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G = C62.101C23order 288 = 25·32

96th non-split extension by C62 of C23 acting via C23/C2=C22

metabelian, supersoluble, monomial

Aliases: C62.101C23, Dic324C2, C23.13S32, C6.164(S3×D4), D6⋊Dic326C2, C625C44C2, C6.D45S3, (C22×C6).66D6, C6.62(C4○D12), (C3×Dic3).19D4, (C2×Dic3).82D6, (C22×S3).23D6, C6.49(D42S3), C32(C23.12D6), Dic3.7(C3⋊D4), C3210(C4.4D4), (C2×C62).20C22, C37(C23.11D6), C2.14(D6.4D6), C2.23(D6.3D6), (C6×Dic3).73C22, (C6×C3⋊D4).1C2, (C2×C3⋊D4).2S3, C2.37(S3×C3⋊D4), C6.60(C2×C3⋊D4), C22.132(C2×S32), (C3×C6).147(C2×D4), (S3×C2×C6).40C22, (C3×C6).77(C4○D4), (C2×C322Q8)⋊13C2, (C3×C6.D4)⋊13C2, (C2×C6).120(C22×S3), (C2×C3⋊Dic3).63C22, SmallGroup(288,607)

Series: Derived Chief Lower central Upper central

C1C62 — C62.101C23
C1C3C32C3×C6C62S3×C2×C6D6⋊Dic3 — C62.101C23
C32C62 — C62.101C23
C1C22C23

Generators and relations for C62.101C23
 G = < a,b,c,d,e | a6=b6=c2=e2=1, d2=a3, ab=ba, ac=ca, dad-1=a-1, ae=ea, cbc=b-1, bd=db, be=eb, dcd-1=b3c, ece=a3b3c, ede=b3d >

Subgroups: 602 in 169 conjugacy classes, 48 normal (44 characteristic)
C1, C2 [×3], C2 [×2], C3 [×2], C3, C4 [×6], C22, C22 [×6], S3, C6 [×6], C6 [×8], C2×C4 [×5], D4 [×2], Q8 [×2], C23, C23, C32, Dic3 [×2], Dic3 [×8], C12 [×4], D6 [×3], C2×C6 [×2], C2×C6 [×14], C42, C22⋊C4 [×4], C2×D4, C2×Q8, C3×S3, C3×C6 [×3], C3×C6, Dic6 [×4], C2×Dic3 [×3], C2×Dic3 [×6], C3⋊D4 [×2], C2×C12 [×3], C3×D4 [×2], C22×S3, C22×C6 [×2], C22×C6 [×2], C4.4D4, C3×Dic3 [×2], C3×Dic3 [×2], C3⋊Dic3 [×2], S3×C6 [×3], C62, C62 [×3], C4×Dic3 [×2], D6⋊C4 [×2], C6.D4, C6.D4 [×5], C3×C22⋊C4, C2×Dic6 [×2], C2×C3⋊D4, C6×D4, C322Q8 [×2], C6×Dic3 [×3], C3×C3⋊D4 [×2], C2×C3⋊Dic3 [×2], S3×C2×C6, C2×C62, C23.11D6, C23.12D6, Dic32, D6⋊Dic3 [×2], C3×C6.D4, C625C4, C2×C322Q8, C6×C3⋊D4, C62.101C23
Quotients: C1, C2 [×7], C22 [×7], S3 [×2], D4 [×2], C23, D6 [×6], C2×D4, C4○D4 [×2], C3⋊D4 [×2], C22×S3 [×2], C4.4D4, S32, C4○D12, S3×D4, D42S3 [×3], C2×C3⋊D4, C2×S32, C23.11D6, C23.12D6, D6.3D6, D6.4D6, S3×C3⋊D4, C62.101C23

Smallest permutation representation of C62.101C23
On 48 points
Generators in S48
(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)
(1 16 3 18 5 14)(2 17 4 13 6 15)(7 48 9 44 11 46)(8 43 10 45 12 47)(19 28 23 26 21 30)(20 29 24 27 22 25)(31 41 35 39 33 37)(32 42 36 40 34 38)
(1 34)(2 35)(3 36)(4 31)(5 32)(6 33)(7 27)(8 28)(9 29)(10 30)(11 25)(12 26)(13 37)(14 38)(15 39)(16 40)(17 41)(18 42)(19 43)(20 44)(21 45)(22 46)(23 47)(24 48)
(1 23 4 20)(2 22 5 19)(3 21 6 24)(7 34 10 31)(8 33 11 36)(9 32 12 35)(13 29 16 26)(14 28 17 25)(15 27 18 30)(37 46 40 43)(38 45 41 48)(39 44 42 47)
(7 10)(8 11)(9 12)(19 26)(20 27)(21 28)(22 29)(23 30)(24 25)(31 42)(32 37)(33 38)(34 39)(35 40)(36 41)(43 46)(44 47)(45 48)

G:=sub<Sym(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), (1,16,3,18,5,14)(2,17,4,13,6,15)(7,48,9,44,11,46)(8,43,10,45,12,47)(19,28,23,26,21,30)(20,29,24,27,22,25)(31,41,35,39,33,37)(32,42,36,40,34,38), (1,34)(2,35)(3,36)(4,31)(5,32)(6,33)(7,27)(8,28)(9,29)(10,30)(11,25)(12,26)(13,37)(14,38)(15,39)(16,40)(17,41)(18,42)(19,43)(20,44)(21,45)(22,46)(23,47)(24,48), (1,23,4,20)(2,22,5,19)(3,21,6,24)(7,34,10,31)(8,33,11,36)(9,32,12,35)(13,29,16,26)(14,28,17,25)(15,27,18,30)(37,46,40,43)(38,45,41,48)(39,44,42,47), (7,10)(8,11)(9,12)(19,26)(20,27)(21,28)(22,29)(23,30)(24,25)(31,42)(32,37)(33,38)(34,39)(35,40)(36,41)(43,46)(44,47)(45,48)>;

G:=Group( (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), (1,16,3,18,5,14)(2,17,4,13,6,15)(7,48,9,44,11,46)(8,43,10,45,12,47)(19,28,23,26,21,30)(20,29,24,27,22,25)(31,41,35,39,33,37)(32,42,36,40,34,38), (1,34)(2,35)(3,36)(4,31)(5,32)(6,33)(7,27)(8,28)(9,29)(10,30)(11,25)(12,26)(13,37)(14,38)(15,39)(16,40)(17,41)(18,42)(19,43)(20,44)(21,45)(22,46)(23,47)(24,48), (1,23,4,20)(2,22,5,19)(3,21,6,24)(7,34,10,31)(8,33,11,36)(9,32,12,35)(13,29,16,26)(14,28,17,25)(15,27,18,30)(37,46,40,43)(38,45,41,48)(39,44,42,47), (7,10)(8,11)(9,12)(19,26)(20,27)(21,28)(22,29)(23,30)(24,25)(31,42)(32,37)(33,38)(34,39)(35,40)(36,41)(43,46)(44,47)(45,48) );

G=PermutationGroup([(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)], [(1,16,3,18,5,14),(2,17,4,13,6,15),(7,48,9,44,11,46),(8,43,10,45,12,47),(19,28,23,26,21,30),(20,29,24,27,22,25),(31,41,35,39,33,37),(32,42,36,40,34,38)], [(1,34),(2,35),(3,36),(4,31),(5,32),(6,33),(7,27),(8,28),(9,29),(10,30),(11,25),(12,26),(13,37),(14,38),(15,39),(16,40),(17,41),(18,42),(19,43),(20,44),(21,45),(22,46),(23,47),(24,48)], [(1,23,4,20),(2,22,5,19),(3,21,6,24),(7,34,10,31),(8,33,11,36),(9,32,12,35),(13,29,16,26),(14,28,17,25),(15,27,18,30),(37,46,40,43),(38,45,41,48),(39,44,42,47)], [(7,10),(8,11),(9,12),(19,26),(20,27),(21,28),(22,29),(23,30),(24,25),(31,42),(32,37),(33,38),(34,39),(35,40),(36,41),(43,46),(44,47),(45,48)])

42 conjugacy classes

class 1 2A2B2C2D2E3A3B3C4A4B4C4D4E4F4G4H6A···6F6G···6Q6R6S12A···12F
order122222333444444446···66···66612···12
size11114122246666121818362···24···4121212···12

42 irreducible representations

dim11111112222222224444444
type+++++++++++++++-+-
imageC1C2C2C2C2C2C2S3S3D4D6D6D6C4○D4C3⋊D4C4○D12S32S3×D4D42S3C2×S32D6.3D6D6.4D6S3×C3⋊D4
kernelC62.101C23Dic32D6⋊Dic3C3×C6.D4C625C4C2×C322Q8C6×C3⋊D4C6.D4C2×C3⋊D4C3×Dic3C2×Dic3C22×S3C22×C6C3×C6Dic3C6C23C6C6C22C2C2C2
# reps11211111123124441131222

Matrix representation of C62.101C23 in GL8(𝔽13)

10000000
01000000
001200000
000120000
00001000
00000100
000000121
000000120
,
120000000
012000000
001200000
000120000
000012100
000012000
00000010
00000001
,
12000000
012000000
001200000
00210000
00000100
00001000
00000010
00000001
,
510000000
88000000
0012120000
00210000
00001000
00000100
00000001
00000010
,
10000000
1212000000
00100000
0011120000
000012000
000001200
00000010
00000001

G:=sub<GL(8,GF(13))| [1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,12,0,0,0,0,0,0,1,0],[12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,12,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,2,12,0,0,0,0,0,0,0,0,12,2,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[5,8,0,0,0,0,0,0,10,8,0,0,0,0,0,0,0,0,12,2,0,0,0,0,0,0,12,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0],[1,12,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,11,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1] >;

C62.101C23 in GAP, Magma, Sage, TeX

C_6^2._{101}C_2^3
% in TeX

G:=Group("C6^2.101C2^3");
// GroupNames label

G:=SmallGroup(288,607);
// by ID

G=gap.SmallGroup(288,607);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,253,64,590,219,1356,9414]);
// Polycyclic

G:=Group<a,b,c,d,e|a^6=b^6=c^2=e^2=1,d^2=a^3,a*b=b*a,a*c=c*a,d*a*d^-1=a^-1,a*e=e*a,c*b*c=b^-1,b*d=d*b,b*e=e*b,d*c*d^-1=b^3*c,e*c*e=a^3*b^3*c,e*d*e=b^3*d>;
// generators/relations

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