 43 35)(10 51 40 62)(12 49 38 64)(14 46 27 59)(16 48 25 57)(18 47 29 60)(20 45 31 58)(21 52 36 63)(23 50 34 61)

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

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

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

38 conjugacy classes

class 1 2A···2G2H2I4A···4H4I···4X4Y4Z4AA4AB
order12···2224···44···44444
size11···1442···24···48888

38 irreducible representations

dim11111111111224
type+++++++++++-+
imageC1C2C2C2C2C2C2C2C2C2C2Q8C4○D42+ 1+4
kernelC23.422C24C4×C22⋊C4C4×C4⋊C4C23.7Q8C23.8Q8C23.63C23C24.C22C23.65C23C23.Q8C23.81C23C23.4Q8C22⋊C4C2×C4C22
# reps111113221214162

Matrix representation of C23.422C24 in GL6(𝔽5)

400000
040000
004000
000400
000010
000001
,
100000
010000
004000
000400
000010
000001
,
100000
010000
001000
000100
000040
000004
,
040000
100000
000100
004000
000020
000002
,
200000
030000
003000
000200
000001
000010
,
100000
010000
001000
000400
000010
000004
,
300000
020000
001000
000400
000010
000001

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

C23.422C24 in GAP, Magma, Sage, TeX

C_2^3._{422}C_2^4
% in TeX

G:=Group("C2^3.422C2^4");
// GroupNames label

G:=SmallGroup(128,1254);
// by ID

G=gap.SmallGroup(128,1254);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,560,253,120,758,723,675,192]);
// Polycyclic

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

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