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G = C24.301C23order 128 = 27

141st non-split extension by C24 of C23 acting via C23/C2=C22

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

Aliases: C24.301C23, C23.389C24, C22.1422- 1+4, C4⋊C4.227D4, C428C426C2, C2.41(D46D4), C2.33(Q85D4), C4.42(C4.4D4), C23.40(C4○D4), (C22×C4).72C23, (C2×C42).517C22, (C23×C4).374C22, C23.11D4.6C2, C22.269(C22×D4), C23.7Q8.42C2, (C22×Q8).115C22, C23.83C2319C2, C23.67C2349C2, C24.C22.18C2, C2.C42.142C22, C2.47(C23.36C23), C2.36(C22.46C24), C2.16(C22.35C24), (C4×C4⋊C4)⋊67C2, (C2×C4).60(C2×D4), (C2×C42.C2)⋊8C2, (C4×C22⋊C4).45C2, C2.15(C2×C4.4D4), (C2×C22⋊Q8).29C2, (C2×C4).811(C4○D4), (C2×C4⋊C4).259C22, C22.266(C2×C4○D4), (C2×C22⋊C4).463C22, SmallGroup(128,1221)

Series: Derived Chief Lower central Upper central Jennings

C1C23 — C24.301C23
C1C2C22C23C22×C4C23×C4C4×C22⋊C4 — C24.301C23
C1C23 — C24.301C23
C1C23 — C24.301C23
C1C23 — C24.301C23

Generators and relations for C24.301C23
 G = < a,b,c,d,e,f,g | a2=b2=c2=d2=1, e2=d, f2=c, g2=b, ab=ba, eae-1=ac=ca, faf-1=ad=da, ag=ga, bc=cb, bd=db, fef-1=geg-1=be=eb, bf=fb, bg=gb, cd=dc, ce=ec, cf=fc, cg=gc, de=ed, df=fd, dg=gd, fg=gf >

Subgroups: 420 in 238 conjugacy classes, 104 normal (42 characteristic)
C1, C2 [×7], C2 [×2], C4 [×4], C4 [×16], C22 [×7], C22 [×10], C2×C4 [×12], C2×C4 [×40], Q8 [×4], C23, C23 [×2], C23 [×6], C42 [×8], C22⋊C4 [×12], C4⋊C4 [×4], C4⋊C4 [×16], C22×C4 [×6], C22×C4 [×8], C22×C4 [×6], C2×Q8 [×6], C24, C2.C42 [×2], C2.C42 [×12], C2×C42 [×2], C2×C42 [×2], C2×C22⋊C4 [×2], C2×C22⋊C4 [×4], C2×C4⋊C4 [×3], C2×C4⋊C4 [×6], C22⋊Q8 [×4], C42.C2 [×4], C23×C4, C22×Q8, C4×C22⋊C4, C4×C4⋊C4, C23.7Q8, C428C4 [×2], C24.C22 [×2], C23.67C23 [×2], C23.11D4 [×2], C23.83C23 [×2], C2×C22⋊Q8, C2×C42.C2, C24.301C23
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], C2×D4 [×6], C4○D4 [×8], C24, C4.4D4 [×4], C22×D4, C2×C4○D4 [×4], 2- 1+4 [×2], C2×C4.4D4, C23.36C23, C22.35C24, D46D4, Q85D4, C22.46C24 [×2], C24.301C23

Smallest permutation representation of C24.301C23
On 64 points
Generators in S64
(2 52)(4 50)(5 64)(6 8)(7 62)(10 24)(12 22)(14 28)(16 26)(17 19)(18 30)(20 32)(29 31)(33 35)(34 38)(36 40)(37 39)(42 56)(44 54)(45 47)(46 58)(48 60)(57 59)(61 63)
(1 9)(2 10)(3 11)(4 12)(5 38)(6 39)(7 40)(8 37)(13 41)(14 42)(15 43)(16 44)(17 45)(18 46)(19 47)(20 48)(21 49)(22 50)(23 51)(24 52)(25 53)(26 54)(27 55)(28 56)(29 57)(30 58)(31 59)(32 60)(33 63)(34 64)(35 61)(36 62)
(1 51)(2 52)(3 49)(4 50)(5 62)(6 63)(7 64)(8 61)(9 23)(10 24)(11 21)(12 22)(13 27)(14 28)(15 25)(16 26)(17 31)(18 32)(19 29)(20 30)(33 39)(34 40)(35 37)(36 38)(41 55)(42 56)(43 53)(44 54)(45 59)(46 60)(47 57)(48 58)
(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 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 59 51 45)(2 32 52 18)(3 57 49 47)(4 30 50 20)(5 16 62 26)(6 41 63 55)(7 14 64 28)(8 43 61 53)(9 31 23 17)(10 60 24 46)(11 29 21 19)(12 58 22 48)(13 33 27 39)(15 35 25 37)(34 56 40 42)(36 54 38 44)
(1 13 9 41)(2 42 10 14)(3 15 11 43)(4 44 12 16)(5 20 38 48)(6 45 39 17)(7 18 40 46)(8 47 37 19)(21 53 49 25)(22 26 50 54)(23 55 51 27)(24 28 52 56)(29 61 57 35)(30 36 58 62)(31 63 59 33)(32 34 60 64)

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

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

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

38 conjugacy classes

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

38 irreducible representations

dim111111111112224
type++++++++++++-
imageC1C2C2C2C2C2C2C2C2C2C2D4C4○D4C4○D42- 1+4
kernelC24.301C23C4×C22⋊C4C4×C4⋊C4C23.7Q8C428C4C24.C22C23.67C23C23.11D4C23.83C23C2×C22⋊Q8C2×C42.C2C4⋊C4C2×C4C23C22
# reps1111222221141242

Matrix representation of C24.301C23 in GL6(𝔽5)

100000
040000
001000
004400
000010
000004
,
100000
010000
004000
000400
000010
000001
,
100000
010000
004000
000400
000040
000004
,
400000
040000
001000
000100
000040
000004
,
300000
030000
004300
000100
000001
000040
,
010000
100000
002000
003300
000001
000040
,
100000
010000
002000
003300
000040
000004

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

C24.301C23 in GAP, Magma, Sage, TeX

C_2^4._{301}C_2^3
% in TeX

G:=Group("C2^4.301C2^3");
// GroupNames label

G:=SmallGroup(128,1221);
// by ID

G=gap.SmallGroup(128,1221);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,448,253,232,758,723,675,80]);
// Polycyclic

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

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